# Traveling waves & finite gap potentials for the Calogero–Sutherland
Derivative nonlinear Schrödinger equation

Rana Badreddine  Université Paris-Saclay, Laboratoire de mathématiques
d’Orsay, UMR 8628 du CNRS, Bâtiment 307, 91405 Orsay Cedex, France
<EMAIL_ADDRESS>
(Date: June 30, 2023.)

###### Abstract.

We consider the Calogero–Sutherland derivative nonlinear Schrödinger equation

$i\partial_{t}u+\partial_{x}^{2}u\,\pm\,\frac{2}{i}\,\partial_{x}\Pi(|u|^{2})u=0\,,\qquad
x\in\mathbb{T}\,,$ (CS)

where $\Pi$ is the Szegő projector

$\Pi\Big{(}\sum_{n\in\mathbb{Z}}\widehat{u}(n)\mathrm{e}^{inx}\Big{)}=\sum_{n\geq
0}\widehat{u}(n)\mathrm{e}^{inx}\,.$

First, we characterize the traveling wave $u_{0}(x-ct)$ solutions to the
defocusing equation (CS-) , and prove for the focusing equation (CS+), that
all the traveling waves must be either the constant functions, or plane waves,
or rational functions. A noteworthy observation is that the (CS)–equation is
one of the fewest nonlinear PDE enjoying nontrivial traveling waves with
arbitrary small and large $L^{2}$–norms. Second, we study the finite gap
potentials, and show that they are also rational functions, containing the
traveling waves, and they can be grouped into sets that remain invariant under
the evolution of the system.

###### Key words and phrases:

Calogero–Sutherland–Moser systems, Derivative nonlinear Schrödinger equation
(DNLS), Finite gap potentials, Hardy space, Integrable systems, Intermediate
nonlinear Schrödinger equation, Stationary waves, Traveling wave solutions

###### 1991 Mathematics Subject Classification:

35C07,37K10, 35Q55

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 About the Calogero–Sutherland DNLS equation
      1. 1.2.1 Local and global well–posedness results
      2. 1.2.2 Integrability of the (CS)-equation
      3. 1.2.3 Traveling waves on $\mathbb{R}$
    3. 1.3 Outline of the paper
  2. 2 Spectral properties for the Lax operators
  3. 3 Traveling waves for the defocusing (CS-)
    1. 3.1 Spectral Characterization
    2. 3.2 Explicit formulas of the traveling waves
    3. 3.3 The $L^{2}$–norm and the speed
  4. 4 Traveling waves for the focusing (CS+)
    1. 4.1 Toward the characterization of the traveling waves for (CS+)
    2. 4.2 The $L^{2}$–norm and the speed
  5. 5 The finite gap potentials
  6. 6 Remark on the regularity of $u$
  7. 7 Open problems

##  1\. Introduction

In recent decades, the theory of traveling wave solutions has been the subject
of intense research in theoretical and numerical analysis. Indeed, many
nonlinear PDEs exhibit these type of waves [Ch04, CH13, An09] . Their
importance resides as they are explicit solutions for nonlinear PDEs, and they
can sometimes provide information regarding the dynamics of the equation.
However, the problem of proving the existence of these waves can be more or
less challenging depending on the nonlinear part of the PDE. In this paper, we
consider a type of derivative nonlinear Schrödinger equation with a nonlocal
nonlinearity, called the Calogero–Sutherland derivative nonlinear Schrödinger
equation

$i\partial_{t}u+\partial_{x}^{2}u\,\pm\,2D\Pi(|u|^{2})u=0\,,\qquad
x\in\mathbb{T}:=\mathbb{R}/(2\pi\mathbb{Z})\,,$ (CS)

where $D=-i\partial_{x}\,,$ and $\Pi$ denotes the Szegő projector

$\Pi\left(\sum_{n\in\mathbb{Z}}\,\widehat{u}(n)\operatorname{e}^{inx}\right):=\sum_{n\geq
0}\,\widehat{u}(n)\operatorname{e}^{inx}\,,$ (1.1)

which is an orthogonal projector from $L^{2}(\mathbb{T})$ into the Hardy space

$L^{2}_{+}(\mathbb{T}):=\left\\{u\in L^{2}(\mathbb{T})\ |\
\widehat{u}(n)=0\,,\,\forall n\in\mathbb{Z}_{\leq-1}\right\\}\,.$ (1.2)

Note that the operator $\Pi$ can also be read as
$\Pi=\frac{\operatorname{Id}+i\mathcal{H}+\left\langle\cdot\,|\,1\right\rangle}{2}$
on the circle $\mathbb{T}$, where $\left\langle
u\,|\,1\right\rangle=\int_{0}^{2\pi}u\,\frac{dx}{2\pi}$ and $\mathcal{H}$ is
the Hilbert transform

$\mathcal{H}u(x):=\sum_{n\in\mathbb{Z}}-i\operatorname{sign}(n)\,\widehat{u}(n)\mathrm{e}^{inx}\,,\qquad\operatorname{sign}(0)=0\,.$
(1.3)

We are interested in studying the traveling waves $u_{0}(x-ct)$ of this
equation in the focusing (with sign $+$ in front of the nonlinearity) and
defocusing case (with sign$-$) in the periodic setting, namely when
$x\in\mathbb{T}$ . As noted in [An09], the presence of the dispersion operator
$\partial_{x}\mathcal{H}$ appearing in the nonlinearity can make the problem
of existence of traveling waves more complicated. The approach addressed in
this paper to characterize the traveling waves is based on studying them, at a
first stage, spectrally i.e. by means of spectral property of the Lax operator
related to this equation (see below), before deriving, at a second stage,
their explicit formulas. 111It should be noted that the idea of using the
spectral theory to derive the traveling waves of (CS), draws inspiration from
[GK21, Appendix B] , where the authors provide an alternative proof to the
characterization of the traveling waves for the Benjamin–Ono equation [AT91,
Be67] by characterizing them first spectrally.

###  1.1. Main results

Settings and notation. In the sequel, our study takes place with potentials in
the Hardy Sobolev spaces of the torus

$H^{s}_{+}(\mathbb{T}):=H^{s}(\mathbb{T})\cap L^{2}_{+}(\mathbb{T})\,,\qquad
s\geq 0\,,$

where $L^{2}_{+}(\mathbb{T})$ is defined in (1.2) and $H^{s}$ refers to the
Sobolev space. We equip $L^{2}_{+}(\mathbb{T})$ with the standard inner
product of $L^{2}(\mathbb{T})$ ,

$\left\langle
u\,|\,v\right\rangle=\int_{0}^{2\pi}u\bar{v}\,\frac{dx}{2\pi}\,.$

We recall also, that via the following isometric isomorphism

$\displaystyle z\in\mathbb{D}\,,\quad u(z)=\sum_{k\geq 0}\widehat{u}(k)z^{k}$
$\displaystyle\qquad\longmapsto\qquad u^{*}(x):=\sum_{k\geq
0}\widehat{u}(k)\operatorname{e}^{ikx}\,,\quad x\in\mathbb{T}\,,$
$\displaystyle\sum_{k\geq 0}\,\lvert\widehat{u}(k)\rvert^{2}<\infty\,,$

one can interpret any element of the Hardy space as an analytic function on
the open unit disc $\mathbb{D}$ , whose trace on the boundary
$\partial\mathbb{D}$ is in $L^{2}\;$ .222 For a simple introduction to the
different definitions of Hardy space, we refer to [GMR16, Chapter 3.] We will
frequently utilize this property in various proofs. Furthermore, we denote by
$\mathbb{D}$ the open unit disc on $\mathbb{C}$ ,
$\mathbb{D}^{*}:=\left\\{z\in\mathbb{C}\;;\;0<|z|<1\right\\}\,$. Moreover,
$\mathbb{N}$ denotes the positive integers $1,2,3,\ldots$ And for all
$a\in\mathbb{N}\cup\left\\{0\right\\}\,,$ $\mathbb{N}_{\geq a}$ refers to the
set of integer numbers $\left\\{n\in\mathbb{Z}\,;\,n\geq a\right\\}\,.$

***

First, we deal with the defocusing Calogero–Sutherland DNLS equation

$i\partial_{t}u+\partial_{x}^{2}u-2D\Pi(\lvert u\rvert^{2})u=0\,.$ (CS-)

We denote by $\mathcal{G}_{1}$ the set of the trivial traveling waves, made up
from the constant functions and the plane wave solutions

$\mathcal{G}_{1}=\left\\{C\operatorname{e}^{iN(x-Nt)}\,\mid\,C\in\mathbb{C}\,,\,N\in\mathbb{N}_{\geq
0}\right\\}\,.$ (1.4)

######  Theorem 1.1 (Characterization of the traveling waves of (CS-)).

A potential $u$ is a traveling wave of (CS-) if and only if
$u\in\mathcal{G}_{1}$ or

$u(t,x):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,$ (1.5)

where $N\in\mathbb{N}\,,$ $\,c:=-N\Big{(}1+\frac{2\alpha}{\beta}\Big{)},$ and
$(\alpha,\beta)$ are two real constants satisfying

$\alpha\beta+\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=-N\,.$ (1.6)

######  Remark 1.1.

The condition (1.6) implies that the real constants $\alpha$ and $\beta$ must
be of opposite signs.

Second, we pass to the focusing Calogero–Sutherland DNLS equation

$i\partial_{t}u+\partial_{x}^{2}u+2D\Pi(\lvert u\rvert^{2})u=0\,.$ (CS+)

By changing the sign in front of the nonlinearity, the strategy adopted in the
defocusing case to exhibit the traveling waves becomes significantly more
complicated. However, we can ensure the existence of a larger set of traveling
wave solutions for (CS+) comparing to (CS-) , and that all the non–trivial
traveling waves $u(t,x):=u_{0}(x-ct)$ of (CS+) are also rational functions.

######  Theorem 1.2.

The traveling waves $u_{0}(x-ct)$ of (CS+) are either rational functions or
trivial waves in $\mathcal{G}_{1}$ . In addition, the potentials

$u(t,x):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,N\in\mathbb{N}\,,$

where $c=-N\left(1+\frac{2\alpha}{\beta}\right)$ ,
$\,(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ such that

$\alpha\beta+\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=N\,,$ (1.7)

and the potentials

$u(t,z)=\operatorname{e}^{i\theta}\operatorname{e}^{im(x-mt)}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{i(x-mt)}}\right)\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,m\in\mathbb{N}\,,$

where $(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ such that

$\alpha\beta+\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}=1\,,\qquad\beta(m-1)=2\alpha\,,$

are parts of the set of the traveling waves of (CS+) .

######  Remark 1.2.

It is worth noting that the condition on $(\alpha,\beta)$ appearing in (1.7)
for the focusing case, allows to obtain a larger set of traveling waves in
comparison to the condition (1.6) of the defocusing case. Indeed, (1.7)
enables, for instance $\alpha$ or $\alpha+\beta$ to vanish, which leads
respectively to the following traveling waves

$u(t,x)=\operatorname{e}^{i\theta}\frac{\sqrt{N(1-\lvert
p\rvert^{2})}}{1-p\operatorname{e}^{iN(x+Nt)}}\quad\text{and}\quad
u(t,x)=\operatorname{e}^{i\theta}\frac{\sqrt{N(1-\lvert
p\rvert^{2})}\,\operatorname{e}^{iN(x-Nt)}}{1-p\operatorname{e}^{iN(x-Nt)}}\,.$

Contrary to the focusing case, no traveling waves $u(t,x):=u_{0}(x-ct)$ with a
profile $\displaystyle u_{0}(x):=\frac{\beta}{1-p\operatorname{e}^{iNx}}\;$ or
$\;\displaystyle
u_{0}(x):=\frac{\alpha\operatorname{e}^{iNx}}{1-p\operatorname{e}^{iNx}}\>$
can be found for the (CS-)–equation because otherwise, thanks to (1.6) ,

$\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=-N\qquad\text{or}\qquad\frac{\lvert
p\rvert^{2}}{1-\lvert p\rvert^{2}}\,\beta^{2}=-N$

which is clearly impossible for $p\in\mathbb{D}^{*}\,.$

######  Remark 1.3.

(The $L^{2}$–norm and the speed of the traveling waves of (CS) )

  1. (1)

As will be established in Subsection 3.3 for the defocusing equation and in
Subsection 4.2 for the focusing equation, the $L^{2}$–norm of the non–trivial
traveling waves of (CS) can be arbitrarily small or large in
$L^{2}_{+}(\mathbb{T})\,.$ More rigorously, for any $r>0\,,$ there exists a
non–trivial traveling wave $u(t,x):=u_{0}(x-ct)$ of (CS) where

$\|u_{0}\|_{L^{2}}=r\,.$

  2. (2)

For the defocusing (CS-)–equation. The nontrivial traveling waves $u$ of the
form (1.5) propagate to the right with a speed $c>N\,,$ where $N$ is the
degree appearing in the denominator of $u\,.$ In addition, when
$\|u\|_{L^{2}}\to+\infty\,,$ we have $c\to+\infty\,,$ and when
$\|u\|_{L^{2}}\to 0$ then $c\to N\,.$ (See Remark 3.3 and Subsection 3.3 for
the proofs). For the focusing (CS+)–equation. Contrary to the defocusing
equation, the (CS+)’s nontrivial traveling waves do not necessarily propagate
at a relatively high speed (i.e. $c\to\infty$) when $\|u\|_{L^{2}}^{2}$ is
large (i.e. $\|u\|_{L^{2}}\to\infty)$. In fact, the speed of the traveling
waves in the focusing case is independent of the size of its
$L^{2}_{+}(\mathbb{T})$–norm . We refer to Remark 4.2 for an example.

In light of the previous remarks, we infer that the Calogero–Sutherland DNLS
equation enjoys a significantly richer dynamic in the focusing case. In
particular, one can observe that the (CS+) admits non-trivial stationary waves
$u(t,x):=u_{0}(x)$, which is not the case of the defocusing equation. An
example of nontrivial stationary waves for (CS+) is

$u(t,x):=\operatorname{e}^{i\theta}\sqrt{\frac{N(1-\lvert
p\rvert^{2})}{2(1+\lvert
p\rvert^{2})}}\left(1-\frac{2}{1-p\operatorname{e}^{iNx}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\ \theta\in\mathbb{T}\,,\ N\in\mathbb{N}\,.$

At a second stage, we study the finite gap potentials of the
Calogero–Sutherland DNLS equation (CS) , i.e. potentials satisfying that, from
a certain rank, all the gaps between the consecutive eigenvalues of the Lax
operator are equal to $1$ (see Subsection 1.2.2 for the Lax operator). It
turns out that these potentials are multiphase solutions containing the
stationary and traveling waves of (CS) . The following theorem aims to
characterize the finite gap potentials on $\mathbb{T}\,$ in the state space.

######  Theorem 1.3 (Characterization in the state space of the (CS)’s finite
gap potentials).

The finite gap potentials of (CS) are either the functions
$u(x)=C\operatorname{e}^{iNx}$ , $C\in\mathbb{C}^{*}\,,$ $N\in\mathbb{N}_{\geq
0}\,,$ or the rational function

$u(x)=\operatorname{e}^{im_{0}\,x}\prod_{j=1}^{r}\left(\frac{\mathrm{e}^{ix}-\overline{p_{j}}}{1-p_{j}\mathrm{e}^{ix}}\right)^{m_{j}-1}\left(a+\sum_{j=1}^{r}\frac{c_{j}}{1-p_{j}\mathrm{e}^{ix}}\right),\quad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,$

where, for $N\in\mathbb{N}\,,$ $\,m_{0}\in\llbracket 0,N-1\rrbracket\,,\,$
$m_{1},\ldots,m_{r}\in\llbracket 1,N\rrbracket$ such that
$m_{0}+\sum_{j=1}^{r}m_{j}=N\,,$ and
$(a,\,c_{1}\,,\,\ldots\,,\,c_{r})\in\mathbb{C}\times\mathbb{C}^{r}$ satisfy
for all $j=1\,,\,\ldots\,,r\,,$

  1. (i)

In the defocusing case,

$\overline{a}\,c_{j}\,+\,\sum_{k=1}^{r}\,\frac{c_{j}\,\overline{c_{k}}}{1-p_{j}\overline{p_{k}}}=-m_{j}\,,$
(1.8)

  2. (ii)

In the focusing case,

$\overline{a}\,c_{j}\,+\,\sum_{k=1}^{r}\,\frac{c_{j}\,\overline{c_{k}}}{1-p_{j}\overline{p_{k}}}=m_{j}\,,$
(1.9)

with $a\neq 0$ if $m_{0}\neq 0\,.$ Moreover, these finite gap potentials can
be regrouped into sets that remain invariant under the evolution of (CS) .

In order to establish the results mentioned above, it is necessary to provide
an overview regarding the integrability of the Calogero–Sutherland derivative
nonlinear Schrödinger equation (CS).

###  1.2. About the Calogero–Sutherland DNLS equation

The Calogero– Sutherland DNLS equation (CS) has been actively studied by
physicists and engineers. In particular, we cite the works of Tutiya [Tu09],
Berntson–Fagerlund [BF22], Stone–Anduaga–Xing [SAX08], Polychronakos [Po95a,
Po95a] and Matsuno [Ma00, Ma01a, Ma01b, Ma02a, Ma02b, Ma03, Ma04a, Ma04b,
Ma23]… Mathematically, recent progress has been made with regard to this
equation. In this subsection, we provide a brief overview of some established
results concerning (CS) .

####  1.2.1. Local and global well–posedness results

To the best of the author’s knowledge, the first LWP result for (CS) equation
traced back to De Moura [deM07] who established the LWP 333 Actually, they
prove the local well–posedness of a family of nonlocal nonlinear Schrödinger
equation [PG96] that includes also the (CS)–equation.  of (CS) for small
initial data in $H^{s}(\mathbb{R})$ with $s\geq 1\,$, and extend his LWP’s
result to a GWP by means of the gauge transformation. More recently,
Barros–DeMoura–Santos present in [BMS19] the LWP of (CS) for small initial
data in the Besov space $B_{2}^{\frac{1}{2},1}(\mathbb{R})\,$. Besides,
observe that the Calogero–Sutherland DNLS equations (CS) is invariant under
the scaling

$u_{\lambda}(t,x)=\lambda^{\frac{1}{2}}u(\lambda
t,\lambda^{2}x)\,,\qquad\lambda>0\,.$

This suggests that (CS) is $L^{2}$–critical. In the Hardy Sobolev spaces
settings, i.e. in $H^{s}_{+}:=H^{s}\cap L^{2}_{+}\,,$ where recall $L^{2}_{+}$
is the Hardy space defined in (1.2) in the periodic case, and as follows in
the non-periodic case

$L_{+}^{2}(\mathbb{R})=\left\\{u\in L^{2}(\mathbb{R})\ ;\
\operatorname{supp}\hat{u}\subset[0,\infty)\right\\}\,,$

Gérard–Lenzmann [GL22] obtained the LWP in $H^{s}_{+}(\mathbb{R})$ with
$s>\frac{1}{2}\,,$ by following the arguments of [deMP10] . Furthermore, by
the virtue of a Lax pair structure associated with the Calogero–Sutherland
DNLS equation (CS) (see below), they inferred the global well–posedness of the
equation in all $H^{k}_{+}(\mathbb{R})\,$, $k\in\mathbb{N}_{\geq 1}$ for small
initial data $\|u_{0}\|_{L^{2}(\mathbb{R})}<\sqrt{2\pi}$ in the focusing case.
Moving to the periodic setting, i.e. when $x\in\mathbb{T}\,,$ a recent work of
the author shows the GWP of (CS) in all $H^{s}_{+}(\mathbb{T})$ , $s\geq 0\,,$
for small critical initial data in the focusing case, namely when
$\|u_{0}\|_{L^{2}(\mathbb{T})}<1\,,$ and for arbitrary initial data in the
defocusing case. In particular, the extension of the flow to the critical
space $L^{2}_{+}(\mathbb{T})$ has been achieved after deriving the explicit
formula for the solution of the Calogero–Sutherland DNLS equation (CS) [Ba23,
Proposition 2.5]. Moreover, under the same assumptions, the relative
compactness of the trajectories has been established in
$H^{s}_{+}(\mathbb{T})$ , for all $s\geq 0$ [Ba23].

####  1.2.2. Integrability of the (CS)-equation

One of the most remarkable features of the Calogero–Sutherland DNLS equation
is its integrability as a PDE on $\mathbb{R}$ and on $\mathbb{T}$. In fact, it
enjoys a Lax pair structure in the focusing and defocusing case [GL22, Ba23] :
for any $u\in H^{s}_{+}(\mathbb{T})\,,$ $s>\frac{3}{2}\,,$ there exists two
operators $(L_{u},B_{u})$ satisfying the Lax equation

$\frac{dL_{u}}{dt}=[B_{u},L_{u}]\,,\qquad[B_{u},L_{u}]:=B_{u}L_{u}-L_{u}B_{u}\,,$

where

  1. (i)

In the focusing case,

$L_{u}=D-T_{u}T_{\bar{u}}\,,\qquad
B_{u}=T_{u}T_{\partial_{x}\bar{u}}-T_{\partial_{x}{u}}T_{\bar{u}}+i(T_{u}T_{\bar{u}})^{2}\,.$
(1.10)

  2. (ii)

In the defocusing case,

$\tilde{L}_{u}=D+T_{u}T_{\bar{u}}\,,\qquad\tilde{B}_{u}=-T_{u}T_{\partial_{x}\bar{u}}+T_{\partial_{x}u}T_{\bar{u}}+i(T_{u}T_{\bar{u}})^{2}\,.$
(1.11)

The differential operator $D$ is $-i\partial_{x}\,,$ and $T_{u}$ is the
Toeplitz operator of symbol $u$ defined for any $u\in L^{\infty}$ by

$T_{u}f=\Pi(uf)\,,\qquad\forall f\in L^{2}_{+}\,,$ (1.12)

where $\Pi$ is the Szegő projector introduced in (1.1) . Note that since we
are working in the Hardy space, $L_{u}$ is a semi-bounded operator from below
and $\tilde{L}_{u}$ is a nonnegative operator. In addition, as noted in [Ba23,
Proposition 2.3] , the Lax operators $L_{u}$ and $\tilde{L}_{u}$ are
self–adjoint operators of domain $H^{1}_{+}(\mathbb{T})\,,$ and are of compact
resolvent. Therefore, their spectra are made up of a sequence of eigenvalues
going to $+\infty\,,$

$\displaystyle\sigma(L_{u}):=$
$\displaystyle\left\\{\nu_{0}(u)\leq\ldots\leq\nu_{n}(u)\leq\ldots\right\\},\qquad\nu_{0}(u)\geq-\|u\|_{L^{\infty}}^{2}\,,$
(1.13) $\displaystyle\sigma(\tilde{L}_{u}):=$
$\displaystyle\left\\{\lambda_{0}(u)\leq\ldots\leq\lambda_{n}(u)\leq\ldots\right\\}\,,\qquad\lambda_{0}(u)\geq
0\;.$

Recall that any Lax operator satisfies the isospectral property

$L_{u_{0}}=U(t)^{-1}L_{u(t)}U(t)\,,$ (1.14)

where $u_{0}$ is the initial data, $u(t)$ is the evolution of the solution
starting from $u_{0}\,,$ and $U(t)$ is a family of operators solving the
Cauchy problem

$\begin{cases}\frac{d}{dt}U(t)=B_{u(t)}\,U(t)\\\
U(0)=\operatorname{Id}\end{cases}\,.$

The identity (1.14) implies that the spectrum of $L_{u(t)}$ is invariant by
the evolution, i.e. $\nu_{n}(u(t))=\nu_{n}(u_{0})$ and
$\lambda_{n}(u(t))=\lambda_{n}(u_{0})$ for all $n\,$. Therefore, in the
sequel, we omit the variable $u$ in $\nu_{n}(u)$ and $\lambda_{n}(u)$ when it
does not make confusion. Further information regarding the spectrum of the Lax
operators will be provided in Section 2.

