# The Calogero–Moser derivative
nonlinear Schrödinger equation

Patrick Gérard  P. Gérard, Laboratoire de Mathématiques d’Orsay, CNRS,
Université Paris-Saclay, 91405 Orsay, France.  patrick.gerard@universite-
paris-saclay.fr and  Enno Lenzmann  E. Lenzmann, University of Basel,
Department of Mathematics and Computer Science, Spiegelgasse 1, CH-4051 Basel,
Switzerland<EMAIL_ADDRESS>
###### Abstract.

We study the Calogero–Moser derivative NLS equation

$i\partial_{t}u+\partial_{xx}u+(D+|D|)(|u|^{2})u=0$

posed on the Hardy-Sobolev space $H^{s}_{+}(\mathbb{R})$ with suitable $s>0$.
By using a Lax pair structure for this $L^{2}$-critical equation, we prove
global well-posedness for $s\geq 1$ and initial data with sub-critical or
critical $L^{2}$-mass $\|u_{0}\|_{L^{2}}^{2}\leq 2\pi$. Moreover, we prove
uniqueness of ground states and also classify all traveling solitary waves.
Finally, we study in detail the class of multi-soliton solutions $u(t)$ and we
prove that they exhibit energy cascades in the following strong sense such
that $\|u(t)\|_{H^{s}}\sim_{s}|t|^{2s}$ as $t\to\pm\infty$ for every $s>0$.

###### 2010 Mathematics Subject Classification:

37K15 primary, 47B35 secondary

###### Contents

  1. 1 Introduction and Main Results
  2. 2 Well-Posedness, Lax Structure, and Conservation Laws
  3. 3 Nonexistence of Minimal Mass Blowup
  4. 4 Ground States and Traveling Solitary Waves
  5. 5 Analysis of the Lax Operator
  6. 6 Dynamics of Multi-Solitons
  7. A Definition of the Lax Operator for $u\in L^{2}_{+}(\mathbb{R})$
  8. B Variational Properties of $E(u)$
  9. C Useful Identities and Gauge Transformation

##  1\. Introduction and Main Results

This paper is devoted to the study of the Calogero–Moser derivative nonlinear
Schrödinger equation, which can be written as

$i\partial_{t}u+\partial_{xx}u+(D+|D|)(|u|^{2})u=0$ (CM-DNLS)

for $u:I\times\mathbb{R}\to\mathbb{C}$ with some time interval
$I\subset\mathbb{R}$. Here and in what follows, we use the standard notation
$D=-i\partial_{x}$ and hence $|D|$ denotes the Fourier multiplier with symbol
$|\xi|$.

We remark that equation (CM-DNLS) was introduced in [1] as a formal continuum
limit of classical Calogero–Moser systems [27, 28]. Also, prior to [1], a
defocusing version given by

$i\partial_{t}u+\partial_{xx}u-(D+|D|)(|u|^{2})=0$ (INLS)

was introduced in [30] under the name intermediate nonlinear Schrödinger
equation (INLS), as describing envelope waves in a deep stratified fluid. We
will concentrate on (CM-DNLS), because it offers richer dynamics, e. g. multi-
soliton solutions with turbulence in Sobolev norms (see Theorem 1.3 below).
However, part of our results can be extended to this defocusing version above.

###  1.1. Symmetries, Phase Space, and Hamiltonian Features

We observe that equation (CM-DNLS) admits the invariance by phase, scaling and
translation,

$u(t,x)\mapsto\mathrm{e}^{i\theta}\lambda^{1/2}u(\lambda^{2}t,\lambda
x+x_{0}),\quad x_{0}\in\mathbb{R},\theta\in\mathbb{R},\lambda>0,$

which makes it a $L^{2}$-critical equation on the line. It also enjoys the
Galilean invariance

$u(t,x)\mapsto\mathrm{e}^{i\eta
x-it\eta^{2}}u(t,x-2t\eta),\quad\eta\in\mathbb{R},$

as well as the pseudo–conformal symmetry found by Ginibre and Velo for the
$L^{2}$-critical NLS. Recall that a special case of this space-time transform
reads

$u(t,x)\mapsto\frac{1}{t^{1/2}}\mathrm{e}^{i\frac{x^{2}}{4t}}u\left(-\frac{1}{t},\frac{x}{t}\right).$

In what follows, we are interested in solutions of (CM-DNLS) satisfying the
additional condition that

$u(t)\in H^{s}_{+}(\mathbb{R}):=\\{f\in
H^{s}(\mathbb{R}):\mathrm{supp}(\hat{f})\subset[0,+\infty)\\},$

where $H^{s}(\mathbb{R})$ denotes the usual Sobolev space based on
$L^{2}(\mathbb{R})$. The spaces $H^{s}_{+}(\mathbb{R})$ will serve as phase
spaces on which we study (CM-DNLS) as a Hamiltonian system.