####  1.2.3. Traveling waves on $\mathbb{R}$

Let us mention that the focusing Calogero–Sutherland DNLS equation (CS+) also
enjoys traveling waves and stationary waves in the nonperiodic case (i.e.
$x\in\mathbb{R}$). They are of the form

$u(t,x)=\mathrm{e}^{i\theta}\mathrm{e}^{iv(x-vt)}\,\lambda^{\frac{1}{2}}\,\mathcal{R}\big{(}\lambda(x-2vt)+y\big{)}\,,\quad\lambda>0,\
y\in\mathbb{R},\ \theta\in\mathbb{T}\,,\,v\in\mathbb{R}\,,$

where the profile

$\mathcal{R}(x)=\mathrm{e}^{i\theta}\frac{\sqrt{2\,\mathrm{Im}p}}{x+p}\,,\qquad
p\in\mathbb{C}_{+}\,,\ \theta\in\mathbb{R}\,,$ (1.15)

is obtained as ground states (minimizers) for the energy functional [GL22,
Section 4]. Notice that all these waves are of $L^{2}$–norm equal to
$\sqrt{2\pi}\,$. Therefore, this situation differs from the torus
$\mathbb{T}$, where in the latter case, there is no $L^{2}$-threshold that
would prevent the existence of small or large traveling waves in
$L^{2}(\mathbb{T})$ . Essentially, the main reason that leads to a more
diverse class of traveling waves in the periodic setting compared to the
non–periodic seting, is due to the spectral property carried by the Lax
operator in both cases. Indeed, on $\mathbb{R}$ , the Lax operator has an
absolute continuous spectrum and a finite number of eigenvalues [GL22, Section
5]. In contrast with $\mathbb{T}$, the Lax operator present only point
spectrum formed by eigenvalues [Ba23, Section 2].

To summarize, we refer to the following table (Table 1) .

|  Focusing (CS+) on $\mathbb{R}$ |  Defocusing (CS+) on $\mathbb{R}$
---|---|---
Stationary waves |  ✓ |
Traveling waves |  ✓ |
Wave speed |  $c\in\mathbb{R}$ |
$L^{2}$–norm of the traveling waves |  $\|u\|_{L^{2}}=\sqrt{2\pi}$ |
|  Focusing (CS+) on $\mathbb{T}$ |  Defocusing (CS+) on $\mathbb{T}$
Non–trivial stationary waves |  ✓ |  ✗
Traveling waves |  ✓ |  ✓
Wave speed |  $c\in\mathbb{R}$ |  $\hskip 49.79231ptc\geq N$
$L^{2}$–norm of the non–trivial traveling waves |  $\|u\|_{L^{2}}\in(0,+\infty)$ |  $\hskip 22.76228pt\|u\|_{L^{2}}\in(0,+\infty)$
Table 1.  Where $N\in\mathbb{N}$ is the denominator’s degree of a traveling
wave of the form (1.5). In addition, by a non–trivial traveling wave, we mean
the traveling wave that does not belong to $\mathcal{G}_{1}$ where
$\mathcal{G}_{1}$ is the set of trivial traveling waves defined in (1.4) .
Moreover, by non–trivial stationary waves we mean the solution
$u(t,x)=u_{0}(x)$ that are not constant functions.

###  1.3. Outline of the paper

The paper is organized as follows. In Section 2 , we present some spectral
properties concerning the eigenvalues and the eigenfunctions of the Lax
operators $L_{u}$ and $\tilde{L}_{u}\,.$ Moving on to Section 3, we focus on
the traveling waves of the defocusing Calogero-Sutherland DNLS equation (CS-).
This section follows a two-step process. Subsection 3.1 provides a spectral
characterization of these waves, while Subsection 3.2 derives their explicit
formulas. Moreover, Subsection 3.3 includes remarks concerning the speed and
$L^{2}$-norm of these traveling waves for the defocusing (CS-)–equation. In
Section 4 , we delve into the analysis of traveling waves for the focusing
Calogero–Sutherland DNLS equation (CS+) . Thus, we describe the set of
traveling waves of (CS+) in Subsubsection 4.1, and we highlight the presence
of a larger set of traveling waves in the focusing case comparing to the
defocusing case. Similarly to the defocusing case, some remarks related to the
speed and the $L^{2}$–norm of the traveling waves of (CS+) are discussed in
Subsection 4.2 , and in particular we establish the existence of stationary
wave solutions for the focusing Calogero-Sutherland DNLS equation (CS+) . Note
that in order to describe the traveling waves of (CS+), one need to understand
the set of finite gap potentials. To this end, Section 5 is dedicated to the
study of finite gap potentials for the Calogero–Sutherland DNLS equation (CS).
Throughout this paper, we have assumed sufficient regularity on the solutions.
However, in Section 6, we discuss how the same analysis can be extended to
solutions with lower regularity. Lastly, in Section 7, we present some open
problems for further exploration.

### Acknowledges

The author would like to thank her Ph.D. advisor Patrick Gérard  for proposing
this problem and suggesting [GK21, Appendix B] as a useful reference to start
the investigation.

##  2\. Spectral properties for the Lax operators

As mentioned in the introduction, our aim is to describe the traveling waves
of the Calogero-Sutherland DNLS equation (CS). In order to accomplish this
goal, our strategy relies on characterizing them first in the state space, by
means of some spectral tools of the Lax operators $L_{u}$ and $\tilde{L}_{u}$
introduced in (1.10) and in (1.11) , respectively. Therefore, we need to delve
deeper into the spectral properties of the Lax operators.

In the sequel, we assume, for more convenience, that $u$ is any function of
the state space with enough regularity, for example, $u\in
H^{2}_{+}(\mathbb{T})\,.$ But, it is worth mentioning that the analysis can be
easily extended to potentials with less regularity as well (see Section 6).
Besides, recall from (1.13) , that the Lax operators $\tilde{L}_{u}$ and
$L_{u}$ have point spectra, bounded from below

$\displaystyle\sigma(\tilde{L}_{u}):=$
$\displaystyle\left\\{\lambda_{0}\leq\ldots\leq\lambda_{n}\leq\ldots\right\\}\,,\qquad\lambda_{0}\geq
0\;,$ $\displaystyle\sigma(L_{u}):=$
$\displaystyle\left\\{\nu_{0}\leq\ldots\leq\nu_{n}\leq\ldots\right\\},\qquad\nu_{0}\geq-\|u\|_{L^{\infty}}^{2}\,.$

The following proposition aims to give more information, regarding the
multiplicity of the eigenvalues $(\nu_{n})$ and $(\lambda_{n})\,$. But before,
we need to recall two useful commutator identities. We denote by $S$ the shift
operator defined as

$S:L^{2}_{+}(\mathbb{T})\to L^{2}_{+}(\mathbb{T})\,,\qquad
Sh(x)=\operatorname{e}^{ix}h(x)\,.$ (2.1)

Thus, for all $u\in H^{2}_{+}(\mathbb{T})\,,$ we have from [Ba23, Lemma 2.3] ,

$\displaystyle\tilde{L}_{u}S=S\tilde{L}_{u}+S+\left\langle\,\cdot\,|\,S^{*}u\right\rangle
u\,,$ (2.2) $\displaystyle
L_{u}S=SL_{u}+S-\left\langle\,\cdot\,|\,S^{*}u\right\rangle u\,,$

where $S^{*}$ denotes the adjoint operator of $S$

$S^{*}:L^{2}_{+}(\mathbb{T})\to L^{2}_{+}(\mathbb{T})\,,\qquad
S^{*}h(x)=\Pi(\operatorname{e}^{-ix}h(x))\,,$ (2.3)

with $\Pi$ is the Szegő projector defined in (1.1) , and $L_{u}$ and
$\tilde{L}_{u}$ are defined in (1.10) and (1.11) . In addition, we also have
from the same lemma [Ba23, Lemma 2.3] ,

$\displaystyle[S^{*},B_{u}]=i\Big{(}S^{*}L_{u}^{2}\,-\,(L_{u}+\operatorname{Id})^{2}S^{*}\Big{)}\,,$
(2.4)
$\displaystyle[S^{*},\tilde{B}_{u}]=i\Big{(}S^{*}\tilde{L}_{u}^{2}\,-\,(\tilde{L}_{u}+\operatorname{Id})^{2}S^{*}\Big{)}\,,$

where $[S^{*},B_{u}]$ denotes the commutator $S^{*}B_{u}-B_{u}S^{*}\,,$
$B_{u}$ and $\tilde{B}_{u}$ are the two skew-adjoint operators of the Lax
pairs, defined respectively in (1.10) and (1.11) .

######  Proposition 2.1 (Multiplicity of $(\lambda_{n})$ and $(\nu_{n})$).

Defocusing case. The eigenvalues $(\lambda_{n})$ of $\tilde{L}_{u}$ are all
simple. More precisely,

$\lambda_{n+1}\geq\lambda_{n}+1\,,\qquad n\in\mathbb{N}_{\geq 0}\,.$ (2.5)

Focusing case. The eigenvalues $(\nu_{n})$ of $L_{u}$ are of multiplicity at
most two

$\nu_{n+2}\geq\nu_{n}+1\,\qquad n\in\mathbb{N}_{\geq 0}\,.$ (2.6)

Moreover, when $n$ is large enough, the eigenvalues of $L_{u}$ are simple.
More precisely,

$\liminf_{n\to\infty}\ \nu_{n+1}-\nu_{n}\geq 1\,.$ (2.7)

Furthermore, for all $0\leq\alpha<1$ such that $\|u\|_{L^{2}}^{2}<1-\alpha\,,$
we have for all $n\in\mathbb{N}_{\geq 0}\,,$

$\nu_{n+1}>\nu_{n}+\alpha\,\,.$ (2.8)

######  Remark 2.1.

  1. (1)

It should be noted that for any potential $u$, the eigenvalues $(\nu_{n})$ of
$L_{u}$ cannot be all simple. For instance, take $u(x)=\operatorname{e}^{ix}$
, one can easily check that for $L_{u}=D-T_{u}T_{\overline{u}}\,,$

$L_{u}1=L_{u}e^{ix}=0\,.$

  2. (2)

Inequality (2.7) implies that as $n>\\!>0$, the lower bound of the distance
between two consecutive eigenvalues $\nu_{n}$ gets closer to $1\,$.

###### Proof.

All the presented inequalities are a direct consequence of the max–min
principle

$\displaystyle\lambda_{n}=\max_{\underset{\dim F\leq n}{F\subseteq
L^{2}_{+}}}\,\min\left\\{\left\langle\tilde{L}_{u}h\,|\,h\right\rangle\,;\,h\in
F^{\perp}\cap H^{\frac{1}{2}}_{+}(\mathbb{T})\,,\ \|h\|_{L^{2}}=1\right\\}\,.$
$\displaystyle\nu_{n}=\max_{\underset{\dim F\leq n}{F\subseteq
L^{2}_{+}}}\,\min\left\\{\left\langle L_{u}h\,|\,h\right\rangle\,;\,h\in
F^{\perp}\cap H^{\frac{1}{2}}_{+}(\mathbb{T})\,,\ \|h\|_{L^{2}}=1\right\\}\,.$

Spectrum of $\tilde{L}_{u}\,$. Let $F$ be any subspace of
$L^{2}_{+}(\mathbb{T})$ of dimension $n\,,$ and consider $E:=\mathbb{C}1\oplus
S(F)\,,$ where $S$ is the shift operator, then

$\lambda_{n+1}\geq\min\\{\langle\tilde{L}_{u}h\mid h\rangle\
;\|h\|_{L^{2}}=1,\,h\in E^{\bot}\cap H^{\frac{1}{2}}_{+}\\}$

Observe that $E^{\perp}=S\left(F^{\perp}\right)\,$, thus by (2.2) ,

$\displaystyle\lambda_{n+1}\geq$
$\displaystyle\,\min\left\\{\langle\tilde{L}_{u}g\mid
g\rangle+1+\lvert\left\langle Sg\,|\,u\right\rangle\rvert^{2}\ ;\
\|g\|_{L^{2}}=1,\ g\in F^{\bot}\cap H^{\frac{1}{2}}_{+}\right\\}\,.$

In addition, since $\lvert\left\langle Sg\,|\,u\right\rangle\rvert^{2}\geq 0$
, we infer for all $n\in\mathbb{N}_{\geq 0}\,,$

$\lambda_{n+1}\geq\lambda_{n}+1\,.$

Spectrum of $L_{u}\,$–Inequality (2.6). let $F$ be any subspace of
$L^{2}_{+}(\mathbb{T})$ of dimension $n$, and take $G:=\mathbb{C}1\oplus
S(F)+~{}\mathbb{C}u$ . Then,

$\nu_{n+2}(u)\geq\min\\{\langle L_{u}h\mid h\rangle\ ;\|h\|_{L^{2}}=1,\ h\in
G^{\bot}\cap H^{\frac{1}{2}}_{+}\\}\,.$

Since $G^{\perp}=S\left(F^{\perp}\cap(S^{*}u)^{\perp}\right)$, then

$\nu_{n+2}\geq\min\left\\{\langle L_{u}Sg\mid Sg\rangle\ ;\ \|g\|_{L^{2}}=1,\
g\in F^{\bot}\cap(S^{*}u)^{\bot}\cap
H^{\frac{1}{2}}_{+}(\mathbb{T})\right\\}\,.$

Note that $g\perp S^{*}u$, then by (2.2),

$\displaystyle\nu_{n+2}\geq$ $\displaystyle\,\min\left\\{\langle L_{u}g\mid
g\rangle+1;\ \|g\|_{L^{2}}=1,\ g\in F^{\bot}\cap(S^{*}u)^{\bot}\cap
H^{\frac{1}{2}}_{+}(\mathbb{T})\right\\}\,,$

leading to

$\nu_{n+2}\geq\nu_{n}+1\,.$

Inequality (2.7) . For any $n\,,$ let
$F_{n}=\operatorname{span}\\{f_{0},f_{1},\ldots,f_{n-1}\\}$ be the subspace of
$L^{2}_{+}(\mathbb{T})$ of dimension $n$ made up of the first $n$
eigenfunctions of $L_{u}\,.$ For this choice of $F_{n}$ ,

$\min\left\\{\langle L_{u}h\mid h\rangle\ ;\ \|h\|_{L^{2}}=1,\ h\in
F^{\bot}_{n}\cap H^{\frac{1}{2}}_{+}\right\\}=\nu_{n}\,.$ (2.9)

Let us consider the subspace $E:=\mathbb{C}1\oplus S(F_{n})$ of
$L^{2}_{+}(\mathbb{T})$ of dimension $n+1$ , then

$\nu_{n+1}\geq\,\min\\{\langle L_{u}g\mid g\rangle\ ;\|g\|_{L^{2}}=1\,,\;g\in
E^{\bot}\cap H^{\frac{1}{2}}_{+}\\}\,.$

Note that $E^{\perp}\cap H_{+}^{\frac{1}{2}}=S\left(F^{\perp}_{n}\cap
H_{+}^{\frac{1}{2}}\right)$ . Therefore, by (2.2) ,

$\displaystyle\nu_{n+1}\geq\,\min\left\\{\langle
L_{u}\phi\mid\phi\rangle+1-\lvert\left\langle
S\varphi\,|\,u\right\rangle\rvert^{2}\ ;\ \|\varphi\|_{L^{2}}=1,\ \varphi\in
F^{\bot}_{n}\cap H^{1}_{+}\right\\}.$

It results, for all $n\in\mathbb{N}_{\geq 0}\,,$

$\nu_{n+1}\geq\,\nu_{n}\,+\,1\,-\,\sup_{\underset{\varphi\in
F^{\bot}_{n}}{\left\|{\varphi}\right\|_{L^{2}(\mathbb{T})}=1}}\left|\langle
S\varphi\right|u\rangle|^{2}\ .$ (2.10)

To conclude the proof, it remains to prove $\sup_{\underset{\varphi\in
F^{\bot}_{n}}{\left\|{\varphi}\right\|_{L^{2}(\mathbb{T})}=1}}\left|\langle
S\varphi\right|u\rangle|^{2}\underset{n\to\infty}{\longrightarrow}0\,.$

######  Lemma 2.2.

Let $F_{n}$ be the be the subspace of $L^{2}_{+}(\mathbb{T})$ defined as
above. Then

$\sup_{\underset{\varphi\in
F^{\bot}_{n}}{\left\|{\varphi}\right\|_{L^{2}(\mathbb{T})}=1}}\lvert\langle
S\varphi\mid u\rangle\rvert\longrightarrow 0\ \ \text{as}\ n\to\infty\ .$

###### Proof.

Suppose for the sake of contradiction, that for all $n\in\mathbb{N}_{\geq
0}\,,$

$\sup_{\underset{\varphi\in
F^{\bot}_{n}}{\left\|{\varphi}\right\|_{L^{2}(\mathbb{T})}=1}}\lvert\left\langle
S\varphi\,|\,u\right\rangle\rvert\geq\varepsilon\,,\qquad\varepsilon>0\,.$

Namely, there exists $\varphi_{n}\in F^{\bot}_{n}$ , $\
\left\|{\varphi_{n}}\right\|_{L^{2}(\mathbb{T})}=1$ such that $\left|\langle
S\varphi_{n}\right|u\rangle|\geq\frac{\varepsilon}{2}\ .$ Hence, since
$\left\|{\varphi_{n}}\right\|_{L^{2}(\mathbb{T})}=1\,,$ then up to a sub-
sequence $\varphi_{n}\rightharpoonup\varphi$ in $L^{2}_{+}(\mathbb{T})$ as
$n\to\infty$ , which yields to

$\lvert\left\langle
S\varphi_{n}\,|\,u\right\rangle\rvert\underset{n\to\infty}{\longrightarrow}\lvert\left\langle
S\varphi\,|\,u\right\rangle\rvert\,,$

and so $\left\langle S\varphi\,|\,u\right\rangle\geq\frac{\varepsilon}{2}$ .
On the other hand, since $\varphi_{n}\perp F_{n}$ then

$\left\langle\varphi_{n}\,|\,f_{p}\right\rangle=0\,,\qquad\forall\,0\leq p\leq
n-1\,.$

Taking $n\to\infty$ , we infer

$\left\langle\varphi\,|\,f_{p}\right\rangle=0\ ,\qquad\forall
p\in\mathbb{N}_{\geq 0}\,.$

Note that the eigenfunctions $(f_{p})$ of the self–adjoint operator $L_{u}$
form an orthonormal basis of $L^{2}_{+}(\mathbb{T})\,.$ Therefore, we have
$\varphi=0$ , which is a contradiction with $\left\langle
S\varphi\,|\,u\right\rangle\geq~{}\frac{\varepsilon}{2}\,$. ∎

Inequality (2.8) . It is a consequence of inequality (2.10) after applying the
Cauchy–Schwarz inequality and considering the fact that
$\|u\|_{L^{2}}^{2}<1-\alpha\,.$

∎

In what follows, we make a slight abuse of notation by using $(f_{n})$ to
denote both an orthonormal basis of $L^{2}_{+}(\mathbb{T})$ consisting of the
eigenfunctions of the self-adjoint operator $L_{u}$, and an orthonormal basis
of $L^{2}_{+}(\mathbb{T})$ consisting of the eigenfunctions of
$\tilde{L}_{u}$. Nonetheless, we will specify the context in which we are
working to avoid confusion and ensure that $(f_{n})$ is understood
appropriately as either the eigenfunctions of $L_{u}$ or $\tilde{L}_{u}$.

######  Lemma 2.3.

Given $u\in H^{2}_{+}(\mathbb{T})$ , then for all $n,p\in\mathbb{N}_{\geq
0}\,,$

  * •

Defocusing case.

$\displaystyle\left\langle 1\,|\,u\right\rangle\langle u\mid
f_{n}\rangle=\lambda_{n}\,\langle 1\mid f_{n}\rangle\,,$
$\displaystyle(\lambda_{n}-\lambda_{p}-1)\,\langle Sf_{p}\mid
f_{n}\rangle=\langle Sf_{p}\mid u\rangle\,\langle u\mid f_{n}\rangle\,.$

  * •

Focusing case.

$\displaystyle\left\langle 1\,|\,u\right\rangle\left\langle
u\,|\,f_{n}\right\rangle=-\nu_{n}\left\langle 1\,|\,f_{n}\right\rangle\,,$
$\displaystyle(\nu_{n}-\nu_{p}-1)\left\langle
Sf_{p}\,|\,f_{n}\right\rangle=-\left\langle
Sf_{p}\,|\,u\right\rangle\left\langle u\,|\,f_{n}\right\rangle\,.$

###### Proof.

We prove first the identities for the defocusing case. By definition of
$\tilde{L}_{u}=D+T_{u}T_{\bar{u}}$ , we have

$\tilde{L}_{u}1=\left\langle 1\,|\,u\right\rangle u\,.$

Then taking the inner product of both sides with $f_{n}\,,$ and using the fact
that $\tilde{L}_{u}$ is a self–adjoint operator, lead to the first identity.
For the second one, thanks to the commutator relation between $\tilde{L}_{u}$
and $S$

$\tilde{L}_{u}Sf_{p}=S\tilde{L}_{u}f_{p}+Sf_{p}+\left\langle
Sf_{p}\,|\,u\right\rangle u\,,$

of equation (2.2), we infer by taking the inner product with $f_{n}$ the
second identity. Besides, by considering the focusing case with
$L_{u}=D-T_{u}T_{\bar{u}}$ it follows that $L_{u}1=-\left\langle
1\,|\,u\right\rangle u\,.$ This explains the sign $-$ appearing in the first
statement. As for the second one, since by (2.2)

$L_{u}Sf_{p}=SL_{u}f_{p}+Sf_{p}-\left\langle Sf_{p}\,|\,u\right\rangle u\,.$

than taking once more the inner product with $f_{n}$ leads to the desired
identity. ∎

In light of the previous lemma and based on the commutator identities (2.2),
one can investigate further information regarding the spectral data (i.e. the
eigenvalues and the eigenvectors) of $L_{u}$ and $\tilde{L}_{u}\,,$ especially
when the quantities $\left\langle u\,|\,f_{n}\right\rangle$ vanishes. The
following lemmas/propositions aim to achieve this. For the following, we
denote by $E_{\nu_{n}}$ the eigenspace of $L_{u}$ corresponding to the
eigenvalue $\nu_{n}$ . In addition, the notation
$f\mathbin{\\!/\mkern-5.0mu/\\!}g$ means that the two vectors $f$ and $g$ are
collinear in $L^{2}_{+}(\mathbb{T})\,$.

######  Proposition 2.4.

For all $n\in\mathbb{N}\,,$ such that $\nu_{n}\neq 0\,,$ we have

$\nu_{n}=\nu_{n-1}+1\implies[Sf_{n-1}\in E_{\nu_{n}}]\text{ or }\,[f_{n}\in
SE_{\nu_{n-1}}]\>,$

Moreover, for the defocusing case,

$\lambda_{n}=\lambda_{n-1}+1\implies
Sf_{n-1}\mathbin{\\!/\mkern-5.0mu/\\!}f_{n}\>.$

######  Remark 2.2.

  1. (1)

The condition $\nu_{n}\neq 0$ cannot be omitted. For an example, we refer to
the Appendix 1 .

  2. (2)

For the defocusing case, the condition of non–vanishing eigenvalues
$\lambda_{n}\neq 0$ is already satisfied for all $n\in\mathbb{N}\,,$ since
$\tilde{L}_{u}$ is a non–negative operator on the Hardy space, and for all
$n\in\mathbb{N}$ , we have by (2.5) , $\lambda_{n}\geq\lambda_{n-1}+1\,.$

###### Proof.

The key is to use Lemma 2.3 and the commutator identities (2.2) . In view of
the second identity of Lemma 2.3, we have

$\left\langle Sf_{n-1}\,|\,u\right\rangle\left\langle
u\,|\,f_{n}\right\rangle=0\,.$ (2.11)

If $\left\langle u\,|\,Sf_{n-1}\right\rangle=0\,,$ then by (2.2),

$\displaystyle L_{u}Sf_{n-1}=$ $\displaystyle\,SL_{u}f_{n-1}+Sf_{n-1}$
$\displaystyle=$ $\displaystyle\,(\nu_{n-1}+1)\,Sf_{n-1}$ $\displaystyle=$
$\displaystyle\,\nu_{n}\,Sf_{n-1}\,,$

as $\nu_{n}=\nu_{n-1}+1$ . Namely, $Sf_{n-1}\in E_{\nu_{n}}$ . Let us move to
the second case where $\left\langle u\,|\,f_{n}\right\rangle=0\,.$ By the
first identity of Lemma 2.3 ,

$\nu_{n}\left\langle f_{n}\,|\,1\right\rangle=0\,.$

Therefore, since $\nu_{n}\neq 0\,,$ there exists $g_{n}\in
H^{1}_{+}(\mathbb{T})$ such that $f_{n}=Sg_{n}$ . Using again the commutator
identity (2.2), we have

$SL_{u}g_{n}=(\nu_{n}-1)Sg_{n}\,.$

Applying $S^{*}$ to both sides of the latter identity, and using the fact that
$S^{*}S=\operatorname{Id}\,$, and as $\nu_{n}=\nu_{n-1}+1\,$, we find

$L_{u}g_{n}=\nu_{n-1}\,g_{n}\,.$

That is, $g_{n}\in E_{\nu_{n-1}}$ , and so $f_{n}\in SE_{\nu_{n-1}}$ .
Besides, note that for the defocusing equation, the vector spaces
$E_{\lambda_{n}}$ are of dimension one, thanks to Proposition 2.1.
Consequently, the results $[Sf_{n-1}\in E_{\lambda_{n}}]$ or $[f_{n}\in
SE_{\lambda_{n-1}}]\,,$ leads to
$Sf_{n-1}\mathbin{\\!/\mkern-5.0mu/\\!}f_{n}\,.$

∎

In the sequel, we denote by $\mathcal{I}(u)$ the set of

$\mathcal{I}(u):=\left\\{n\in\mathbb{N}\;|\;\left\langle
Sf_{n-1}\,|\,f_{n}\right\rangle=0\right\\}\,.$ (2.12)

######  Lemma 2.5.

Defocusing case. For any $u\in H^{2}_{+}(\mathbb{T})\,$, the set
$\mathcal{I}(u)$ is empty.

Focusing case. Given $u\in H^{2}_{+}(\mathbb{T})\,,$ let
$m\in\mathcal{I}(u)\,.$ Assume that the eigenvalues $\nu_{m}$ and $\nu_{m-1}$
are simple, Then, either

$\nu_{m-1}+1=\nu_{m+1}\,,\qquad\text{with}\qquad Sf_{m-1}\in E_{\nu_{m+1}}$

or

$\nu_{m-2}+1=\nu_{m}\,,\qquad\text{with}\qquad S^{*}f_{m}\in E_{\nu_{m-2}}$

or

$\nu_{m}=0\,,\qquad\text{with}\qquad f_{m}\mathbin{\\!/\mkern-5.0mu/\\!}1\,.$

######  Remark 2.3.

  1. (i)

Observe that in the focusing case, if $\|u\|_{L^{2}}^{2}<\frac{1}{2}$ , then
by inequality (2.8) ,

$\nu_{n}>\nu_{n-1}+\frac{1}{2}\,,\qquad\forall n\in\mathbb{N}\,.$

Hence, for such $u\,,$ if $m\in\mathcal{I}(u)$ then the only possible choice
is to have $\nu_{m}=0$ with $f_{m}\,\mathbin{\\!/\mkern-5.0mu/\\!}\,1\,.$ In
other words, for $\|u\|_{L^{2}}^{2}<\frac{1}{2}\,,$ we have, either
$\mathcal{I}(u)=\varnothing\,,$ or $\mathcal{I}(u)=\left\\{m\right\\}$ and in
such case $\nu_{m}=0$ and $f_{m}\mathbin{\\!/\mkern-5.0mu/\\!}1\,.$

  2. (ii)

For any $u\in H^{2}_{+}(\mathbb{T})\,,$ the set $\mathcal{I}(u)$ in the
focusing case is of finite cardinal, since by inequality (2.7) we have
$\nu_{n}>\nu_{n-1}+\frac{1}{2}$ and $\nu_{n}\neq 0$ for all $n$ large enough.

###### Proof.

Focusing case. Let $m\in\mathcal{I}(u)\,.$ By the second identity of Lemma 2.3
,

$\left\langle Sf_{m-1}\,|\,u\right\rangle\left\langle
u\,|\,f_{m}\right\rangle=0\,.$

If $\left\langle Sf_{m-1}\,|\,u\right\rangle=0\,,$ then applying the
commutator identity (2.2) ,

$L_{u}Sf_{m-1}=(\nu_{m-1}+1)Sf_{m-1}\,.$ (2.13)

Namely, $\nu_{m-1}+1$ is an eigenvalue of $L_{u}$ and $Sf_{m-1}$ is a
corresponding eigenfunction. Since $\nu_{m}$ is simple, then $Sf_{m-1}$ cannot
be collinear to $f_{m}$ as $\left\langle Sf_{m-1}\,|\,f_{m}\right\rangle=0$
for $m\in\mathcal{I}(u)\,.$ Therefore, by (2.6) ,

$\nu_{m-1}+1=\nu_{m+1}\,.$

If $\left\langle u\,|\,f_{m}\right\rangle=0\,,$ then by applying the adjoint
of the commutator identity (2.2) ,

$S^{*}{L}_{u}={L}_{u}S^{*}+S^{*}+\left\langle\,\cdot\,|\,u\right\rangle
S^{*}u\,,$

we infer,

${L}_{u}\,S^{*}f_{m}=(\nu_{m}-1)S^{*}f_{m}\,.$

That is, if $S^{*}f_{m}\neq 0\,,$ then $S^{*}f_{m}$ is an eigenfunction of
$L_{u}$ associated with the eigenvalue $\nu_{m}-1\,.$ Recall, we have by
assumption that $\nu_{m-1}$ is simple, and since $S^{*}f_{m}$ cannot be
collinear to $f_{m-1}$ as $m\in\mathcal{I}(u)$ , then

$\nu_{m}-1=\nu_{m-2}\,,$

thanks to (2.6) . It remains to study the case where $S^{*}f_{m}=0\,,$ i.e.,
$f_{m}\mathbin{\\!/\mkern-5.0mu/\\!}1\,.$ For that case, we have thanks to the
first identity of Lemma 2.3 , $\nu_{m}=0$ as $\left\langle
u\,|\,f_{m}\right\rangle=0\,.$ The defocusing case. Suppose that there exists
$m\in\mathcal{I}(u)$ . Then, using the same analysis as in the focusing case,
we infer that, either $\lambda_{m-1}+1=\lambda_{m+1}$ or
$\lambda_{m-2}+1=\lambda_{m}$ or $\lambda_{m}=0\,.$ However, recall that
$\lambda_{n}\geq\lambda_{n-1}+1$ for all $n\in\mathbb{N}$ (inequality 2.5) ,
thus the first two cases cannot occur. In addition, since $\tilde{L}_{u}$ is a
non negative operator, where all the eigenvalues satisfy the inequality (2.5)
, then $\lambda_{m}=0$ implies $m=0\notin\mathcal{I}(u)\,.$

∎

######  Corollary 2.6.

For all $n\geq 1\,,$

$\lambda_{n}=\lambda_{n-1}+1\iff\left\langle u\,|\,f_{n}\right\rangle=0\,.$

In addition,

$\nu_{n}=\nu_{n-1}+1\,,\ \forall n\geq N_{1}\quad\iff\quad\left\langle
u\,|\,f_{n}\right\rangle=0\,,\ \forall n\geq N_{2}\,.$

######  Remark 2.4.

We refer to the Appendix 2 for an example that shows that $N_{2}$ is not
necessarily equal to $N_{1}$ .

###### Proof.

For the defocusing case. Suppose that $\lambda_{n}=\lambda_{n-1}+1\,.$ Then,
from one side we have by Proposition 2.4 ,
$Sf_{n-1}\mathbin{\\!/\mkern-5.0mu/\\!}f_{n}\,,$ and from the other hand, we
infer by the second identity of Lemma 2.3 ,

$\left\langle Sf_{n-1}\,|\,u\right\rangle\left\langle
u\,|\,f_{n}\right\rangle=0\,.$

That is, $\left\langle u\,|\,f_{n}\right\rangle=0\,.$ The converse is a direct
consequence of the second identity of Lemma 2.3 and the previous lemma.

For the focusing case, the same analysis can be applied. However, it should be
noted that, since not all the eigenvalues $(\nu_{n})$ satisfy
$\nu_{n}>\nu_{n-1}+\frac{1}{2}\,,$ and $\nu_{n}\neq 0\,,$ for all
$n\in\mathbb{N}\,$, but only for large $n\,,$ thanks to Proposition 2.1 , then
the equivalence holds for $n$ sufficiently large.

∎

##  3\. Traveling waves for the defocusing (CS-)

###  3.1. Spectral Characterization

One way to understand the behavior of a linear PDE’s solution is to consider
its Fourier transform. Specifically, on the periodic domain $\mathbb{T}$, this
consists of computing the inner product with
$\left\langle\,\cdot\,|\,\operatorname{e}^{inx}\right\rangle$ for all
$n\in\mathbb{Z}$. The main idea behind this approach is to “diagonalize” the
problem in the $(\operatorname{e}^{inx})$–basis, which facilitates solving the
equation. However, by considering the Calogero–Sutherland DNLS equation (CS-)
, we are dealing with a nonlinear integrable PDE, which can also be
“diagonalized” in some coordinate system (think about the Birkhoff
coordinates). Thus, by imitating the idea of the linear case, we suggest
taking the inner product of the (CS-)–equation with an appropriate orthonormal
basis of $L^{2}_{+}(\mathbb{T})\,.$ Before proceeding, observe that the
defocusing Calogero–Sutherland DNLS equation can be rewritten in terms of the
Lax pair as [Ba23, Lemma 2.4, Lemma 5.2]

$\partial_{t}u=\tilde{B}_{u}u-i\tilde{L}^{2}_{u}u\,.$ (3.1)

This motivates the choice of the following orthonormal basis of
$L^{2}_{+}(\mathbb{T})\,$.

######  Definition 3.1.

Given $u\in\mathcal{C}_{\,t}H^{2}_{+}(\mathbb{T})_{x}\,,$ let $(g_{n}^{\,t})$
be the evolving orthonormal basis of $L^{2}_{+}(\mathbb{T})$ defined along the
curve $t\mapsto u(t)$ as

$\displaystyle\begin{cases}\partial_{t}\,g_{n}^{\,t}&=\,\tilde{B}_{u(t)}\,g_{n}^{\,t}\\\
{g_{n}^{\,t}\,}_{|_{t=0}}&=\,f_{n}^{\,u_{0}}\end{cases}\,,\qquad\forall
n\in\mathbb{N}_{\geq 0}\,,$ (3.2)

where $(f_{n}^{\,u_{0}})$ is an orthonormal basis of $L^{2}_{+}(\mathbb{T})$
made up of the eigenfunctions of $\tilde{L}_{u_{0}}$ at $t=0\,,$ and
$\tilde{B}_{u(t)}$ is the skew–adjoint operator defined in (1.11) .

######  Remark 3.1.

Note that the $(g_{n}^{\,t})$ satisfies for all $n\in\mathbb{N}_{\geq 0}$
[Ku06, Lemma 4.1] ,

$\tilde{L}_{u(t)}\,g_{n}^{\,t}=\lambda_{n}\,g_{n}^{\,t}\,.$ (3.3)

Therefore, as it was established in [Ba23, Lemma 3.6] , by taking the inner
product of (3.1) with the $g_{n}^{\,t}$ and using that $\tilde{L}_{u}$ is a
self–adjoint operator and $\tilde{B}_{u}$ is skew–symmetric, we find

$\partial_{t}\left\langle
u(t)\,|\,g_{n}^{\,t}\right\rangle=-i\lambda_{n}^{2}\left\langle
u(t)\,|\,g_{n}^{\,t}\right\rangle\,,$

or

$\left\langle u(t)\,|\,g_{n}^{\,t}\right\rangle=\left\langle
u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{-i\lambda_{n}^{2}\,t}\,.$
(3.4)

######  Lemma 3.2.

For any $u\in\mathcal{C}_{\,t}H^{2}_{+}(\mathbb{T})_{x}$ solution of (CS-) and
for all $n\,,p\in\mathbb{N}_{\geq 0}\,,$

$\displaystyle\left\langle 1\,|\,g_{n}^{\,t}\right\rangle=$
$\displaystyle\,\left\langle
1\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{-i\lambda_{n}^{2}t}\ ,$
(3.5) $\displaystyle\left\langle Sg_{p}^{\,t}\,|\,g_{n}^{\,t}\right\rangle=$
$\displaystyle\,\left\langle
Sf_{p}^{\,u_{0}}\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{i((\lambda_{p}+1)^{2}-\lambda_{n}^{2})\,t}\,.$

###### Proof.

By Definition 3.1 , and since $\tilde{B}_{u}$ is skew-symmetric operator,

$\partial_{t}\left\langle 1\,|\,g_{n}^{\,t}\right\rangle=\left\langle
1\,|\,\tilde{B}_{u}g_{n}^{\,t}\right\rangle=-\left\langle\tilde{B}_{u}1\,|\,g_{n}^{\,t}\right\rangle\,,$

where by (1.11) ,

$\displaystyle\tilde{B}_{u}1=$
$\displaystyle\,-T_{u}T_{\partial_{x}\bar{u}}1+T_{\partial_{x}u}T_{\bar{u}}1+i(T_{u}T_{\bar{u}})^{2}1$
$\displaystyle=$ $\displaystyle\,\left\langle
1\,|\,u\right\rangle\big{(}\partial_{x}u+iT_{u}T_{\overline{u}}\,u\big{)}\,.$

Note that $\tilde{L}_{u}1=-i\partial_{x}1+T_{u}T_{\bar{u}}1=\left\langle
1\,|\,u\right\rangle u\,.$ Therefore, $\tilde{B}_{u}1=i\tilde{L}_{u}^{2}1$ and

$\partial_{t}\left\langle 1\,|\,g_{n}^{\,t}\right\rangle=-i\left\langle
L^{2}_{u}1\,|\,g_{n}^{\,t}\right\rangle=-i\lambda_{n}^{2}\left\langle
1\,|\,g_{n}^{\,t}\right\rangle\,.$

This achieves the proof of the first point. To prove the second one, we
proceed with the same manner. By Definition 3.1 ,

$\partial_{t}\left\langle
g_{n}^{\,t}\,|\,Sg_{p}^{\,t}\right\rangle=\left\langle\tilde{B}_{u}g_{n}^{\,t}\,|\,Sg_{p}^{\,t}\right\rangle+\left\langle
g_{n}^{\,t}\,|\,S\tilde{B}_{u}g_{p}^{\,t}\right\rangle=\left\langle[S^{*},\tilde{B}_{u}]\,g_{n}^{\,t}\,|\,\,g_{p}^{\,t}\right\rangle.$

Hence, applying the commutator identity (2.4) , and since $\tilde{L}_{u}$ is a
self–adjoint operator, we infer

$\displaystyle\partial_{t}\left\langle
g_{n}^{\,t}\,|\,Sg_{p}^{\,t}\right\rangle=$
$\displaystyle\,i\,\big{\langle}\big{(}S^{*}\tilde{L}_{u}^{2}\,-\,(\tilde{L}_{u}+\operatorname{Id})^{2}S^{*}\big{)}g_{n}^{\,t}\;\big{|}\;g_{p}^{\,t}\big{\rangle}$
$\displaystyle=$
$\displaystyle\,i\,(\lambda_{n}^{2}+(\lambda_{p}+1)^{2})\left\langle
g_{n}^{\,t}\,|\,Sg_{p}^{\,t}\right\rangle\,.$

Therefore,

$\left\langle g_{n}^{\,t}\,|\,Sg_{p}^{\,t}\right\rangle=\left\langle
f_{n}^{\,u_{0}}\,|\,Sf_{p}^{\,u_{0}}\right\rangle\,\operatorname{e}^{i(\lambda_{n}^{2}-(\lambda_{p}+1)^{2})t}\,.$

∎

######  Remark 3.2.

The consideration of the evolution of $\left\langle
u\,|\,g_{n}^{\,t}\right\rangle$, $\left\langle
1\,|\,g_{n}^{\,t}\right\rangle$, and $\left\langle
Sg_{p}^{\,t}\,|\,g_{n}^{\,t}\right\rangle$ is motivated by the fact that any
element $u$ of the Hardy space can be written as follows.

######  Lemma 3.3 ([GK21, GMR16]).

For any $u\in L^{2}_{+}(\mathbb{T})\,,$

$u(z)=\left\langle(\operatorname{Id}-zS^{*})^{-1}\,u\,|\,1\right\rangle\,,\qquad
z\in\mathbb{D}\,,$

where $S^{*}$ is the adjoint operator of $S$ in $L^{2}_{+}(\mathbb{T})\,.$

Therefore, by expressing the operator $S^{*}\,$, and the two vectors $u$ and
$1$ in their matrix representations with respect to the $(g_{n}^{\,t})$–basis,
we obtain

$u(t,z)=\left\langle(\operatorname{Id}-zM)^{-1}\,X\,|\,Y\right\rangle\,,\qquad
z\in\mathbb{D}\,,$ (3.6)

where $X\,,$ $Y$ are infinite column vectors and $M$ is the infinite matrix
representation :

$X:=\big{(}\left\langle u\,|\,g_{n}^{\,t}\right\rangle\big{)}\,,\quad
Y:=\big{(}\left\langle 1\mid g_{n}^{\,t}\right\rangle\big{)}\,,\quad
M:=\Big{(}\left\langle g_{n}^{\,t}\mid Sg_{n}^{\,t}\right\rangle\Big{)}\,.$

###### Proof of Lemma 3.3 .

([GK21]) The idea is to observe that any element $u$ of the Hardy space
$L^{2}_{+}(\mathbb{T})$ can be read as an analytic function on the open unit
disc $\mathbb{D}$ , whose trace on the boundary $\partial\mathbb{D}$ is in
$L^{2}\;$ 444For a simple introduction to the different definitions of Hardy
space, we refer to [GMR16, Chapter 3.]. Thus, for any $z\in\mathbb{D}\,,$

$u(z)=\sum_{k\in\mathbb{N}_{\geq
0}}\widehat{u}(k)z^{k}=\sum_{k\in\mathbb{N}_{\geq 0}}\left\langle
u\,|\,S^{k}1\right\rangle z^{k}=\sum_{k\in\mathbb{N}_{\geq
0}}\left\langle(S^{*})^{k}u\,|\,1\right\rangle z^{k}\,.$

As a result, by Neumann series,

$u(z)=\left\langle(\operatorname{Id}-zS^{*})^{-1}u\mid 1\right\rangle.$

∎

At this stage, we consider $u(t):=u_{0}(x-ct)$ to be a traveling wave to the
Calogero–Sutherland DNLS equation (CS-). We denote, for all
$c,t\in\mathbb{R}\,,$ by $\tau_{ct}$ the isometric linear map

$\tau_{ct}:L^{2}_{+}(\mathbb{T})\to
L^{2}_{+}(\mathbb{T})\,,\qquad\tau_{ct}\,u_{0}(x)=u_{0}(x-ct)\,.$

Our aim for this subsection is to prove the following Theorem.

######  Theorem 3.4.

Let $u(t):=\tau_{ct}\,u_{0}\,$ be a traveling wave to the (CS-)–equation. Then
there exists at most one $N\in\mathbb{N}$ such that

$\left\langle u_{0}\,|\,f_{N}^{\,u_{0}}\right\rangle\neq 0\,.$

Moreover, the speed $c$ is given by

$c=1+\frac{2}{N}\sum_{k=0}^{N-1}\lambda_{k}\,.$

To this end, we shall need two key elements. Firstly, Lemma 3.2 and identity
(3.4). Secondly, we will utilize the existence of a relationship (identity
(3.7)) connecting the eigenfunctions $(g_{n}^{\,t})$ of $\tilde{L}_{u(t)}$
introduced in Definition 3.1 , with the functions
$(\tau_{ct}f_{n}^{\,u_{0}})$, where recall $(f_{n}^{\,u_{0}})$ represents the
eigenfunctions of $\tilde{L}_{u_{0}}$. To establish this connection, we
present the following proposition, which also describes the behavior of the
eigenfunctions $(f_{n}^{\,u_{0}})$ of $\tilde{L}_{u_{0}}$ under the action of
the translation map on the spatial variable

$f_{n}^{\,u_{0}}\longmapsto\tau_{ct}\,f_{n}^{\,u_{0}}\,,\qquad
c,t\in\mathbb{R}\,.$

######  Proposition 3.5.

Let $u(t):=\tau_{ct}\,u_{0}\,$ be a solution to (CS-) . There exists a
sequence $(\theta_{n}(t))\subseteq\mathbb{R}\,,$ such that

$\tau_{ct}\,f_{n}^{\,u_{0}}=\operatorname{e}^{i\theta_{n}(t)}\,g_{n}^{\,t}\;,\qquad\forall
n\in\mathbb{N}_{\geq 0}\,.$ (3.7)

In other words, the $(\tau_{ct}f_{n}^{\,u_{0}})$ are also eigenfunctions of
$\tilde{L}_{u(t)}\,.$

###### Proof.

By definition of $\tilde{L}_{u}=D+u\,\Pi\big{(}\bar{u}\,\cdot\big{)}\,,$ and
since $u(t)=\tau_{ct}u_{0}$

$\displaystyle\tilde{L}_{u(t)}\;\tau_{ct}\,f_{n}^{\,u_{0}}=$
$\displaystyle\,Df_{n}^{\,u_{0}}(x-ct)\,+\,u_{0}(x-ct)\,\Pi\big{(}\bar{u}_{0}(x-ct)\;f_{n}^{\,u_{0}}(x-ct)\big{)}\,,$
$\displaystyle=$
$\displaystyle\,\tau_{ct}\left(\tilde{L}_{u_{0}}f_{n}^{\,u_{0}}\right)$
$\displaystyle=$
$\displaystyle\,\lambda_{n}(u_{0})\,\tau_{ct}f_{n}^{\,u_{0}}\,,\hskip
120.92421pt\forall n\in\mathbb{N}_{\geq 0}\,.$

In other words, $\tau_{ct}f_{n}^{\,u_{0}}$ is an eigenfunction of
$\tilde{L}_{u(t)}$ associated with the eigenvalue $\lambda_{n}(u_{0})\,.$ On
the other hand, recall that all the eigenvalues $\lambda_{n}(u_{0})$ of
$\tilde{L}_{u(t)}$ are simple, as stated in Proposition 2.1 . Additionally,
according to Remark 3.1, the $(g_{n}^{\,t})$ are eigenfunctions of
$\tilde{L}_{u(t)}$ associated to the eigenvalues $\lambda_{n}(u_{0})\,$.
Therefore, for all $n\in\mathbb{N}_{\geq 0}\,$, the two vectors
$\tau_{ct}f_{n}^{\,u_{0}}$ and $g_{n}^{\,t}$ are collinear. Since both vectors
belongs to an orthonormal basis of $L^{2}_{+}(\mathbb{T})$, then each one has
an $L^{2}$-norm equal to one. Thus, we infer for all $n\in\mathbb{N}_{\geq
0}\,,$ there exists $\theta_{n}(t)\in\mathbb{R}$ such that for all
$t\in\mathbb{R}\,,$

$\tau_{ct}f_{n}^{\,u_{0}}=\operatorname{e}^{i\theta_{n}(t)}g_{n}^{\,t}\,.$

∎

######  Corollary 3.6.

For all $n\,,p\in\mathbb{N}_{\geq 0}\,,$ and for all $t\in\mathbb{R}\,,$ we
have

  1. (1)

If $\langle 1\,|f_{n}^{\,u_{0}}\rangle\neq 0$ then

$\theta_{n}(t)=-\lambda_{n}^{2}t\,.$

  2. (2)

If $\langle u_{0}\,|f_{n}^{\,u_{0}}\rangle\neq 0$ then

$\theta_{n}(t)=-\lambda_{n}^{2}t\,.$

  3. (3)

If $\langle Sf_{p}^{\,u_{0}}|f_{n}^{\,u_{0}}\rangle\neq 0$ then

$\theta_{n}(t)=((\lambda_{p}+1)^{2}-\lambda_{n}^{2})t-ct+\theta_{p}(t)\,,$

where $\theta_{n}(t)$ is the angle obtain in (3.7) .

###### Proof.

By combining identity (3.4) and the two identities of Lemma 3.2 , with
identity (3.7) of the previous proposition, we infer

$\left|\begin{array}[]{l}\mathrm{e}^{i\theta_{n}(t)}\left\langle
1\,|\,\tau_{ct}f_{n}^{\,u_{0}}\right\rangle=\left\langle
1\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{-i\lambda_{n}^{2}t}\
,\\\\[5.69046pt]
\mathrm{e}^{i\theta_{n}(t)}\left\langle\tau_{ct}u_{0}\,|\,\tau_{ct}f_{n}^{\,u_{0}}\right\rangle=\left\langle
u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{-i\lambda_{n}^{2}\,t}\\\\[5.69046pt]
\operatorname{e}^{-i\theta_{p}(t)}\operatorname{e}^{i\theta_{n}(t)}\left\langle
S\tau_{ct}f_{p}^{\,u_{0}}\,|\,\tau_{ct}f_{n}^{\,u_{0}}\right\rangle=\left\langle
Sf_{p}^{\,u_{0}}\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{i((\lambda_{p}+1)^{2}-\lambda_{n}^{2})\,t}\,.\end{array}\right.$

Note that,
$S\,\tau_{ct}\;(\cdot)=\operatorname{e}^{ict}\tau_{ct}(S\;\cdot)\,$, and since
we are dealing with periodic functions , we deduce,

$\left|\begin{array}[]{l}\mathrm{e}^{i\theta_{n}(t)}\left\langle
1\,|\,f_{n}^{\,u_{0}}\right\rangle=\left\langle
1\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{-i\lambda_{n}^{2}t}\
,\\\\[5.69046pt]
\mathrm{e}^{i\theta_{n}(t)}\operatorname{e}^{i\varphi(t)}\left\langle
u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle=\left\langle
u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{-i\lambda_{n}^{2}\,t}\\\\[5.69046pt]
\operatorname{e}^{-i\theta_{p}(t)}\operatorname{e}^{i\theta_{n}(t)}\operatorname{e}^{ict}\left\langle
Sf_{p}^{\,u_{0}}\,|\,f_{n}^{\,u_{0}}\right\rangle=\left\langle
Sf_{p}^{\,u_{0}}\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{i((\lambda_{p}+1)^{2}-\lambda_{n}^{2})\,t}\,.\end{array}\right.$
(3.8)

leading to the result.

∎

At this point, we are ready to prove the spectral characterization of the
traveling waves for (CS-), namely Theorem 3.4 .

###### Proof of Theorem 3.4.

The proof relies on the spectral property of $\tilde{L}_{u}$ discussed in
Section 2 and on Corollary 3.6 . Indeed, observe first by Lemma 2.5 , we have
$\left\langle Sf_{n-1}^{\,u_{0}}\,|\,f_{n}^{\,u_{0}}\right\rangle\neq 0$ for
all $n\in\mathbb{N}\,.$ Hence, applying the third identity of Corollary 3.6
with $p=n-1$ , leads to the recurrence relation

$\theta_{n}(t)=\big{(}(\lambda_{n-1}+1)^{2}-\lambda_{n}^{2}\big{)}t-ct+\theta_{n-1}(t)\,,\qquad
n\geq 1\,.$

Taking the sum of all these expressions from $n=1$ to $n\in\mathbb{N}$ , we
infer

$\theta_{n}(t)=\lambda_{0}^{2}\,t+2t\sum_{k=0}^{n-1}\,\lambda_{k}+nt-\lambda_{n}^{2}\,t-nct+\theta_{0}(t)\,.$
(3.9)

Our aim is to prove that for all $n\geq 1$ , $\left\langle
u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle=0$ unless at most for one $n$ . For the
sake of contradiction, suppose that there exist two integers $1\leq
n_{1}<n_{2}$ such that $\left\langle
u_{0}\,|\,f_{n_{1}}^{\,u_{0}}\right\rangle\neq 0$ and $\left\langle
u_{0}\,|\,f_{n_{2}}^{\,u_{0}}\right\rangle\neq 0$ . Then by Corollary 3.6, we
infer

$\displaystyle\theta_{n_{1}}(t)=-\lambda_{n_{1}}^{2}t$ (3.10)
$\displaystyle\theta_{n_{2}}(t)=-\lambda_{n_{2}}^{2}t$

Plugging (3.10) in (3.9) we obtain

$\displaystyle
n_{1}ct=n_{1}t+2t\sum_{k=0}^{n_{1}-1}\lambda_{k}+\theta_{0}(t)+\lambda_{0}^{2}t\,.$
(3.11) $\displaystyle
n_{2}\,ct=n_{2}t+2t\sum_{k=0}^{n_{2}-1}\lambda_{k}+\theta_{0}(t)+\lambda_{0}^{2}t\,.$

Besides, notice that

$\theta_{0}(t)=-\lambda_{0}^{2}t\,.$ (3.12)

Indeed, if $\left\langle u_{0}\,|\,f_{0}^{\,u_{0}}\right\rangle\neq 0$ then by
the second point of Corollary 3.6, we have the claimed identity. Otherwise, if
$\left\langle u_{0}\,|\,f_{0}^{\,u_{0}}\right\rangle=0$ then $\left\langle
1\,|\,f_{0}^{\,u_{0}}\right\rangle\neq 0$ , since if it is not the case, i.e.
if there exists $h\in L^{2}_{+}(\mathbb{T})$ such that $f_{0}^{\,u_{0}}=Sh\,,$
then we have by the commutator relation (2.2)

$\lambda_{0}Sh=\tilde{L}_{u_{0}}Sh=S\tilde{L}_{u_{0}}h+Sh+\left\langle
Sh\,|\,u_{0}\right\rangle u_{0}\,,$

implying, as $\left\langle Sh\,|\,u_{0}\right\rangle=\left\langle
f_{0}^{\,u_{0}}\,|\,u_{0}\right\rangle=0$,

$\tilde{L}_{u_{0}}h=(\lambda_{0}-1)h\,.$

That means, $h$ is an eigenvector of $\tilde{L}_{u_{0}}$ associated with an
eigenvalue strictly less than $\lambda_{0}\,,$ which is impossible. Therefore
$\left\langle 1\,|\,f_{0}^{\,u_{0}}\right\rangle\neq 0\,,$ and so by the first
identity of Corollary 3.6, we infer $\theta_{0}(t)=-\lambda_{0}^{2}\,t\,.$
Substituting (3.12) in (3.11) , we obtain

$\displaystyle\left|\begin{array}[]{l}\displaystyle
c=1+\frac{2}{n_{1}}\,\sum_{k=0}^{n_{1}-1}\lambda_{k}\,.\\\\[14.22636pt]
\displaystyle
c=1+\frac{2}{n_{2}}\,\sum_{k=0}^{n_{2}-1}\lambda_{k}\,.\end{array}\right.$

That is,

$n_{2}\,\sum_{k=0}^{n_{1}-1}\lambda_{k}=n_{1}\,\sum_{k=0}^{n_{2}-1}\lambda_{k}\,,$

or

$(n_{2}-n_{1})\,\sum_{k=0}^{n_{1}-1}\lambda_{k}\,=\,n_{1}\,\sum_{k=n_{1}}^{n_{2}-1}\lambda_{k}$

But recall by (2.5) , $\lambda_{n+1}>\lambda_{n}\,,$ for all $n\,.$ Combining
this fact with the last equality, we conclude

$n_{1}(n_{2}-n_{1})\,\lambda_{n_{1}-1}\,>\,n_{1}(n_{2}-n_{1})\,\lambda_{n_{1}}\,,$

leading to a contradiction. As a consequence, for any traveling wave solution
$u(t,x):=u_{0}(x-ct)$ of (CS-) , there exists at most one $N\in\mathbb{N}$
such that

$\left\langle u_{0}\,|\,f_{N}^{\,u_{0}}\right\rangle\neq 0\,,$

where $(f_{n}^{\,u_{0}})$ is any orthonormal basis of $L^{2}_{+}(\mathbb{T})$
consisting of the eigenfunctions of $\tilde{L}_{u_{0}}$. Moreover, $u$ travels
with the speed

$c=1+\frac{2}{N}\,\sum_{k=0}^{N-1}\lambda_{k}\,.$ (3.13)

∎

######  Remark 3.3.

In view of the previous Theorem and Corollary 2.6 , it follows that any
traveling wave solution $u$ of (CS-) propagates with a speed

$c=N+2\lambda_{0}\,.$ (3.14)

Indeed, since $\left\langle u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle=0$ for all
$1\leq n<N\,,$ then by Corollary 2.6 ,

$\lambda_{n}=\lambda_{n-1}+1\,,\qquad\forall 1\leq n<N\,.$

leading to the fact that (3.13) is equivalent to (3.14) . Besides, since
$\tilde{L}_{u}$ is a non–negative operator, then $\lambda_{0}\geq 0$ , which
implies that the speed of the traveling wave solution satisfies $c\geq N\,.$
However, as will be observed in Subsection 3.3, the speed $c=N$ can only be
reached by traveling waves of the form $u(t,x)=\operatorname{e}^{iN(x-Nt)}\,.$

###  3.2. Explicit formulas of the traveling waves

Recall by Remark 3.2 , any elements of the Hardy space, in particular
$u_{0}\,,$ can be written as

$u_{0}(z)=\left\langle(\operatorname{Id}-zM)^{-1}\,X\,|\,Y\right\rangle\,,$
(3.15)

where $X\,,$ $Y$ are infinite column vectors, $M$ is an infinite matrix :

$X:=\big{(}\left\langle u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle\big{)}\,,\quad
Y:=\big{(}\left\langle 1\mid f_{n}^{\,u_{0}}\right\rangle\big{)}\,,\quad
M=\Big{(}\left\langle f_{p}^{\,u_{0}}\mid
Sf_{n}^{\,u_{0}}\right\rangle\Big{)}\,.$ (3.16)

In the following, we denote by $\mathcal{G}_{1}$ the set of the semi–trivial
traveling waves, made up from the constant and the plane wave solutions

$\mathcal{G}_{1}=\left\\{C\operatorname{e}^{iN(x-Nt)}\,\mid\,C\in\mathbb{C}\,,\,N\in\mathbb{N}_{\geq
0}\right\\}\,.$ (3.17)

######  Theorem (1.1).

The traveling waves $u(t,x)=u_{0}(x-ct)$ of (CS-) are the potentials
$u(t,x)\in\mathcal{G}_{1}$ and

$u(t,x):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,$

where $N\in\mathbb{N}\,,$ $\,c:=-N\Big{(}1+\frac{2\alpha}{\beta}\Big{)},$ and
$(\alpha,\beta)$ are two real constants satisfying

$\alpha\beta+\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=-N\,.$ (3.18)

###### Proof.

The proof is based on the inversion spectral formula

$u_{0}(z)=\left\langle(\operatorname{Id}-zM)^{-1}\,X\,|\,Y\right\rangle\,,$

of (3.15) , and on the spectral characterization of $u_{0}$ described in
Theorem 3.4. In the sequence, to make the notation less cluttered, we denote
$f_{n}:=f_{n}^{\,u_{0}}\,.$ Let $u(t,x):=u_{0}(x-ct)\,$. As a first step, we
prove that the infinite matrices $X\,,Y$ and $M$ reduce to finite matrices in
the context of a traveling wave solution. Indeed, by Theorem 3.4 , there
exists at most one $N\in\mathbb{N}\,,$ such that $\left\langle
u_{0}\,|\,f_{N}\right\rangle\neq 0$ . We focus on the case where such an $N$
exists, that is:

$\begin{cases}\displaystyle\left\langle u_{0}\,|\,f_{N}\right\rangle\neq
0\\\\[3.41418pt] \displaystyle\left\langle
u_{0}\,|\,f_{n}\right\rangle=0\,,\;\forall
n\in\mathbb{N}\backslash\left\\{N\right\\}\end{cases}\,.$ (3.19)

The case where $\left\langle u\,|\,f_{n}\right\rangle=0$ for all
$n\in\mathbb{N}$ can be handled similarly, leading also to the reduction of
the study to finite matrices. From now on, we suppose (3.19) holds. Therefore,
it follows by Lemma 2.3 , that $\lambda_{n}\left\langle
1\,|\,f_{n}\right\rangle=0\,,$ implying that

$\left\langle 1\,|\,f_{n}\right\rangle=0\,,\qquad\forall
n\in\mathbb{N}\backslash\left\\{N\right\\}\,,$

as the eigenvalues $\lambda_{n}$ are all positive for any $n\in\mathbb{N}$
since $\tilde{L}_{u}$ is a non–negative operator. Therefore, the two infinite
column vectors $X$ and $Y$ of (3.16) reduces to

$X=\left(\begin{array}[]{c}\left\langle u_{0}\,|\,f_{0}\right\rangle\\\ 0\\\
\vdots\\\ 0\\\ \left\langle u_{0}\,|\,f_{N}\right\rangle\\\ 0\\\
\vdots\end{array}\right)\,,\qquad Y=\left(\begin{array}[]{c}\left\langle
1\,|\,f_{0}\right\rangle\\\ 0\\\ \vdots\\\ 0\\\ \left\langle
1\,|\,f_{N}\right\rangle\\\ 0\\\ \vdots\end{array}\right)\,.$ (3.20)

On the other hand, since $\left\langle u_{0}\,|\,f_{n}\right\rangle=0\,$, for
all $n\in\mathbb{N}\backslash\left\\{N\right\\}$ , then by Corollary 2.6 , we
have $\lambda_{n}=\lambda_{n-1}+1$ for all
$n\in\mathbb{N}\backslash\left\\{N\right\\}\,$. Whence,
$Sf_{n-1}\,\mathbin{\\!/\mkern-5.0mu/\\!}\,f_{n}$ for all
$n\in\mathbb{N}\backslash\left\\{N\right\\}\,,$ thanks to Proposition 2.4 .
More specifically,

$\begin{cases}f_{n}\,\mathbin{\\!/\mkern-5.0mu/\\!}\,S^{n}f_{0}\,\hskip
17.07182pt,\qquad 1\leq n\leq N-1\\\
f_{n}\,\mathbin{\\!/\mkern-5.0mu/\\!}\,S^{n-N}f_{N}\,,\qquad n\geq
N\end{cases}\,.$ (3.21)

As a consequence, the set
$\big{\\{}(S^{n}f_{0})_{n=0\,,\ldots\,,N-1}\,,(S^{n}f_{N})_{n\geq 0}\big{\\}}$
is an orthonormal basis of $L^{2}_{+}(\mathbb{T})$ and the matrix
$M=\big{(}\left\langle f_{p}\,|\,Sf_{n}\right\rangle\big{)}$ reduces to

$M=\left(\begin{array}[]{ccccc|cccc}0&1&0&\ldots&0&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&&\vdots\\\ 0&&\ddots&1&0&&\\\
\langle f_{0}\mid S^{N}f_{0}\rangle&0&\ldots&0&\left\langle
f_{N}\,|\,S^{N}f_{0}\right\rangle&0&\\\ 0&\ldots&&\ldots&0&1&0\\\ \hline\cr
0&\ldots&&\ldots&0&0&1&0\\\
\vdots&&&&\vdots&\vdots&\ddots&\ddots\end{array}\right).$

Hence, the infinite matrices $X\,,\,Y$ and $M$ in formula (3.15) can be
restrained to finite matrices involving only the first $N+1$ coordinates of
$X\,$, $Y$, and $M$ [GK21] . Indeed, denoting
$\bm{\xi}:=\left(\operatorname{Id}-zM\right)^{-1}X$ , we have

$\left(\operatorname{Id}-zM\right)\bm{\xi}=X\,.$

That is,

${\tiny\left(\begin{array}[]{ccccc|cccc}1&-z&0&\ldots&0&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&&\vdots\\\ 0&&\ddots&-z&0&&\\\
-\langle f_{0}\mid S^{N}f_{0}\rangle z&0&\ldots&1&-\left\langle
f_{N}\,|\,S^{N}f_{0}\right\rangle z&0&\\\ 0&\ldots&&\ldots&1&-z&0\\\ \hline\cr
0&\ldots&&\ldots&0&1&-z&0\\\
\vdots&&&&\vdots&\vdots&\ddots&\ddots\end{array}\right)\cdot\left(\begin{array}[]{c}\xi_{0}\\\
\vdots\\\ \xi_{N-1}\\\ \xi_{N}\\\ \xi_{N+1}\\\
\vdots\end{array}\right)=\left(\begin{array}[]{c}\left\langle
u_{0}\,|\,f_{0}\right\rangle\\\ 0\\\ \vdots\\\ 0\\\ \left\langle
u_{0}\,|\,f_{N}\right\rangle\\\ 0\\\ \vdots\end{array}\right)}$

Thus, for all $n\geq N+1\,,$ the $n^{\text{th}}$ coordinate of $\bm{\xi}$ is
$\xi_{n}=z\,\xi_{n+1}\,$, i.e.

$\xi_{N+1}=z^{n-N-1}\xi_{n}\,,\qquad\forall n\geq N+1\,.$

And since $\sum_{n\geq 0}|\xi_{n}|^{2}<\infty\,$, then

$\xi_{n}=0\,,\qquad\forall n\geq N+1\,.$

As a result,

$\left\langle\left(\operatorname{Id}-z(M_{mn})_{m,n\geq
N+1}\right)^{-1}(X_{n})_{n\geq N+1}\,|\,(Y_{m})_{m\geq N+1}\right\rangle=0\,,$
(3.22)

and therefore

$u_{0}(z)=\left\langle\left(\operatorname{Id}-zM_{\leq N}\right)^{-1}X_{\leq
N}\,|\,Y_{\leq N}\right\rangle_{\mathbb{C}^{N+1}\times\mathbb{C}^{N+1}}\,,$

where $M_{\leq N}:=(M_{mn})_{\,0\leq m,n\leq N}\,,$ $X_{\leq
N}:=(X_{n})_{\,0\leq n\leq N}$ and $Y_{\leq N}:=(Y_{N})_{\,0\leq n\leq N}\,.$
Consequently, $u_{0}$ is a rational function

$u_{0}(z)=\frac{P(z)}{\det\left(\operatorname{Id}-zM_{\leq N}\right)}\;,\;$

where $P(z)=Y_{\leq N}^{*}\cdot\,\mathrm{Com}(\operatorname{Id}-zM_{\leq
N})^{T}\,\cdot X_{\leq N}$. Computing the numerator $P$ and the denominator of
$u_{0}$ via these finite matrices, we obtain that $u_{0}$ is of the form

$u_{0}(z)=\frac{az^{N}+b}{1-pz^{N}}\,,\qquad a,b\in\mathbb{C}\,,$ (3.23)

where $p=\left\langle f_{0}\,|\,S^{N}f_{0}\right\rangle\,,$ $\lvert
p\rvert<1\,.$ If $p=0\,$. Namely, if $\left\langle
f_{0}\,|\,S^{N}f_{0}\right\rangle=0$ , then

$S^{N}f_{0}=\sum_{n\geq n}\left\langle S^{N}f_{0}\,|\,f_{n}\right\rangle
f_{n}=\left\langle S^{N}f_{0}\,|\,f_{N}\right\rangle f_{N}\,,$

since by (3.21) , the set
$\big{\\{}(S^{n}f_{0})_{n=0\,,\ldots\,,N-1}\;,\,(S^{n}f_{N})_{N\geq
0}\big{\\}}$ is an orthonormal basis of $L^{2}_{+}(\mathbb{T})$ . Thus, the
two vectors $f_{N}$ and $S^{N}f_{0}$ are collinear, leading to : for all
$n\in\mathbb{N}_{\geq 0}\,,$

$f_{n}\mathbin{\\!/\mkern-5.0mu/\\!}S^{n}f_{0}\,,$

thanks to (3.21) . Consequently, $\left\\{S^{n}f_{0}\,,n\in\mathbb{N}_{\geq
0}\right\\}$ is an orthonormal basis of $L^{2}_{+}(\mathbb{T})$ , which means,
the vector $f_{0}$ is necessarily collinear to $1\,.$ Besides, recall from
(3.20) ,

$\displaystyle u_{0}=$ $\displaystyle\,\left\langle
u_{0}\,|\,f_{0}\right\rangle f_{0}+\left\langle u_{0}\,|\,f_{N}\right\rangle
f_{N}$ $\displaystyle=$ $\displaystyle\,\left\langle
u_{0}\,|\,f_{0}\right\rangle f_{0}+\left\langle
u_{0}\,|\,S^{N}f_{0}\right\rangle S^{N}f_{0}\;,$ $\displaystyle=$
$\displaystyle\,\left\langle u_{0}\,|\,1\right\rangle+\left\langle
u_{0}\,|\,\mathrm{e}^{iNx}\right\rangle\mathrm{e}^{iNx}$

and, as $p=0$ i.e. $\left\langle f_{0}\,|\,S^{N}f_{0}\right\rangle=0\,,$ we
have by the second identity of Lemma 2.3 , either

$\left\langle u_{0}\,|\,f_{0}\right\rangle=0\qquad\text{or}\qquad\left\langle
u_{0}\,|\,S^{N}f_{0}\right\rangle=0\,.$

i.e.

$\left\langle u_{0}\,|\,1\right\rangle=0\qquad\text{or}\qquad\left\langle
u_{0}\,|\,\mathrm{e}^{iNx}\right\rangle=0\,.$

Therefore, either $u_{0}(x)$ is a complex constant, or
$u_{0}(x)=C\mathrm{e}^{iNx}\,,$ with $C\in\mathbb{C}$ , $N\in\mathbb{N}\,.$
Taking, $u(t,x)=u_{0}(x-ct)=C\operatorname{e}^{iN(x-ct)}$, and substituting it
in the defocusing Calogero–Sutherland DNLS equation (CS-) , we infer, since
the nonlinearity
$D\Pi(\lvert\operatorname{e}^{iN(x-ct)}\rvert^{2})\operatorname{e}^{iN(x-ct)}$
vanishes,

$Nc\operatorname{e}^{iN(x-ct)}-N^{2}\operatorname{e}^{iN(x-ct)}=0\,,$

and thus $c=N\,.$ As a result, if $p=0$ then the traveling waves
$u(t,x):=u_{0}(x-ct)$ are

$u(t,x)=C\operatorname{e}^{iN(x-Nt)}\,,\qquad
C\in\mathbb{C}\,,\,N\in\mathbb{N}_{\geq 0}\,.$

Let us move, to the case where $p\neq 0\,.$ The potential $u_{0}$ of (3.23)
can be rewritten as

$u_{0}=\alpha+\frac{\beta}{1-pz^{n}}\,,\qquad\alpha,\beta\in\mathbb{C}\,,\quad
p,z\in\mathbb{D}\,,$

In order to find the relation between $\alpha,\beta$ and obtain the speed
$c\,,$ we substitute $u(t,z):=u_{0}(\operatorname{e}^{-ict}z)$ into the
defocusing Calogero–Sutherland DNLS equation (CS-). This equation can be
rewritten as

$i\partial_{t}u-(z\partial_{z})^{2}u-2z\partial_{z}\Pi(\lvert
u\rvert^{2})u=0\,,$ (3.24)

after observing that $D=-i\partial_{x}$ can be expressed as $D\equiv
z\partial_{z}$. Thus, starting from

$u:=u(t,z)=\alpha+\frac{\beta}{1-p\operatorname{e}^{-iNct}z^{N}}\,,$

and computing $i\partial_{t}u$ and $(z\partial_{z})^{2}u\,,$ we find

$i\partial_{t}u=-c\beta
N\left(\frac{1}{1-p\operatorname{e}^{-iNct}z^{N}}-\frac{1}{(1-p\operatorname{e}^{-iNct}z^{N})^{2}}\right)\,,$
$(z\partial_{z})^{2}u=\beta
N^{2}\left(\frac{1}{1-p\operatorname{e}^{-iNct}z^{N}}-\frac{3}{(1-p\operatorname{e}^{-iNct}z^{N})^{2}}+\frac{2}{(1-p\operatorname{e}^{-iNct}z^{N})^{3}}\right)\,.$

For the nonlinear part,

$\displaystyle\lvert
u\rvert^{2}=\lvert\alpha\rvert^{2}+\alpha\bar{\beta}+\frac{\alpha\bar{\beta}\,\bar{p}\operatorname{e}^{iNct}}{z^{N}-\bar{p}\operatorname{e}^{iNct}}+\frac{\alpha\bar{\beta}}{1-p\operatorname{e}^{-iNct}z^{N}}+\frac{\lvert\beta\rvert^{2}\,z^{N}}{(1-p\operatorname{e}^{-iNct}z^{N})(z^{N}-\bar{p}\operatorname{e}^{iNct})}\,.$

Recall that $\Pi$ is an orthonormal projector into the Hardy space (in
particular to a subspace of the holomorphic functions on $\mathbb{D}$). Thus,
applying $\Pi$ , it follows

$\Pi(\lvert
u\rvert^{2})=\lvert\alpha\rvert^{2}+\alpha\bar{\beta}+\frac{\alpha\bar{\beta}}{1-p\operatorname{e}^{-iNct}z^{N}}+\frac{\lvert\beta\rvert^{2}}{1-\lvert
p\rvert^{2}}\frac{1}{1-p\operatorname{e}^{-iNct}z^{N}}\,.$

And hence,

$z\partial_{z}\Pi(\lvert u\rvert^{2})\cdot
u=A\left(\frac{-\alpha}{1-p\operatorname{e}^{-iNct}z^{N}}+\frac{-\beta+\alpha}{(1-p\operatorname{e}^{-iNct}z^{N})^{2}}+\frac{\beta}{(1-p\operatorname{e}^{-iNct}z^{N})^{3}}\right)\,,$

where

$A=N\left(\bar{\alpha}\beta+\frac{\lvert\beta\rvert^{2}}{1-\lvert
p\rvert^{2}}\right)\,.$

Substituting the expressions of $i\partial_{t}u\,,\,$ $(z\partial_{z})^{2}u\,$
and $\,z\partial_{z}\Pi(\lvert u\rvert^{2})u$ into (3.24) , and comparing the
terms $\displaystyle\frac{1}{(1-p\operatorname{e}^{-iNct}z^{N})^{n}}$ for
$n=1,2,3$ , we deduce

  * •

With $n=3\,,$ $A=-N^{2}\,.$ That is,

$\bar{\alpha}\beta+\frac{\lvert\beta\rvert^{2}}{1-\lvert p\rvert^{2}}=-N$

  * •

With $n=2$ and $n=1\,,$

$c=-N\left(1+\frac{2\alpha}{\beta}\right)\,.$

As a result, for $p\neq 0\,,$

$u(t,z):=\alpha+\frac{\beta}{1-p\operatorname{e}^{-iNct}z^{N}}\,,\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,$

where $N\in\mathbb{N}\,,$ $\,c:=-N\Big{(}1+\frac{2\alpha}{\beta}\Big{)},$ and
$(\alpha,\beta)\in\mathbb{C}\times\mathbb{C}$ satisfy

$\alpha\beta+\frac{\lvert\beta\rvert^{2}}{1-\lvert p\rvert^{2}}=-N\,.$ (3.25)

Finally, observe by (3.25) , the two complex constants $(\alpha,\beta)$
satisfy $\bar{\alpha}\beta\in\mathbb{R}\,.$ Thus, by making a slight abuse of
notation on $\alpha$ and $\beta$, we have obtained that the traveling waves of
(CS-) with $p\neq 0$ are given by

$u(t,z):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{-iNct}z^{N}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,$

where $N\in\mathbb{N}\,,$ $\,c:=-N\Big{(}1+\frac{2\alpha}{\beta}\Big{)},$ and
$(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ satisfy

$\alpha\beta+\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=-N\,.$

∎

###  3.3. The $L^{2}$–norm and the speed

In this subsection, we analyze how the traveling waves of (CS-) behaves, by
providing information regarding their $L^{2}$–norm and their speed $c$. Recall
that the set of traveling wave solutions of the defocusing Calogero–Sutherland
DNLS equation are made up by the trivial solutions

$\mathcal{G}_{1}=\left\\{C\operatorname{e}^{iN(x-Nt)}\,\mid\,C\in\mathbb{C}\,,\,N\in\mathbb{N}_{\geq
0}\right\\}\,,$

and by the set of functions

$u(t,x):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,$ (3.26)

where $N\in\mathbb{N}\,,$ $\,c:=-N\Big{(}1+\frac{2\alpha}{\beta}\Big{)},$ and
$(\alpha,\beta)$ are two real constants satisfying (3.18) . For
$u\in\mathcal{G}_{1}\,,$ it is easy to see that the $L^{2}$–norm of the
semi–trivial solution can be arbitrarily small or large in $[0,+\infty)\,,$
and its speed $c$ is given as $c=N\in\mathbb{N}_{\geq 0}\,.$ The following
proposition aims to provide those for the nontrivial traveling waves of (CS-)
.

######  Proposition 3.7 ($L^{2}$ norm of a non–trivial traveling wave and the
speed).

  1. (i)

For any $r>0\,,$ there exists a non–trivial traveling waves
$u(t,x):=u_{0}(x-ct)$ for (CS-) with

$\|u_{0}\|_{L^{2}}=r\,.$

In other words, the traveling waves of (CS-) can be arbitrarily small or large
in $L^{2}_{+}(\mathbb{T})\,.$

  2. (ii)

Let $u$ be a traveling wave for (CS-) of the form (3.26) , then $u$ propagates
to the right with a speed $c>N\,.$ In addition, when $\|u\|_{L^{2}}\to\infty$
then $c\to\infty$ and when $\|u\|_{L^{2}}\to 0$ then $c\to N\,.$

######  Remark 3.4 (Non-existence of stationary solution for (CS-)).

Since for any traveling wave $u_{0}(x-ct)$ of the defocusing
Calogero–Sutherland DNLS equation (CS-) we have $c\geq N\,,$ where $N$ is the
numerator’s degree of $u_{0}\,,$ then there is no stationary solution (i.e.
$u(t,x)=u_{0}(x)$) for the (CS-)–equation . Another way to see this, is by
observing that if $c=0\,$, which occurs when $\alpha=-\frac{\beta}{2}$
according to Theorem 1.1, then we have by (3.18)

$\frac{1+\lvert p\rvert^{2}}{1-\lvert p\rvert^{2}}\,\beta^{2}=-N\,,$

which is impossible as $p\in\mathbb{D}^{*}$.

###### Proof.

(i) The $L^{2}$–norm of the non–trivial traveling wave can be arbitrarily
small or large. Let $u$ be a traveling wave of the form (3.26)

$u(t,x):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,N\in\mathbb{N}\,,$

where $(\alpha,\beta)\in\mathbb{R}^{2}$ satisfies the identity (3.18) . Recall
that any function $u$ in the Hardy space can be seen as an analytic function
on the open unit disc $\mathbb{D}\,,$ whose trace on the boundary
$\partial\mathbb{D}$ is in $L^{2}\,.$ Hence,

$\|u\|_{L^{2}}^{2}=\,\int_{z\in\mathscr{C}(0,1)}\lvert
u(z)\rvert^{2}\,\frac{dz}{2\pi iz}\,,$

where

$\displaystyle\lvert u(z)\rvert^{2}=$
$\displaystyle\,\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{-iNct}z^{N}}\right)\left(\alpha+\frac{\beta\,z^{N}}{z^{N}-\bar{p}\operatorname{e}^{iNct}}\right)$
$\displaystyle=$
$\displaystyle\;\alpha^{2}+\alpha\beta+\frac{\alpha\beta\operatorname{e}^{iNct}\bar{p}}{z^{N}-\bar{p}\operatorname{e}^{iNct}}+\frac{\alpha\beta}{1-p\operatorname{e}^{-iNct}z^{N}}+\frac{\beta^{2}\,z^{N}}{(1-p\operatorname{e}^{-iNct}z^{N})(z^{N}-\bar{p}\operatorname{e}^{iNct})}$

Writing

$\frac{\beta^{2}\,z^{N}}{(1-p\operatorname{e}^{-iNct}z^{N})(z^{N}-\bar{p}\operatorname{e}^{iNct})}=\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}\left(\frac{1}{1-p\operatorname{e}^{-iNct}z^{N}}+\frac{\bar{p}\operatorname{e}^{iNct}}{z^{N}-\bar{p}\operatorname{e}^{iNct}}\right)\,,$

we infer

$\lvert
u(z)\rvert^{2}=\alpha^{2}+\alpha\beta+\left(\alpha\beta+\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}\right)\frac{1}{1-p\operatorname{e}^{-iNct}z^{N}}+\left(\alpha\beta+\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}\right)\frac{\operatorname{e}^{iNct}\bar{p}}{z^{N}-\bar{p}\operatorname{e}^{iNct}}\,.$

Therefore,

$\displaystyle\|u\|_{L^{2}}^{2}=\alpha^{2}+\alpha\beta+\alpha\beta+\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}\;,$ (3.27)

since for $N\in\mathbb{N}\,,$

$\left\langle
1\;\Big{|}\;\frac{1}{z^{N}-\bar{p}\operatorname{e}^{iNct}}\right\rangle=\int_{z\in\mathscr{C}(0,1)}\frac{z^{N}}{1-p\operatorname{e}^{-iNct}z^{N}}\frac{dz}{2\pi
iz}=0\,.$

Consequently, by (3.18) , 555As we shall see in Corollary 5.5 of Section 5,
this corresponds to $\|u\|_{L^{2}}^{2}=\lambda_{N}-N$ where
$\lambda_{N}>N+\lambda_{0}>N\,.$

$\|u\|_{L^{2}}^{2}=\alpha^{2}+\alpha\beta-N\,.$ (3.28)

In addition, since by (3.18)

$\alpha=-\frac{N}{\beta}-\frac{\beta}{1-\lvert p\rvert^{2}}\,,$

then,

$\displaystyle\|u\|_{L^{2}}^{2}=$
$\displaystyle\,\left(-\frac{N}{\beta}-\frac{\beta}{1-\lvert
p\rvert^{2}}\right)^{2}+\left(-\frac{N}{\beta}-\frac{\beta}{1-\lvert
p\rvert^{2}}\right)\beta-N$ $\displaystyle=$ $\displaystyle\,\frac{\lvert
p\rvert^{2}}{1-\lvert p\rvert^{2}}\left(\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}+2N\right)+\frac{N^{2}}{\beta^{2}}\,.$

Observe that, $\|u\|_{L^{2}}^{2}$ is a continuous function of $\lvert
p\rvert^{2}$ and $\beta^{2}$ . Moreover, by taking $\beta\to 0$ then
$\|u\|_{L^{2}}^{2}\to\infty\,.$ And if we take $\lvert p\rvert^{2}\to 0$ then

$\|u\|_{L^{2}}^{2}\underset{\lvert p\rvert^{2}\to
0}{\sim}\;\frac{N^{2}}{\beta^{2}}$

which can be arbitrary small when $\beta>\\!>1$.

(ii) Speed : $\mathbf{c>\textit{N}}\,$. By Theorem 1.1 , the speed of the
traveling waves of the form (3.26) is given by
$c=-N(1+\frac{2\alpha}{\beta})\,.$ Besides, recall from (3.18) ,

$\frac{\alpha}{\beta}=-\frac{N}{\beta^{2}}-\frac{1}{1-\lvert p\rvert^{2}}\,.$

Substituting the latter identity in the expression of $c$ , it follows

$c\,=\,N\left(\frac{1+\lvert p\rvert^{2}}{1-\lvert
p\rvert^{2}}+\frac{2N}{\beta^{2}}\right)\,>\,N\,.$ (3.29)

It remains to prove that

  * •

when $\|u\|_{L^{2}}\to+\infty$ , we have $c\to+\infty\,,$

  * •

and when $\|u\|_{L^{2}}\to 0$ then $c\to N\,.$

Indeed, observe that $\|u\|_{L^{2}}^{2}\to\infty$ when $\beta^{2}\to 0$ or
$\lvert p\rvert^{2}\to 1\,,$ and in both cases

$c\to\infty\,.$

On the other side, $\|u\|_{L^{2}}^{2}$ is arbitrary small when $\lvert
p\rvert^{2}\to 0$ and $\beta$ is big enough. Hence, by passing to the limit
$\lvert p\rvert^{2}\to 0$ in (3.29) , we infer

$c\underset{\lvert p\rvert^{2}\to
0}{\sim}N\left(1+\frac{2N}{\beta^{2}}\right)\,,$

which can arbitrary close to $N$ as $\|u\|^{2}_{L^{2}}$ is arbitrary close to
$0$ .

∎

##  4\. Traveling waves for the focusing (CS+)

###  4.1. Toward the characterization of the traveling waves for (CS+)

Recall that to characterize the traveling waves of the defocusing equation
(CS-), a spectral analysis was initially conducted, followed by the derivation
of explicit formulas. Here, we aim to footstep the same strategy. But before
proceeding, we shall require some analogous lemmas to the defocusing case.

######  Lemma 4.1 (The analog of Lemma 3.2).

Let $u\in\mathcal{C}_{t}H^{2}_{+}(\mathbb{T})_{x}$ solution of (CS+). Then,
for all $n\,,p\in\mathbb{N}_{\geq 0}\,,$

$\displaystyle\left\langle 1\,|\,g_{n}^{\,t}\right\rangle=$
$\displaystyle\,\left\langle
1\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{-i\nu_{n}^{2}t}\,,$
$\displaystyle\left\langle u\,|\,g_{n}^{\,t}\right\rangle=$
$\displaystyle\,\left\langle
u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle\operatorname{e}^{-i\nu_{n}^{2}t}$ (4.1)
$\displaystyle\left\langle Sg_{p}^{\,t}\,|\,u\right\rangle=$
$\displaystyle\,\left\langle
Sf_{p}^{\,u_{0}}\,|\,u_{0}\right\rangle\operatorname{e}^{i(\nu_{p}+1)^{2}\,t}\,,$
$\displaystyle\left\langle Sg_{p}^{\,t}\,|\,g_{n}^{\,t}\right\rangle=$
$\displaystyle\,\left\langle
Sf_{p}^{\,u_{0}}\,|\,f_{n}^{\,u_{0}}\right\rangle\,\operatorname{e}^{i((\nu_{p}+1)^{2}-\nu_{n}^{2})\,t}\,,$

where the $(g_{n}^{\,t})$ denotes the orthonormal basis of
$L^{2}_{+}(\mathbb{T})$ solution to the Cauchy problem

$\begin{cases}\partial_{t}\,g_{n}^{\,t}&={B}_{u(t)}\,g_{n}^{\,t}\\\
{g_{n}^{\,t}\,}_{|_{t=0}}&=f_{n}^{\,u_{0}}\end{cases}\,,\qquad\forall
n\in\mathbb{N}_{\geq 0}\,,$

and $(f_{n}^{\,u_{0}})$ is an orthonormal basis of $L^{2}_{+}(\mathbb{T})$
made up of the eigenfunctions of $L_{u_{0}}$ and ${B}_{u(t)}$ is the
skew–adjoint operator defined in (1.11) .

###### Proof.

Since the focusing Calogero–Sutherland DNLS equation can also be rewritten in
terms of its Lax operators [Ba23, Lemma 2.4]

$\partial_{t}u=B_{u}u-iL^{2}_{u}u\,,$

then one can repeat exactly the same proof of Lemma 3.2 and obtain the same
results. ∎

######  Lemma 4.2 (The analog of Proposition 3.5).

Let $u:=\tau_{ct}u_{0}$ be a traveling wave of (CS+) such that the eigenvalue
$\nu_{n}(u_{0})$ is simple. Then, there exists $\theta_{n}(t)\in\mathbb{R}$
such that

$\tau_{ct}f_{n}^{\,u_{0}}=\operatorname{e}^{i\theta_{n}(t)}g_{n}^{\,t}\;,$
(4.2)

where the $(g_{n}^{\,t})$ denotes the orthonormal basis defined in the
previous lemma.

######  Lemma 4.3 (The analog of Corollary 3.6 in the focusing case).

Let $u_{0}$ be a function such that the eigenvalues $(\nu_{n}(u_{0}))$ are
simple. Then, for all $n\,,p\in\mathbb{N}_{\geq 0}\,,$ $t\in\mathbb{R}\,,$ we
have

  1. (1)

If $\langle 1\,|f_{n}^{\,u_{0}}\rangle\neq 0$ then

$\theta_{n}(t)=-\nu_{n}^{2}t\,.$

  2. (2)

If $\langle u_{0}\,|f_{n}^{\,u_{0}}\rangle\neq 0$ then

$\theta_{n}(t)=-\nu_{n}^{2}t\,.$

  3. (3)

If $\langle Sf_{p}^{\,u_{0}}|f_{n}^{\,u_{0}}\rangle\neq 0$ then

$\theta_{n}(t)=((\nu_{p}+1)^{2}-\nu_{n}^{2})t-ct+\theta_{p}(t)\,,$

where $\theta_{n}(t)$ is the angle obtained in (4.2) .

At this stage, we are equipped with the necessary tools to footstep the proof
of the defocusing equation. However, it is important to emphasize two
fundamental differences between the Lax operators $L_{u}$ and
$\tilde{L}_{u}\,,$ which ultimately offer a considerably expanded set of
traveling waves for (CS+) in comparison to (CS-) :

  * •

The gap between the eigenvalues differ between the focusing and the defocusing
case (Proposition 2.1).

  * •

The fact that the eigenvalues $\lambda_{n}$ of $\tilde{L}_{u}$ are not zero
for any $n\in\mathbb{N}$ .

Indeed, in the defocusing case, since all the eigenvalues satisfy
$\lambda_{n}>\lambda_{n-1}+\frac{1}{2}$ (Inequality (2.5)) and
$\lambda_{n}\neq 0$ for all $n\in\mathbb{N}$ , then we obtained in Lemma 2.5 ,

$\mathcal{I}(u)=\varnothing\,,\qquad\forall u\in H^{2}_{+}(\mathbb{T})\,,$

where $\mathcal{I}(u)$ was defined in (2.12) . As a consequence, we inferred
that if $u(t,x)=u_{0}(x-ct)$ is a traveling wave of (CS-) , then there exists
at most one $N\in\mathbb{N}$ such that $\left\langle
u_{0}\,|\,f_{n}^{\,u_{0}}\right\rangle=0$ for all
$n\in\mathbb{N}\backslash\\{N\\}\,.$ Now, for the focusing equation, recall we
have previously observed in the second point of Remark 2.3 , that
$\mathcal{I}(u)$ is of finite cardinal for all $u\in H^{2}_{+}(\mathbb{T})\,$.
In particular, for $u_{0}\in H^{2}_{+}(\mathbb{T})\,,$ we denote by
$m_{1},\ldots,m_{n}$ its elements

$\mathcal{I}(u_{0})=\\{m_{1}\,,\ldots\,,m_{n}\\}\,.$

######  Theorem 4.4 (Toward the Characterization of the traveling waves of
(CS+)).

The traveling waves $u_{0}(x-ct)$ of (CS+) are either rational functions or
the plane waves $u(t,x)=C\operatorname{e}^{iN(x-Nt)}$ . In addition, the
potentials

$u(t,x):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,N\in\mathbb{N}\,,$ (4.3)

where $c=-N\left(1+\frac{2\alpha}{\beta}\right)$ ,
$\,(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ such that

$\alpha\beta+\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=N\,,$ (4.4)

and the potentials

$u(t,z)=\operatorname{e}^{i\theta}\operatorname{e}^{im(x-ct)}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{i(x-ct)}}\right)\qquad
p\in\mathbb{D}^{*}\,,\,\theta\in\mathbb{T}\,,\,m\in\mathbb{N}\,,$ (4.5)

where $c=m$ , $(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ such that

$\alpha\beta+\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}=1\,,\qquad\beta(m-1)=2\alpha\,,$

are parts of the set of the traveling waves of (CS+) .

###### Proof.

Step 1. (Spectral characterization of the traveling waves of (CS+)). Let
$u(t,x):=u_{0}(x-ct)$ be a traveling wave for (CS+) . Our goal is to prove
that there exists $N\in\mathbb{N}$ such that $\left\langle
u_{0}\,|\,f_{n}^{u_{0}}\right\rangle=0$ for all $n\geq N\,.$ Once more, in
order to simplify the notation, we denote in the following $f_{n}$ instead of
$f_{n}^{u_{0}}$ . Recall by the second point of Remark 2.3 ,
$\mathcal{I}(u_{0})$ is of finite cardinal, that is there exists
$m_{1}<\ldots<m_{j}\in\mathbb{N}$ such that

$\begin{cases}\left\langle Sf_{n-1}\,|\,f_{n}\right\rangle=0\,,\qquad\forall
n\in\\{m_{1}\,,\ldots m_{j}\\}\\\ \left\langle
Sf_{n-1}\,|\,f_{n}\right\rangle\neq 0\,,\qquad\forall
n\in\mathbb{N}\backslash\\{m_{1}\,,\ldots m_{j}\\}\end{cases}\,.$

Suppose that there exists an integer $\ell>\\!>1\,,$ $\ell>m_{j}$ such that
$\left\langle u_{0}\,|\,f_{\ell}\right\rangle\neq 0\,.$ Otherwise, we already
have what we claim to prove. Then,

  * •

For all $n\geq\ell+1\,,$ the quantities $\left\langle
Sf_{n-1}\,|\,f_{n}\right\rangle\neq 0$ .

  * •

Since $\ell>\\!>1\,,$ then by inequality (2.7) , the eigenvalues
$(\nu_{n})_{n\geq\ell+1}$ are simple. This implies that Lemma 4.2 holds for
$n\geq\ell+1\,.$

Therefore, using the third point of Lemma 4.3 , we obtain, for all
$n\geq\ell+1\,,$

$\displaystyle\theta_{n}(t)=$
$\displaystyle\,-\nu_{n}^{2}t+(\nu_{n-1}+1)^{2}t-ct+\theta_{n-1}(t)$
$\displaystyle=$
$\displaystyle\,-\nu_{n}^{2}\,t+(\nu_{n-1}+1)^{2}t-\nu_{n-1}^{2}\,t+(\nu_{n-2}+1)^{2}t-2ct+\theta_{n-2}(t)$
$\displaystyle=$ $\displaystyle\quad\ldots$ $\displaystyle=$
$\displaystyle\,-\nu_{n}^{2}\,t+(n-\ell)t+\nu_{\ell}^{2}\,t-(n-\ell)ct+2t\sum_{k=\ell}^{n-1}\nu_{k}+\theta_{\ell}(t)\,,$

where $\theta_{n}(t)$ is the angle obtained in Lemma 4.2 , and
$\theta_{\ell}(t)=-\nu_{\ell}^{2}\,t$ thanks to the second point of Lemma 4.3
. Hence, for all $n\geq\ell+1\,,$

$\theta_{n}(t)=-\nu_{n}^{2}\,t+(n-\ell)t-(n-\ell)ct+2t\sum_{k=\ell}^{n-1}\nu_{k}\,,$
(4.6)

As a consequence, there exists at most one integer $N\geq\ell$ such that

$\left\langle u_{0}\,|\,f_{N}\right\rangle\neq 0\,.$

Indeed, suppose for the sake of contradiction that there exist
$n_{2}>n_{1}>\ell\,,$ such that $\left\langle
u_{0}\,|\,f_{n_{1}}\right\rangle\neq 0$ and $\left\langle
u_{0}\,|\,f_{n_{2}}\right\rangle\neq 0\,.$ Then, combining the second point of
Lemma 4.3 , and equation (4.6) , we obtain

$\displaystyle\left|\begin{array}[]{l}\displaystyle
c=1+\frac{2}{n_{1}-\ell}\,\sum_{k=\ell}^{n_{1}-1}\nu_{k}\,.\\\\[14.22636pt]
\displaystyle
c=1+\frac{2}{n_{2}-\ell}\,\sum_{k=\ell}^{n_{2}-1}\nu_{k}\,.\end{array}\right.$

Hence,

$(n_{2}-\ell)\,\sum_{k=\ell}^{n_{1}-1}\nu_{k}=(n_{1}-\ell)\,\sum_{k=\ell}^{n_{2}-1}\nu_{k}\,,$

or

$(n_{2}-n_{1})\,\sum_{k=\ell}^{n_{1}-1}\nu_{k}\,=\,(n_{1}-\ell)\,\sum_{k=n_{1}}^{n_{2}-1}\nu_{k}\,.$

As a result,

$(n_{1}-\ell)(n_{2}-n_{1})\,\nu_{n_{1}-1}\,\geq\,(n_{1}-\ell)(n_{2}-n_{1})\,\nu_{n_{1}}\,,$

leading to a contradiction, since for $k\geq\ell\,,$ we have
$\nu_{k+1}>\nu_{k}\,.$ Therefore, there exists $N\in\mathbb{N}$ such that
$\left\langle u_{0}\,|\,f_{n}\right\rangle=0$ for all $n\geq N\,.$ Step 2.
(They are rational functions or potentials in $\mathcal{G}_{1}$) Since
$\left\langle u_{0}\,|\,f_{n}\right\rangle=0$ for all $n\geq N$ , it follows
by Corollary 2.6 that $\nu_{n}=\nu_{n-1}+1$ for all $n\geq N_{2}\,.$ Note that
the potentials satisfying $\nu_{n}=\nu_{n-1}+1$ for all $n\geq N_{2}\,,$ are
referred to be “finite gap potentials” for (CS+), and are studied deeply in
Section 5. In particular, Theorem 1.3 provides a full characterization of
these potentials in the state space. They are either
$u(x)=C\operatorname{e}^{iNx}$ , $C\in\mathbb{R}^{*}\,,$ $N\in\mathbb{N}_{\geq
0}\,,$ or rational functions

$u(x)=\operatorname{e}^{im_{0}\,x}\prod_{j=1}^{r}\left(\frac{\mathrm{e}^{ix}-\overline{p_{j}}}{1-p_{j}\mathrm{e}^{ix}}\right)^{m_{j}-1}\left(\alpha+\sum_{j=1}^{r}\frac{\beta_{j}}{1-p_{j}\mathrm{e}^{ix}}\right),\quad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,$ (4.7)

where, for $N\in\mathbb{N}\,,$ $\,m_{0}\in\llbracket 0,N-1\rrbracket\,,\,$
$m_{1},\ldots,m_{r}\in\llbracket 1,N\rrbracket$ , such that
$m_{0}+\sum_{j=1}^{r}m_{j}=N\,,$ and
$(\alpha,\,\beta_{1}\,,\,\ldots\,,\,\beta_{r})\in\mathbb{C}\times\mathbb{C}^{r}$
satisfy for all $j=1\,,\,\ldots\,,r\,,$

$\overline{\alpha}\,\beta_{j}\,+\,\sum_{k=1}^{r}\,\frac{\alpha_{j}\,\overline{\alpha_{k}}}{1-p_{j}\overline{p_{k}}}=m_{j}\,.$

It remains to verify that (4.3) and (4.5) are traveling waves for (CS+) . To
do so, one can simply substitute them into the equation (CS+)–equation and
check that they satisfy the equation.

∎

######  Remark 4.1.

As was observed in the previous proof all the traveling waves $u_{0}(x-ct)$ of
(CS+) are either $u(t,x)=C\operatorname{e}^{iN(x-Nt)}$ or the rational
functions $u(t,x):=u_{0}(x-ct)$ where $u_{0}$ is defined in (4.7) and the
constants $\alpha,\beta_{j}$ and $c$ can be described by substituting $u$ in
the (CS+)–equation.

###  4.2. The $L^{2}$–norm and the speed

In this subsection, we analyze the $L^{2}$–norm and the speed of the traveling
waves of (CS+) and establish the existence of stationary solutions for the
focusing Calogero-Sutherland DNLS equation (CS+) .

######  Proposition 4.5.

  1. (i)

For any $r>0\,,$ there exists a non–trivial traveling wave
$u(t,x):=u_{0}(x-ct)$ for (CS+) with

$\|u_{0}\|_{L^{2}}=r\,.$

  2. (ii)

Let $u$ be a traveling wave for (CS+) of the form (4.3) , then $u$ can
propagate to the right or to the left with any speed $c\in\mathbb{R}\,.$

######  Remark 4.2.

Contrary to the defocusing case, we do not necessarily have that the traveling
wave propagate with a speed $c\to\infty$ when $\|u\|_{L^{2}}^{2}\to\infty$ .
For instance, take

$u(t,x):=\frac{N}{\beta}-\frac{\beta}{1-\lvert
p\rvert^{2}}+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\,,\qquad\beta^{2}:=\frac{2N}{\frac{1+\lvert
p\rvert^{2}}{1-\lvert p\rvert^{2}}-\frac{c}{N}}\,.$

The proof of this statement will be achieved in the end of the following
proof.

###### Proof of proposition 4.5.

The $L^{2}$–norm . Let $u$ be a traveling wave for (CS+) of the form (4.3) ,

$u(t,x):=\operatorname{e}^{i\theta}\left(\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,.$

Our goal is to prove that the $L^{2}$ norm of these traveling waves can be
arbitrary small or large. The computation of its $L^{2}$-norm has been
performed in the proof of Proposition 3.7. Therefore, by identity (3.27),

$\|u\|_{L^{2}}^{2}=\alpha^{2}+\alpha\beta+\alpha\beta+\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}\;,$ (4.8)

where the two reals $(\alpha,\beta)$ satisfies condition (4.4)

$\alpha\beta+\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=N\,.$

That is,

$\|u\|_{L^{2}}^{2}=\frac{\lvert p\rvert^{2}}{1-\lvert
p\rvert^{2}}\left(\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}-2N\right)+\frac{N^{2}}{\beta^{2}}\,.$ (4.9)

Like for the defocusing case, $\|u\|_{L^{2}}^{2}$ is a continuous function of
$\beta^{2}$ and $\lvert p\rvert^{2}\,.$ And by taking $\beta\to 0$ one has
$\|u\|_{L^{2}}^{2}\to\infty\,.$ In addition, if $\lvert p\rvert\to 0$ then

$\|u\|_{L^{2}}^{2}\underset{\lvert p\rvert^{2}\to
0}{\sim}\;\frac{N^{2}}{\beta^{2}}\,.$

Hence, it is sufficient to take $\beta$ big enough so that $\|u\|_{L^{2}}$ can
be arbitrary small.

Speed : $\mathbf{c\in\mathbb{R}}\,$. By Theorem 4.4 , there exists traveling
waves for (CS+) that propagates with a speed
$c=-N\left(1+\frac{2\alpha}{\beta}\right)$ where $N\in\mathbb{N}$ and the two
reals $(\alpha,\beta)$ satisfy

$\alpha\beta+\frac{\beta^{2}}{1-\lvert p\rvert^{2}}=N\,,\qquad 0<\lvert
p\rvert<1\,.$

That is

$\displaystyle c=$
$\displaystyle\,-N\left(1+\frac{2N}{\beta^{2}}-\frac{2}{1-\lvert
p\rvert^{2}}\right)$ $\displaystyle=$ $\displaystyle\,-N\left(-\frac{1+\lvert
p\rvert^{2}}{1-\lvert p\rvert^{2}}+\frac{2N}{\beta^{2}}\right)\,.$ (4.10)

By taking, for example $\beta=\lvert p\rvert$ , we infer

$c=N\,\frac{\lvert p\rvert^{4}+(2N+1)\lvert p\rvert^{2}-2N}{\lvert
p\rvert^{2}(1-\lvert p\rvert^{2})}$

Assume that $N=1\,,$ and by taking $x=\lvert p\rvert^{2}\in(0,1)$ , we infer
that the continuous function

$c(x):=\frac{x^{2}+3x-2}{x(1-x)}\,,$

satisfies $\displaystyle\inf_{x\in(0,1)}c(x)=-\infty$ and
$\displaystyle\sup_{x\in(0,1)}c(x)=+\infty\,.$

Proof of Remark 4.2 . For a traveling wave $u$ of the form (4.3),

$u(t,x):=\alpha+\frac{\beta}{1-p\operatorname{e}^{iN(x-ct)}}\,,\qquad\alpha\beta+\frac{\beta^{2}}{1-\lvert
p\rvert^{2}}=N\,,$

where $N\in\mathbb{N}$ , one has by (4.2) , that $u$ propagates with a speed

$c=-N\left(-\frac{1+\lvert p\rvert^{2}}{1-\lvert
p\rvert^{2}}+\frac{2N}{\beta^{2}}\right)\,.$

Thus, for any $N\in\mathbb{N}$ , let

$\beta:=\sqrt{\frac{2N}{\frac{1+\lvert p\rvert^{2}}{1-\lvert
p\rvert^{2}}-\frac{\lambda}{N}}}\;,\qquad p\in\mathbb{D}\,,$

where $\lambda$ is a parameter in $\mathbb{R}\,,$ and with $\lvert
p\rvert^{2}$ big enough so that $\beta$ is well defined. Hence, one computes

$\displaystyle c=$ $\displaystyle\,-N\left(-\frac{1+\lvert
p\rvert^{2}}{1-\lvert p\rvert^{2}}+\frac{2N}{\frac{2N}{\frac{1+\lvert
p\rvert^{2}}{1-\lvert
p\rvert^{2}}-\frac{\lambda}{N}}}\right)=\lambda\in\mathbb{R}\,.$

That is, $u$ can propagate with any speed in $\mathbb{R}\,,$ regardless of the
valued attained by the $L^{2}$–norm of $u$

∎

######  Corollary 4.6.

The potentials

$u(t,x):=\operatorname{e}^{i\theta}\sqrt{\frac{N(1-\lvert
p\rvert^{2})}{2(1+\lvert
p\rvert^{2})}}\left(1-\frac{2}{1-p\operatorname{e}^{iNx}}\right)\,,\qquad
p\in\mathbb{D}^{*}\,,\,N\in\mathbb{N}\,,\,\theta\in\mathbb{T}\,,$

are stationary solutions for (CS+) . Conversely, the defocusing (CS-) equation
does not exhibit stationary wave solutions except the complex constant
functions.

###### Proof.

Through a straightforward calculation, one can easily check that the obtained
waves satisfy the (CS+)–equation. On the other side, for the defocusing
equation, we already established via Remark 3.3 or the second point of
Proposition 3.7 , the non–existence of stationary waves $u(t,x)=u_{0}(x)$ for
(CS-) .

∎

##  5\. The finite gap potentials

This section aims to examine the finite gap potentials associated with the
Calogero–Sutherland DNLS equation (CS) in both the focusing and defocusing
cases. Remarkably, these potentials manifest as rational functions containing
the traveling and solitary waves of (CS).

In the following, we adopt a slight abuse of notation, where for all
$n\in\mathbb{N}$, we denote

$\gamma_{n}(u):=\nu_{n}-\nu_{n-1}-1\,,$ (5.1)

the gap between the consecutive eigenvalues in the focusing context, and

$\gamma_{n}(u):=\lambda_{n}-\lambda_{n-1}-1\,,$

as the gap in the defocusing context. At this point, several observations can
be made. First, recall that in the defocusing case, the $(\lambda_{n})$
satisfies inequality (2.5), and thus, for all $n\in\mathbb{N}$ ,
$\gamma_{n}(u)$ is non–negative in the defocusing case . Second, notice that
since the eigenvalues $(\nu_{n})$ and $(\lambda_{n})$ of the Lax operators
$L_{u}$ and $\tilde{L}_{u}$ are invariant by the evolution, then for all
$n\in\mathbb{N}\,,$

$\gamma_{n}(u(t))=\gamma_{n}(u_{0})\,,\qquad\forall t\,.$

######  Definition 5.1 (Finite gap potential).

A function $u\in L^{2}_{+}(\mathbb{T})$ is said to be a finite gap potential
of (CS) if there exists $m\in\mathbb{N}$ such that

$\gamma_{n}(u)=0\,,\qquad\forall n\geq m\,,$ (5.2)

where $\gamma_{n}$ is defined in (5.1) .

Recall that any function in the Hardy space $L^{2}_{+}(\mathbb{T})$ can be
seen as a holomorphic function on the unit disc $\mathbb{D}$ whose trace on
the boundary $\partial\mathbb{D}$ is in $L^{2}\,.$ Hence, in what follows, we
denote by $\mathscr{B}_{N}$ the set of finite Blaschke products of degree $N$
:

$\uppsi(x)=\operatorname{e}^{i\theta}\,\prod_{k=1}^{N}\frac{\operatorname{e}^{ix}-\overline{p_{k}}}{1-p_{k}\operatorname{e}^{ix}}\;,\qquad\theta\in\mathbb{R}\,,\;p_{k}\in\mathbb{D}\,,$

which can be identified as the set of functions

$\uppsi(z)=\operatorname{e}^{i\theta}\,\frac{z^{N}\bar{Q}(\frac{1}{z})}{Q(z)}\,,\qquad
z\in\overline{\mathbb{D}}:=\left\\{\lvert z\rvert\leq
1\right\\}\,,\;\theta\in\mathbb{R}\,.$

where

$Q(z):=\prod_{j=1}^{N}(1-p_{j}z)\,,\qquad p_{j}\in\mathbb{D}\,.$

In other words, $z^{N}\bar{Q}(\frac{1}{z})$ is a Schur polynomial 666 A
polynomial $q(z)=\sum_{k=1}^{N}a_{k}z^{k}$ is called a Schur polynomial if all
its roots are in the open unit disc $\mathbb{D}\,.$ of degree $N$ .

######  Remark 5.1.

By convention, we suppose that a finite Blaschke product of degree $0$ is a
constant in $\mathbb{C}$.

######  Proposition 5.2.

Let $u$ be a finite gap potential of (CS+) . There exist
$(\nu,\uppsi)\in\mathbb{R}\times\mathscr{B}_{n}\,,$ $n\in\mathbb{N}_{\geq 0}$
such that

$L_{u}\,S^{k}\uppsi=(\nu+k)\,S^{k}\uppsi\,,\qquad\forall k\in\mathbb{N}_{\geq
0}\,.$ (5.3)

In addition, the same goes for the defocusing Calogero–Sutherland DNLS
equation (CS-).

###### Proof.

Let $u$ be a finite gap potential, that is $\nu_{n}=\nu_{n-1}+1$ for all
$n\geq m\,.$ We denote by $n_{0}$ the eventual indices where $\nu_{n_{0}}$ may
vanish. Then, by Proposition 2.4 ,

$Sf_{n-1}\mathbin{\\!/\mkern-5.0mu/\\!}f_{n}\,,\qquad\forall
n\geq\,N\coloneqq\max\\{m,n_{0}\\}+2\,,$

as the eigenvalues $(\nu_{n})$ are simple for $n\geq m+1\,.$ Therefore,
letting $\uppsi:=f_{N-1}\,,$ we have

$L_{u}\,S^{k}\uppsi=(\nu_{N-1}+k)S^{k}\uppsi\qquad\forall\,k\in\mathbb{N}_{\geq
0}\,.$ (5.4)

It remains to prove that $\uppsi$ is a finite Blaschke product. Observe that,
by taking the inner product of both sides of the previous identity with
$\uppsi\,,$

$\left\langle S^{k}\uppsi\,|\,\uppsi\right\rangle=0\,,\qquad\forall
k\in\mathbb{N}\,.$ (5.5)

That is

$\left\langle\,\lvert\uppsi\rvert^{2}\,|\,\operatorname{e}^{ikx}\right\rangle=0,\quad\forall
k\in\mathbb{N}\,.$

or

$\left\langle\,\lvert\uppsi\rvert^{2}\,|\,\operatorname{e}^{ikx}\right\rangle=0,\quad\forall
k\in\mathbb{Z}\backslash\\{0\\}\,,$

as $\lvert\uppsi\rvert^{2}$ is real value. Consequently, $|\uppsi|^{2}$ is a
real constant, which can be supposed equal to $1$ since we have assumed that
the eigenfunctions of $L_{u}$ constitute an orthonormal basis of
$L^{2}_{+}(\mathbb{T})$ . Thus, $\lvert\uppsi\rvert=1$ on $\mathbb{T}$. In
order to conclude, we need the following lemma.

######  Lemma 5.3.

Let $\upphi$ be an analytic function on the open unit ball that extends
continuously to an inner function777A bounded analytic function $\uppsi$ on
$\mathbb{D}$ is said to be inner if
$\lvert\uppsi(\operatorname{e}^{ix})\rvert=1$ for almost every $x\,.$ Note
that a Blaschke product is a rational inner function. on the closed unit disc.
Then $\upphi\in\mathscr{B}_{n}\,$.

###### Proof.

Given a holomorphic function $\upphi$ on the open unit ball that extends
continuously to the unit circle while satisfying $\lvert\uppsi\rvert=1>0$ on
$\mathbb{T}$ , we know that its zeros are finite, isolated and all localized
inside the open unit disk $\mathbb{D}$. We denote them by
$\overline{p_{1}},\ldots,\overline{p_{n}}$ . Hence, $\upphi$ can be factorized
as

$\upphi(z)=\upsilon(z)\cdot\prod_{k=1}^{n}\frac{z-\overline{p_{k}}}{1-p_{k}z}\,,$

where $\upsilon$ is a holomorphic function without zeros on $\mathbb{D}$ .
Therefore, $1/\upsilon$ is a holomorphic function on $\mathbb{D}$ , which
continuously extends to the unit circle while satisfying $\lvert
1/\upsilon\rvert=1$ on $\mathbb{T}$ . Thus, by the maximum principle, we infer
that $\left|1/\upsilon\right|\leq 1$ on $\mathbb{D}$ . Using the same argument
on $\upsilon$ instead of $1/\upsilon$ , we deduce that
$\lvert\upsilon\rvert\leq 1$ on the unit disc. As a consequence,
$\lvert\upsilon\rvert=1$ on the close unit disc $\left\\{|z|\leq 1\right\\}$
and so

$\upphi(z)=\operatorname{e}^{i\theta}\cdot\prod_{k=1}^{n}\frac{z-\overline{p_{k}}}{1-p_{k}z}\,,\qquad\theta\in\mathbb{R}\,.$

∎

Coming back to the proof of Proposition 5.2, we denote $\underline{\uppsi}$
the function obtained by the isometric isomorphism map

$\underline{\uppsi}(z)=\sum_{k\geq
0}\widehat{\uppsi}(k)z^{k}\,,\;\;z\in\mathbb{D}\quad\longmapsto\quad\uppsi(x):=\sum_{k\geq
0}\widehat{\uppsi}(k)\operatorname{e}^{ikx}\,,\ \ x\in\mathbb{T}\,,$

In particular, since $\uppsi\in L^{2}_{+}(\mathbb{T})$ then
$\underline{\uppsi}\in\mathbb{H}_{2}(\mathbb{D})\,,$ where

$\mathbb{H}_{2}(\mathbb{D}):=\left\\{u\in\mathrm{Hol}(\mathbb{D})\,;\,\sup_{0\leq
r<1}\int_{0}^{2\pi}\,\lvert
u(r\operatorname{e}^{i\theta})\rvert^{2}\;\dfrac{d\theta}{2\pi}<\infty\right\\}\,,$

Hence, by [Ch, Theorem 4.5.3],

$\underline{\uppsi}(r\operatorname{e}^{ix})=\frac{1}{2\pi}\int_{0}^{2\pi}P_{r}(x-\theta)\uppsi(\operatorname{e}^{i\theta})\,dt\,,\quad
0\leq r<1\,,$

where $P_{r}$ denotes the Poisson Kernel

$P_{r}(x-\theta)=\frac{1-r^{2}}{1-2r\cos(x-\theta)+r^{2}}\;.$

Note that the function $\uppsi\in\mathrm{Dom}(L_{u}):=H^{1}_{+}(\mathbb{T})$
is continuous on $\mathbb{T}$ as

$D\uppsi=L_{u}\uppsi+u\Pi(\bar{u}\uppsi)\in L^{1}_{+}(\mathbb{T})\,.$ (5.6)

Therefore, the Poisson Theorem [Gi04, Theorème 30] implies that the
holomorphic function $\underline{\uppsi}(r\operatorname{e}^{ix})$ extends
continuously to $\uppsi(\operatorname{e}^{ix})$ as $r\to 1\,.$ In addition,
recall that $\lvert\uppsi\rvert=1$ on $\mathbb{T}$ . Thus, applying the
previous lemma, we infer that $\underline{\uppsi}$ is a Blaschke product and
so is $\uppsi\,$.

∎

At this stage, we aim to characterize the finite gap potentials of (CS) . To
this end, we regroup them according to the following procedure : for any
finite gap potential $u$ of (CS) , we denote by $\mathcal{N}(u)$ the
non–negative integer

$\mathcal{N}(u):=\min\left\\{\,n\in\mathbb{N}_{\geq
0}\;\mid\,\exists\,\uppsi\in\mathscr{B}_{n}\;,\;L_{u}\,S^{k}\uppsi=(\nu+k)S^{k}\uppsi\,,\,\forall
k\geq 0\right\\}\,,$ (5.7)

and we define, for $N\in\mathbb{N}_{\geq 0}$ , the set

$\boxed{\ \ \mathcal{U}_{N}:=\left\\{u\text{ finite gap potential
},\;\mathcal{N}(u)={N}\right\\}\,.\ }$

This means that for any $u\in\mathcal{U}_{N}\,,$ there exists a finite
Blaschke product $\uppsi_{u}$ of minimal degree $N\,,$ satisfying

$L_{u}\,S^{k}\uppsi_{u}=(\nu_{u}+k)S^{k}\uppsi_{u}\,,\qquad\forall
k\in\mathbb{N}_{\geq 0}\,,$ (5.8)

where $\nu_{u}$ is the corresponding eigenvalue of $\uppsi_{u}$ . That is
$\left\\{S^{k}\uppsi_{u}\,\mid\,k\in\mathbb{N}_{\geq 0}\right\\}$ are parts of
the orthonormal basis of $L^{2}_{+}(\mathbb{T})\,.$ Besides, observe that,
since $\deg\uppsi_{u}=N\,,$ then there exists $N$ eigenfunctions $f_{0},\ldots
f_{N-1}$ of $L_{u}$ that generate the model space
$(\uppsi_{u}\,L^{2}_{+})^{\perp}\,$ which is of dimension $N$ [GMR16,
Corollary 5.18] . We denote $\nu_{0}\,,\,\ldots\,,\,\nu_{N-1}$ the associated
eigenvalues. Note that the latter $N$ eigenvalues are not necessarily smaller
than $\nu_{u}\,.$ We summarize this discussion by the following diagram. For
any $u\in\mathcal{U}_{N}\,,$

\begin{picture}(20.0,1.3)\par\put(-0.25,0.5){\vector(1,0){14.2}}
\put(2.1,0.21){{\scalebox{0.8}{$\nu_{u}$}}} \put(2.1,0.8){{$\uppsi_{u}$}}
\put(2.2,0.5){{\circle*{0.1}}}
\par\par\put(2.81,0.2){{\scalebox{0.8}{$\nu_{u}+1$}}}
\put(2.9,0.8){{$S\uppsi_{u}$}} \put(3.2,0.5){{\circle*{0.1}}}
\par\put(4.2,0.5){{\circle*{0.1}}}
\put(3.8,0.2){{\scalebox{0.8}{$\nu_{u}+2$}}}
\put(3.8,0.8){{$S^{2}\uppsi_{u}$}} \par\par\put(5.4,0.2){{$\ldots$}}
\par\put(6.11,0.2){{\scalebox{0.8}{$\nu_{u}+n$}}}
\put(6.13,0.8){{$S^{n}\uppsi_{u}$}} \put(6.5,0.5){{\circle*{0.1}}}
\par\put(7.11,0.2){{\scalebox{0.8}{$\nu_{u}+n+1$}}}
\put(7.13,0.8){{$S^{n+1}\uppsi_{u}$}} \put(7.5,0.5){{\circle*{0.1}}}
\par\par\put(7.8,0.4){{$\ldots$}}
\par\put(0.2,0.2){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\nu_{0}$}}
\put(0.2,0.65){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$f_{0}$}}
\put(0.2,0.5){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\circle*{0.1}}}
\par\put(1.8,0.2){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\nu_{1}$}}
\put(1.8,0.65){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$f_{1}$}}
\put(1.8,0.5){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\circle*{0.1}}}
\par\put(4.6,0.2){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\nu_{N-1}$}}
\put(4.6,0.65){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$f_{N-1}$}}
\put(4.7,0.5){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\circle*{0.1}}}
\end{picture}

$\nu_{u}$

$\uppsi_{u}$

$\nu_{u}+1$

$S\uppsi_{u}$

$\nu_{u}+2$

$S^{2}\uppsi_{u}$

$\ldots$

$\nu_{u}+n$

$S^{n}\uppsi_{u}$

$\nu_{u}+n+1$

$S^{n+1}\uppsi_{u}$

$\ldots$

$\nu_{0}$$f_{0}$

$\nu_{1}$$f_{1}$

$\nu_{N-1}$$f_{N-1}$

Of course, the same goes for the defocusing equation with $\tilde{L}_{u}$
instead of $L_{u}\,,$ up to the fact that the remaining $N$ eigenvalues
$\nu_{0}\,,\ldots\,\nu_{N-1}$ are necessarily smaller than $\nu_{u}\,,$ since
the eigenvalues of $\tilde{L}_{u}$ satisfy the property (2.5) . Besides, note
that by taking the minimum in (5.7) we guarantee that:

  1. (1)

If $u\in\left\\{v\text{ finite gap potential
},\;L_{v}\,S^{k}\uppsi=(\nu+k)\,S^{k}\uppsi\,,\;\uppsi\in\mathscr{B}_{N}\right\\}\,,$
then $u\notin\left\\{v\text{ finite gap potential
},\;L_{v}\,S^{k}\uppsi=(\nu+k)\,S^{k}\uppsi\,,\;\uppsi\in\mathscr{B}_{N-1}\right\\}\,.$

  2. (2)

The set $\mathcal{U}_{N}$ is invariant under the evolution of (CS) . (See
Proposition 5.6).

The following theorem aims to characterize the finite gap potentials of the
Calogero–Sutherland DNLS (CS) in the state space.

######  Theorem 5.4.

Let $N\in\mathbb{N}\,.$ A potential $u$ is in $\mathcal{U}_{N}$ if and only if
$u(x)=C\operatorname{e}^{iNx}$ , $C\in\mathbb{C}^{*}\,,$ or $u$ is a rational
function

$u(x)=\operatorname{e}^{im_{0}\,x}\prod_{j=1}^{r}\left(\frac{\mathrm{e}^{ix}-\overline{p_{j}}}{1-p_{j}\mathrm{e}^{ix}}\right)^{m_{j}-1}\left(a+\sum_{j=1}^{r}\frac{c_{j}}{1-p_{j}\mathrm{e}^{ix}}\right)\,,\quad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,$ (5.9)

where $\,m_{0}\in\llbracket 0,N-1\rrbracket\,,\,$
$m_{1},\ldots,m_{r}\in\llbracket 1,N\rrbracket$ , such that
$m_{0}+\sum_{j=1}^{r}m_{j}=N\,,$ and
$(a,\,c_{1}\,,\,\ldots\,,\,c_{r})\in\mathbb{C}\times\mathbb{C}^{r}$ satisfy
for all $j=1\,,\,\ldots\,,N-m\,,$

  1. (i)

In the focusing case,

$\overline{a}\,c_{j}\,+\,\sum_{k=1}^{r}\,\frac{c_{j}\,\overline{c_{k}}}{1-p_{j}\overline{p_{k}}}=m_{j}\,,$
(5.10)

  2. (ii)

In the defocusing case,

$\overline{a}\,c_{j}\,+\,\sum_{k=1}^{r}\,\frac{c_{j}\,\overline{c_{k}}}{1-p_{j}\overline{p_{k}}}=-m_{j}\,,$
(5.11)

with $a\neq 0$ if $m_{0}\in\llbracket 1,N-1\rrbracket\,.$ Besides, if $N=0\,,$
then $u$ is a complex constant function.

######  Remark 5.2.

As we shall see in Step 4 of the proof of Theorem 5.4, if
$u\in\mathcal{U}_{N}\,,$ then the eigenvalue of $L_{u}$ associated

  * •

with the Blaschke product
$\uppsi_{u}=\operatorname{e}^{i\theta}\operatorname{e}^{iNx}$ if
$u=C\operatorname{e}^{iNx}\,,$ is given by

    1. (i)

$\nu_{u}=N-C^{2}$ in the focusing case.

    2. (ii)

$\lambda_{u}=N+C^{2}$ in the defocusing case.

  * •

with the Blaschke product

$\uppsi_{u}=\operatorname{e}^{i\theta}\,\operatorname{e}^{im_{0}\cdot
x}\prod_{j=1}^{r}\left(\frac{\operatorname{e}^{ix}-\overline{p_{j}}}{1-p_{j}\operatorname{e}^{ix}}\right)^{m_{j}}\,,\quad\theta\in\mathbb{R}\,,p_{j}\neq
p_{k}\,,j\neq k\,,$

if $u$ is the rational function (5.9) , is given by

    1. (i)

$\displaystyle\nu_{u}=m_{0}-\lvert
a\rvert^{2}-\sum_{j=1}^{r}\,a\,\overline{c_{j}}$ in the focusing case.

    2. (ii)

$\displaystyle\lambda_{u}=m_{0}+\lvert
a\rvert^{2}+\sum_{j=1}^{r}\,a\,\overline{c_{j}}\,$ in the defocusing case.

In order to establish this theorem, we recall a specific case of formula
(3.15) .

######  Remark 5.3.

Let $(f_{n})$ be an orthonormal basis of $L^{2}_{+}(\mathbb{T})\,.$ For any
$n\geq 0$ ,

$f_{n}(z)=\left\langle(\operatorname{Id}-zM)^{-1}\mathds{1}_{n}\mid
Y\right\rangle_{\ell^{2}}\,,\qquad z\in\mathbb{D}\,,$

where $\mathds{1}_{n}$ and $Y$ are the column vectors

$\mathds{1}_{n}:=\left(\delta_{pn}\right)_{p\geq 0}\ ,\qquad
Y:=\left(\left\langle 1\mid f_{m}\right\rangle\right)_{m\geq 0}$

and $M$ is the matrix representation of the operator $S^{*}$ in the
$\left(f_{m}\right)$–basis

$M=\left(M_{mp}\right)_{mp\geq 0},\qquad M_{mp}=\left\langle f_{p}\mid
Sf_{m}\right\rangle\ .$

In what follows, we denote by $\mathbb{C}_{\leq N}[X]$ the set of polynomials
$P$ in complex coefficients with degree at most $N$ and by $\mathbb{C}_{N}[X]$
those of degree $N\,.$

###### Proof of Theorem 5.4.

We present the proof for the focusing case. Note that the same arguments can
be performed to deduce the result in the defocusing case. The key ingredient
is the inversion spectral formula (3.15)

$u(z)=\left\langle(\operatorname{Id}-zM)^{-1}\,X\,|\,Y\right\rangle\,,$ (5.12)

where $X\,,\,Y$ and $M$ are defined in (3.16) . The proof will be split in 5
steps.

Let $u\in\mathcal{U}_{N}\,,$ then by Proposition 5.2 there exists a finite
Blaschke product

$\uppsi_{u}(z)=\operatorname{e}^{i\theta}\;\frac{\operatorname{e}^{iNx}\bar{Q}(1/z)}{Q(z)}\;;\quad
Q(z):=\prod_{j=1}^{N}(1-p_{j}z)\,,\qquad p_{j}\in\mathbb{D}\,,$ (5.13)

such that (5.3) is satisfied

$L_{u}\,S^{k}\uppsi_{u}=(\nu_{u}+k)\,S^{k}\uppsi_{u}\,,\qquad\forall
k\in\mathbb{N}_{\geq 0}\,.$

Step 1. As a first step, we prove that any $u\in\mathcal{U}_{N}$ must be a
rational function

$u(z)=\frac{P(z)}{Q(z)}\,,\qquad P\in\mathbb{C}_{\leq N}[z]\,,$

where $Q(z)$ is the same denominator of the Blaschke product $\uppsi_{u}(z)$
associated with $u\in\mathcal{U}_{N}\,.$ Indeed, first observe that combining
(5.3) with the commutator identity (2.2) leads to

$\left\langle u\,|\,S^{k}\uppsi_{u}\right\rangle=0\,,\qquad\forall k\geq 1\,.$
(5.14)

Hence, we infer thanks to Lemma 2.3 , that the infinite matrices $M\,,\,X\,,$
and $Y$ of (5.12) written in the basis
$(f_{k})_{k=0}^{N-1}\cup(S^{k}\uppsi_{u})_{k\geq 0}$ are of the form

$M=\left(\begin{array}[]{cccc|cccc}\left\langle
f_{0}\,|\,Sf_{0}\right\rangle&\ldots&\left\langle
f_{n-1}\,|\,Sf_{0}\right\rangle&\left\langle\uppsi_{u}\,|\,Sf_{0}\right\rangle&&&\\\
\vdots&&\vdots&\vdots&&&\\\ \langle f_{0}\mid
Sf_{N-1}\rangle&\ldots&\left\langle
f_{N-1}\,|\,Sf_{N-1}\right\rangle&\left\langle\uppsi_{u}\,|\,Sf_{N-1}\right\rangle&&&\\\
0&\ldots&\ldots&0&1\\\ \hline\cr 0&\ldots&\ldots&0&0&1\\\
\vdots&&&&\vdots&\ddots&\ddots\end{array}\right)\ ,\;n\in\mathbb{N}$
$X=\left(\begin{array}[]{c}\left\langle u\,|\,f_{0}\right\rangle\\\ \vdots\\\
\left\langle u\,|\,f_{N-1}\right\rangle\\\ \left\langle
u\,|\,\uppsi_{u}\right\rangle\\\ 0\\\ \vdots\end{array}\right)\,,\quad
Y=\left(\begin{array}[]{c}\left\langle 1\,|\,f_{0}\right\rangle\\\ \vdots\\\
\left\langle 1\,|\,f_{N-1}\right\rangle\\\ \left\langle
1\,|\,\uppsi_{u}\right\rangle\\\ 0\\\ \vdots\end{array}\right)\,.$

Therefore, following the same procedure presented in the proof of Theorem 1.1
, one can observe that the infinite matrices $M\,,\,X$ and $Y$ can be reduced
to finite matrices that involve only the first $N+1$ coordinates of each of
these matrices. That is,

$u(z)=\left\langle\left(\operatorname{Id}-zM_{\leq N}\right)^{-1}X_{\leq
N}\,|\,Y_{\leq N}\right\rangle_{\mathbb{C}^{N+1}\times\mathbb{C}^{N+1}}\,,$

where $M_{\leq N}=(M_{mn})_{0\leq m,n\leq N}\,,$ $\,X_{\leq
N}=(X_{n})_{\,0\leq n\leq N}$ and $\,Y_{\leq N}=(Y_{n})_{\,0\leq n\leq N}\,.$
As a consequence, $u$ is a rational function

$u(z)=\frac{P(z)}{\det\left(\operatorname{Id}-zM_{\leq N}\right)}\;,\quad
P\in\mathbb{C}_{\leq N}[z]\,.$

Note that $\det\left(\operatorname{Id}-zM_{\leq N}\right)$ coincides with the
denominator of the eigenfunction

$\uppsi_{u}=\operatorname{e}^{i\theta}\,\frac{z^{N}\bar{Q}(1/z)}{Q(z)}\,,$

since by Remark 5.3 , $\,\uppsi_{u}$ is also expressed via the inversion
spectral formula

$\uppsi_{u}(z)=\left\langle(\operatorname{Id}-zM_{\leq
N})^{-1}\mathds{1}_{N}\mid Y_{\leq
N}\right\rangle_{\mathbb{C}^{N+1}\times\mathbb{C}^{N+1}}=\frac{\left\langle\mathrm{Com}(\operatorname{Id}-zM_{\leq
N})^{T}\mathds{1}_{N}\,|\,Y_{\leq
N}\right\rangle}{\det\left(\operatorname{Id}-zM_{\leq N}\right)}\,,$

and hence $\det\left(\operatorname{Id}-zM_{\leq N}\right)=Q(z)\,.$ Thus,

$u(z)=\frac{P(z)}{Q(z)}\,,\qquad P\in\mathbb{C}_{\leq N}[z]\,,\quad
Q(z)=\prod_{k=1}^{N}(1-p_{k}z)\,,\ p_{k}\in\mathbb{D}\,.$ (5.15)

Step 2. In this step, we prove that if $u\in\mathcal{U}_{N}$ then

$\lvert u\rvert^{2}=z\partial_{z}\log\uppsi_{u}-\nu_{u}\,,\qquad\text{ on
}\partial\mathbb{D}\,.$ (5.16)

Indeed, recall that $L_{u}\uppsi_{u}=\nu_{u}\uppsi_{u}$ . Then by definition
of $L_{u}=z\partial_{z}-T_{u}T_{\bar{u}}\,,$

$z\partial_{z}\uppsi_{u}-u\Pi(\overline{u}\,\uppsi_{u})=\nu_{u}\uppsi_{u}\,.$
(5.17)

On $\partial\mathbb{D}\,,$

$\overline{u}\,\uppsi_{u}(z)=\operatorname{e}^{i\theta}\,\frac{z^{N}\,\bar{P}(1/z)}{z^{N}\,\bar{Q}(1/z)}\cdot\frac{z^{N}\bar{Q}(1/z)}{Q(z)}=\operatorname{e}^{i\theta}\,\frac{z^{N}\,\overline{P}(1/z)}{Q(z)}\,,$

extends as a holomorphic function on $\mathbb{D}\,.$ Hence,
$\Pi(\overline{u}\,\uppsi_{u})=\overline{u}\,\uppsi_{u}\,,$ and so identity
(5.17) can be read as

$\frac{z\partial_{z}\uppsi_{u}}{\uppsi_{u}}=|u|^{2}+\nu_{u}\,,$

implying that identity (5.16) holds.

Step 3. In this step, we prove that the rational function $u$ obtained in Step
1 can be rewritten either as $u(z)=Cz^{N}\,,$ $C\in\mathbb{C}^{*}\,,$ or

$u(z)=z^{m_{0}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}-1}\frac{q(z)}{\prod_{j=1}^{r}(1-p_{j}z)}\,,\qquad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,$

where $m_{0}\in\llbracket 0,N-1\rrbracket\,,\,$
$m_{1},\ldots,m_{r}\in\llbracket 1,N\rrbracket$ ,

$m_{0}+\sum_{j=1}^{r}m_{j}=N\,,$

and such that $\deg(q)=r$ if $m_{0}\neq 0\,.$ Indeed, we write (5.13) as
$\uppsi_{u}=\operatorname{e}^{i\theta}z^{N}$ (if all the $p_{k}$ in (5.13)
vanish), or

$\uppsi_{u}=\operatorname{e}^{i\theta}z^{m_{0}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}}\,,\qquad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,\
\theta\in\mathbb{R}$ (5.18)

where $m_{0}\in\llbracket 0,N-1\rrbracket\,,\,$
$m_{1},\ldots,m_{r}\in\llbracket 1,N\rrbracket$ , such that
$m_{0}+\sum_{j=1}^{r}m_{j}=N$ . As a first point, we prove when $m_{0}\geq
1\,,$ then the numerator $P$ of $u$ can be factorized as
$P(z)=z^{m_{0}}P_{N-m_{0}}(z)$ with $P_{N-m_{0}}\in\mathbb{C}_{N-m_{0}}[z]\,.$
Let $m_{0}\geq 1$ then $\left\langle u\,|\,\uppsi_{u}\right\rangle\neq 0\,$,
because otherwise there exists a Balschke product $\upphi_{u}=S^{*}\uppsi_{u}$
of degree $N-1\,,$ such that by the commutator identity (2.2) ,

$L_{u}S^{k}\upphi_{u}=(\nu_{k}-1+k)S^{k}\upphi_{u}\,,\qquad k\geq 0\,,$

meaning that $u\in\mathcal{U}_{N-1}\,,$ which is a contradiction with the fact
that $u\in\mathcal{U}_{N}\,$. Hence, $\left\langle
u\,|\,\uppsi_{u}\right\rangle\neq 0\,.$ This leads to

  1. (i)

The numerator $P$ of $u(z)$ must be degree $N\,.$

  2. (ii)

$\left\langle 1\,|\,u\right\rangle=0$ .

Indeed, for (i) , it is sufficient to note that

$0\neq\left\langle\uppsi_{u}\,|\,u\right\rangle=\int_{z\in\mathscr{C}(0,1)}\operatorname{e}^{i\theta}\,\frac{z^{N}\bar{Q}(1/z)}{Q(z)}\,\frac{\bar{P}(1/z)}{\bar{Q}(1/z)}\,\frac{dz}{2\pi
iz}=\operatorname{e}^{i\theta}\,z^{N}\bar{P}(1/z)_{\,|z=0}$

For (ii), observe by Lemma 2.3 ,

$\left\langle 1\,|\,u\right\rangle\underbrace{\left\langle
u\,|\,\uppsi_{u}\right\rangle}_{\neq 0}=-\nu_{u}\left\langle
1\,|\,\uppsi_{u}\right\rangle\,,$

where the right–hand side vanishes since $\uppsi_{u}=S\upphi$ , $\upphi\in
L^{2}_{+}(\mathbb{T})$ for $m_{0}\geq 1\,.$ Therefore, if $m_{0}=1\,,$ then by
(i) and (ii),

$u(z)=\frac{z\,P_{N-1}(z)}{\prod_{j=1}^{r}(1-p_{j}z)^{m_{j}}}\,,\qquad
P_{N-1}\in\mathbb{C}_{N-1}[z]\,,\quad p_{k}\in\mathbb{D}^{*}\,,\ p_{j}\neq
p_{k}\,.$

Now, if $m_{0}=2\,,$ we have by (ii) $\left\langle u\,|\,1\right\rangle=0\,$,
that is $u=Sv$ with $v\in L^{2}_{+}(\mathbb{T})\,.$ Thus, by the definition of
$L_{u}=z\partial_{z}-u\Pi(\bar{u}\,\cdot)$ ,

$L_{u}\,z=z-\left\langle 1\,|\,v\right\rangle u\,.$

Taking the inner product of the latter identity with $\uppsi_{u}\,,$

$(\nu_{u}-1)\left\langle z\,|\,\uppsi_{u}\right\rangle=-\left\langle
1\,|\,v\right\rangle\underbrace{\left\langle
u\,|\,\uppsi_{u}\right\rangle}_{\neq\,0}\,.$

Note that for $m_{0}=2$ we have $\left\langle
z\,|\,\uppsi_{u}\right\rangle=0\,.$ This implies that $\left\langle
1\,|\,v\right\rangle=0$ leading to $u=S^{2}w$ with $w\in
L^{2}_{+}(\mathbb{T})$ . Therefore, if $m_{0}=2\,,$ then $u$ can be decomposed
as

$u(z)=\frac{z^{2}\,P_{N-2}\,(z)}{\prod_{j=1}^{r}(1-p_{j}z)^{m_{j}}}\,,\qquad
P_{N-2}\in\mathbb{C}_{N-2}[z]\,,\quad p_{k}\in\mathbb{D}^{*}\,,\ p_{j}\neq
p_{k}\,.$

Now, if $m_{0}=3$ then by repeating the same above procedure and taking the
inner product of

$L_{u}z^{2}=2z^{2}-\left\langle 1\,|\,w\right\rangle u$

with $\uppsi_{u}\,,$ one obtains

$(\nu_{u}-2)\underbrace{\left\langle
z^{2}\,|\,\uppsi_{u}\right\rangle}_{=0}=-\left\langle
1\,|\,w\right\rangle\underbrace{\left\langle
u\,|\,\uppsi_{u}\right\rangle}_{\neq\,0}\,.$

That is, $\left\langle 1\,|\,w\right\rangle=0$ i.e, $u=S^{3}\underline{w}$
with $\underline{w}\in L^{2}_{+}(\mathbb{T})\,$… Therefore, for all
$m\in\llbracket 0,N-1\rrbracket\,,$

$u(z)=\frac{z^{m_{0}}\,P_{N-m_{0}}(z)}{\prod_{j=1}^{r}(1-p_{j}z)^{m_{j}}}\,,\qquad
p_{j}\in\mathbb{D}^{*}\,,\ p_{j}\neq p_{k}\,,$

where $P_{N-m_{0}}\in\mathbb{C}_{N-m_{0}}[z]$ if $m_{0}\geq 1$ thanks to (i) .
And if $m_{0}=N$ i.e. all the $p_{j}$ in (5.18) vanish, then $u(z)=Cz^{N}\,,$
$C\in\mathbb{C}^{*}$. Finally, it remains to prove that
$(z-\overline{p_{j}})^{m_{j}-1}$ divides the numerator of $u\,.$ Indeed, by
identity (5.16) of Step 2 ,
$u(z)\bar{u}\left(\frac{1}{z}\right)=z\partial_{z}\log\uppsi_{u}-\nu_{u}$
where one computes by (5.18) ,

$z\partial_{z}\log\uppsi_{u}=m_{0}+\sum_{j=1}^{r}m_{j}\left(\frac{1}{1-p_{j}z}+\frac{\overline{p_{j}}}{z-\overline{p_{j}}}\right)\,.$

That is, for all $m_{0}\in\llbracket 0,N-1\rrbracket\,,$

$\frac{P_{N-m_{0}}(z)}{\prod_{j=1}^{r}(1-p_{j}z)^{m_{j}}}\frac{z^{N-m_{0}}\,\overline{P_{N-m_{0}}}(\frac{1}{z})}{\prod_{j=1}^{r}(z-\overline{p_{j}})^{m_{j}}}=m_{0}-\nu_{u}+\sum_{j=1}^{r}m_{j}\left(\frac{1}{1-p_{j}z}+\frac{\overline{p_{j}}}{z-\overline{p_{j}}}\right)\,,$

where $p_{j}\in\mathbb{D}^{*}\,$, $p_{k}\neq p_{j}\,$ for $k\neq j\,$. Observe
that in the right–hand side, $\frac{1}{p_{j}}$ is a pole of multiplicity one.
Then, the same should hold for the left-hand side as well. Therefore, if
$m_{j}\geq 2\,,$ $j=1,\ldots r\,,$ this implies that $\frac{1}{p_{j}}$ in the
left–hand side must be a root of multiplicity $(m_{j}-1)$ of
$\overline{P_{N-m_{0}}}(\frac{1}{z})$ . That is,

$P_{N-m_{0}}(\overline{p_{j}})=0\ ,\ \ldots\ ,\
P^{\,(m_{j}-2)}_{N-m_{0}}(\overline{p_{j}})=0\,,$

where $P^{(m)}_{N-m_{0}}$ is the $m^{\text{th}}$ derivative of $P_{N-m_{0}}$ .
As a result, $(z-\overline{p_{j}})^{m_{j}-1}$ divides $P_{N-m_{0}}(z)$ , and
so

$u(z)=z^{m_{0}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}-1}\frac{q(z)}{\prod_{j=1}^{r}(1-p_{j}z)}\,,\qquad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,$

with $m_{0}\in\llbracket 0,N-1\rrbracket\,,\,$
$m_{1},\ldots,m_{r}\in\llbracket 1,N\rrbracket$ ,
$m_{0}+\sum_{j=1}^{r}m_{j}=N\,,$ and such that $\deg(q)=r$ if $m_{0}\neq 0$
thanks to (i) .

Step 4. In this step, we write the rational function $u$ obtained in Step 3 on
its partial fractional decomposition

$u(z)=z^{m_{0}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}-1}\left(a+\sum_{j=1}^{r}\frac{c_{j}}{1-p_{j}z}\right)\,,\quad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,$

where $a\neq 0$ if $m_{0}\neq 0$ , and we infer by (5.16) of Step 2 that, for
all $j=1\,,\ldots\,,r\,,$

$\overline{a}c_{j}\,+\,\sum_{k=1}^{r}\,\frac{\overline{c_{k}}\,c_{j}}{1-\overline{p_{k}}\,p_{j}}=m_{j}\,.$
(5.19)

Indeed, by applying $\Pi$ to (5.16) ,

$\Pi(\lvert u\rvert^{2})=\Pi(z\partial_{z}\log\uppsi_{u}-\nu_{u})\,,$

Observe, on the one hand,

$\Pi(z\partial_{z}\log\uppsi_{u}-\nu_{u})=\sum_{j=1}^{r}\,\frac{m_{j}}{1-p_{j}z}+m_{0}-\nu_{u}\,.$

And on the other hand,

$\displaystyle\Pi(\lvert u\rvert^{2})=$
$\displaystyle\,\Pi\left(a+\sum_{j=1}^{r}\frac{c_{j}}{1-p_{j}z}\right)$
$\displaystyle=$ $\displaystyle\,\lvert
a\rvert^{2}+\sum_{j=1}^{r}\bar{a}c_{j}+\,\bar{a}\sum_{j=1}^{r}\frac{c_{j}}{1-p_{j}z}+\sum_{j=1}^{r}\sum_{k=1}^{r}\frac{c_{j}\overline{c_{k}}}{(1-p_{j}\overline{p_{k}})(1-p_{j}z)}\,.$

Therefore, for all $j=1,\ldots,r\,,$ the request conditions (5.19) and

$\nu_{u}=m_{0}-\lvert a\rvert^{2}-\sum_{j=1}^{r}\bar{a}c_{j}\,.$

Step 5. We prove the converse. For $N\in\mathbb{N}$ , let $u=Cz^{N},$
$C\in\mathbb{C}^{*},$ or

$u(z)=z^{m_{0}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}-1}\left(a+\sum_{j=1}^{r}\frac{c_{j}}{1-p_{j}z}\right)\,,\quad
p_{j}\in\mathbb{D}^{*}\,,\ p_{k}\neq p_{j}\,,\ k\neq j\,,$

where $m_{0}\in\llbracket 0,N-1\rrbracket\,,\,$
$m_{1},\ldots,m_{r}\in\llbracket 1,N\rrbracket$ , such that
$m_{0}+\sum_{j=1}^{r}m_{j}=N\,,$ and $\ (a,c_{1},\ldots
c_{r})\in\mathbb{C}\times\mathbb{C}^{r}\,,$ satisfy

$\overline{a}c_{j}\,+\,\sum_{k=1}^{r}\,\frac{\overline{c_{k}}\,c_{j}}{1-\overline{p_{k}}\,p_{j}}=m_{j}\,,$
(5.20)

with $a\neq 0$ if $m\neq 0\,.$ Our aim is to prove that $u\in\mathcal{U}_{N}$
, that is

  * •

$\exists\,\uppsi\in\mathscr{B}_{N}$ such that
$L_{u}\,S^{k}\uppsi=(\mu+k)S^{k}\uppsi$ for all $k\in\mathbb{N}_{\geq 0}\,,$
where $\mu$ is a real constant.

  * •

$\uppsi$ is of minimal degree, i.e. there does not exist
$\upphi\in\mathscr{B}_{\ell}$ with $\ell<N\,,$ such that $\upphi$ satisfies
$L_{u}\,S^{k}\upphi=(\mu_{1}+k)S^{k}\upphi$ for all $k\in\mathbb{N}_{\geq
0}\,.$

For the moment, let us deal with the more complicated case, i.e. $u$ is a
rational function. We start by proving the first point. Let

$\uppsi:=\operatorname{e}^{i\theta}z^{m_{0}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}}\in\mathscr{B}_{N}\,,\quad\theta\in\mathbb{R}\,,\
p_{k}\in\mathbb{D}^{*}\,.$ (5.21)

Observe that $\bar{u}\uppsi$ extends as a holomorphic function on $\mathbb{D}$
as $p_{k}\in\mathbb{D}\,.$ Then, by definition of $L_{u}\,,$

$L_{u}\uppsi=z\partial_{z}\uppsi-\lvert u\rvert^{2}\uppsi\,,$

where

$z\partial_{z}\uppsi=m_{0}\uppsi+\sum_{k=1}^{r}\left(\frac{\overline{p_{k}}}{z-\overline{p_{k}}}+\frac{1}{1-p_{k}z}\right)\uppsi\,,$

and thanks to (5.20) ,

$\displaystyle\lvert u\rvert^{2}=$ $\displaystyle\,\lvert
a\rvert^{2}+\overline{a}\sum_{k=1}^{r}\frac{c_{k}}{1-p_{k}z}+a\sum_{k=1}^{r}\frac{\overline{c_{k}}\,z}{z-\overline{p_{k}}}+\sum_{k=1}^{r}\sum_{j=1}^{r}\frac{c_{k}\overline{c_{j}}\,z}{(1-p_{k}z)(z-\overline{p_{j}})}$
$\displaystyle=$ $\displaystyle\,\lvert
a\rvert^{2}+a\sum_{k=1}^{r}\,\overline{c_{k}}\,+\,\sum_{k=1}^{r}\left(\overline{a}c_{k}+\sum_{j=1}^{r}\frac{c_{k}\overline{c_{j}}}{1-p_{k}\overline{p_{j}}}\right)\frac{1}{1-p_{k}z}$
$\displaystyle\hskip
79.6678pt+\,\sum_{j=1}^{r}\left(\overline{a}c_{j}+\sum_{k=1}^{r}\frac{c_{k}\overline{c_{j}}}{1-p_{k}\overline{p_{j}}}\right)\frac{\overline{p_{j}}}{z-\overline{p_{j}}}$
$\displaystyle=$ $\displaystyle\,\lvert
a\rvert^{2}+a\sum_{k=1}^{r}\,\overline{c_{k}}\,+\,\sum_{k=1}^{r}\frac{m_{k}}{1-p_{k}z}+\sum_{j=1}^{r}\frac{m_{j}\,\overline{p_{j}}}{z-\overline{p_{j}}}\,.$
(5.22)

Therefore,

$L_{u}\uppsi=\Big{(}m_{0}-\lvert
a\rvert^{2}-a\sum_{k=1}^{r}\,\overline{c_{k}}\Big{)}\uppsi$

Additionally, observe that for all $k\in\mathbb{N}\,,$ $\left\langle
S^{k}\uppsi\,|\,u\right\rangle=0\,.$ Hence, by applying the commutator
identity (2.2) , we deduce

$L_{u}\,S^{k}\uppsi=(\mu+k)S^{k}\uppsi\,,\qquad\forall k\in\mathbb{N}_{\geq
0}\,,$ (5.23)

where $\mu:=m_{0}-\lvert a\rvert^{2}-a\sum_{k=1}^{r}\,\overline{c_{k}}\,.$

It remains to prove that $\uppsi$ is of degree minimal. Suppose for the seek
of contradiction that there exists $\upphi\in\mathscr{B}_{\ell}$ , with
$\ell<N\,,$ such that $L_{u}\,S^{j}\upphi=(\mu_{1}+k)S^{j}\upphi$ for all
$j\in\mathbb{N}_{\geq 0}\,.$ By comparing the latter identity to (5.23), and
thanks to (2.7) we infer that there exists $k^{\prime}\,,\,k\in\mathbb{N}$
such that $S^{k}\uppsi=S^{k^{\prime}}\upphi\,,$ i.e.

$\upphi:=\operatorname{e}^{i\tilde{\theta}}z^{m_{0}+k-k^{\prime}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}}\,,\quad
m_{0}+k-k^{\prime}+\sum_{j=1}^{r}m_{j}=\ell<N.$

Therefore, by repeating Step 1 to Step 4, we infer that $u$ must be of the
form

$u(z)=z^{m_{0}+k-k^{\prime}}\prod_{j=1}^{r}\left(\frac{z-\overline{p_{j}}}{1-p_{j}z}\right)^{m_{j}-1}\left(a+\sum_{j=1}^{r}\frac{c_{j}}{1-p_{j}z}\right)\,,\
$

which is a contradiction. ∎

######  Corollary 5.5.

Given $N\in\mathbb{N}$ , let $u\in\mathcal{U}_{N}\,.$ Then,

  1. (i)

In the focusing case, $\|u\|_{L^{2}}^{2}=N-\nu_{u}\,,$

  2. (ii)

In the defocusing case, $\|u\|_{L^{2}}^{2}=\lambda_{u}-N\,,$

where $\nu_{u}$ is the eigenvalue introduced in (5.8) and $\lambda_{u}$ is the
corresponding one in the defocusing case.

######  Remark 5.4.

Based on the previous statement, one can conclude that for any potential
$u\in\mathcal{U}_{N}$, we have

$\begin{cases}\nu_{u}<N\text{ (focusing case) }\\\ \lambda_{u}>N\text{
(defocusing case) }\end{cases}\,.$ (5.24)

###### Proof.

Let $u\in\mathcal{U}_{N}\,.$ Then in light of the previous theorem, we have
either $u=C\operatorname{e}^{iNx}\,,$ $C\in\mathbb{C}^{*}$ or $u$ is the
rational function (5.9) . Thus, if $u=C\operatorname{e}^{iNx}$ then the
results follow easily by Remark 5.2. Now, if $u$ is the rational function
(5.9) , then by computing the $L^{2}$–norm of $u$ in the focusing case, we
infer via (5) ,

$\displaystyle\|u\|_{L^{2}}^{2}=$
$\displaystyle\,\int_{z\in\mathscr{C}(0,1)}\left(\lvert
a\rvert^{2}+a\sum_{k=1}^{r}\,\overline{c_{k}}\,+\,\sum_{k=1}^{r}\frac{1}{1-p_{k}z}+\sum_{k=1}^{r}\frac{\overline{p_{k}}}{z-\overline{p_{k}}}\right)\frac{dz}{2\pi
iz}$ $\displaystyle=$ $\displaystyle\,\lvert
a\rvert^{2}+a\sum_{k=1}^{r}\,\overline{c_{k}}+N-m\,,$

which is equal to $-\nu_{u}+N$ by (i) of Remark 5.2 . For the defocusing case,
we shall have

$\displaystyle\|u\|_{L^{2}}^{2}=$
$\displaystyle\,\int_{z\in\mathscr{C}(0,1)}\left(\lvert
a\rvert^{2}+a\sum_{k=1}^{r}\,\overline{c_{k}}\,-\,\sum_{k=1}^{r}\frac{1}{1-p_{k}z}-\sum_{k=1}^{r}\frac{\overline{p_{k}}}{z-\overline{p_{k}}}\right)\frac{dz}{2\pi
iz}$ $\displaystyle=$ $\displaystyle\,\lvert
a\rvert^{2}+a\sum_{k=1}^{r}\,\overline{c_{k}}-N+m\,,$

which is equal to $\lambda_{u}-N$ by (ii) of Remark 5.2 . ∎

######  Proposition 5.6.

For any $N\in\mathbb{N}_{\geq 0}$ , the set of finite gap potential
$\mathcal{U}_{N}$ is conserved along the flow of the (CS)–equation.

###### Proof.

Let $u_{0}$ be a finite gap potential in $\mathcal{U}_{N}\,,$ that is there
exists $\uppsi_{u_{0}}\in\mathscr{B}_{N}$ of minimal degree $N$ satisfying

$L_{u_{0}}\,S^{k}\uppsi_{u_{0}}=(\nu_{u_{0}}+k)S^{k}\uppsi_{u_{0}}\,,\qquad\forall
k\geq 0\,.$ (5.25)

Our aim is to prove that there exists $\varrho(t)\in\mathscr{B}_{N}$ of
minimal degree 888In the sense, that there does not exist
$\upphi(t)\in\mathscr{B}_{\ell}$ with $\ell<N\,,$ such that $\upphi(t)$
satisfies $L_{u}\,S^{k}\upphi(t)=(\mu_{1}+k)S^{k}\upphi(t)$ for all
$k\in\mathbb{N}_{\geq 0}\,.$ such that

$L_{u(t)}\,S^{k}\varrho(t)=(\nu_{u_{0}}+k)S^{k}\varrho(t)\,,\qquad\forall
k\geq 0\,.$

Let $\varrho(t)$ be a solution of the Cauchy problem

$\begin{cases}\partial_{t}\,\varrho(t)=B_{u(t)}\,\varrho(t)\\\
\varrho(0)=\displaystyle\uppsi_{u_{0}}\end{cases}\,.$

Hence, by Remark 3.1 ,