Recall that $H^{0}_{+}(\mathbb{R})=L^{2}_{+}(\mathbb{R})$ denotes the Hardy
space of holomorphic functions on the complex upper half-plane. If we let
$\Pi_{+}:L^{2}(\mathbb{R})\to L^{2}_{+}(\mathbb{R})$ denote Cauchy–Szegő
orthogonal projection onto $L^{2}_{+}(\mathbb{R})$ given by

$\Pi_{+}(f)(x):=\frac{1}{2\pi}\int_{0}^{\infty}e^{i\xi x}\hat{f}(\xi)\,d\xi,$

then equation (CM-DNLS) can be written as

$i\partial_{t}u+\partial_{xx}u+2D_{+}(|u|^{2})u=0.$

Here $D_{+}:=D\Pi_{+}$ can be seen as the compression of $D=-i\partial_{x}$
onto the Hardy space $L^{2}_{+}(\mathbb{R})$. The positive Fourier frequency
condition $\mathrm{supp}(\hat{f})\subset[0,+\infty)$ is interpreted as a
chirality condition in [1]. In fact, such a condition naturally appears if one
thinks of the Benjamin–Ono equation,

$\partial_{t}v+\partial_{x}|D|v-\partial_{x}(v^{2})=0,$ (BO)

which is known to be well-posed for real valued functions $v$; see [26, 22].
Introducing the new unknown $u=\Pi_{+}v$, the condition $v=\overline{v}$ reads
$v=u+\overline{u}$, so that (BO) is equivalent to

$i\partial_{t}u+\partial_{xx}u+D(u^{2})+2D_{+}(|u|^{2})=0.$

This way, (CM-DNLS) and its defocusing sibling can be seen as $L^{2}$-critical
versions of (BO).

Notice that the pseudo-conformal symmetry does not preserve chirality and that
the Galilean transformation acts on chiral solutions of (CM-DNLS) only if
$\eta\geq 0$.

Now, let us come to the Hamiltonian properties of (CM-DNLS). To this end, we
introduce the following gauge transformation

$v(x):=u(x)\,\mathrm{e}^{-\frac{i}{2}\int_{-\infty}^{x}|u(y)|^{2}\,dy},$ (1.1)

which turns out to be a diffeomorphism of $H^{s}(\mathbb{R})$ into itself for
every $s\geq 0$. An elementary calculation shows that (CM-DNLS) is equivalent
to the equation

$i\partial_{t}v+\partial_{xx}v+|D|(|v|^{2})v-\frac{1}{4}|v|^{4}v=0.$ (1.2)

This is a Hamiltonian PDE with the standard symplectic form
$\omega(h_{1},h_{2})=\mathrm{Im}\langle h_{1},h_{2}\rangle_{L^{2}}$ and the
energy functional

$\widetilde{E}(v):=\frac{1}{2}\|\partial_{x}v\|_{L^{2}}^{2}-\frac{1}{4}\langle|D|(|v|^{2}),|v|^{2}\rangle_{L^{2}}+\frac{1}{24}\|v\|_{L^{6}}^{6}.$

By classical product identities for the Hilbert transform $\mathsf{H}$ (see
Appendix C for details), the energy functional $\widetilde{E}$ can be written
as

$\widetilde{E}(v)=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}v+\frac{1}{2}\mathsf{H}(|v|^{2})v|^{2}\,dx\geq
0.$

Inverting the gauge transformation (1.1) and in view of
$\Pi_{+}=\frac{1}{2}(1+i\mathsf{H})$, we find that $E(u)=\widetilde{E}(v)$ is
an energy functional for (CM-DNLS) given by

$E(u)=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}u-i\Pi_{+}(|u|^{2})u|^{2}\,dx.$
(1.3)

In summary, we deduce that (CM-DNLS) is a Hamiltonian equation generated by
the energy functional $E(u)$ and the symplectic form

$\omega_{u}^{\sharp}(h_{1},h_{2}):=\mathrm{Im}\langle
h_{1},h_{2}\rangle_{L^{2}}+\int\\!\\!\int_{\mathbb{R}\times\mathbb{R}}\mathrm{Re}(\overline{u}h_{1})(x)\mathrm{Re}(\overline{u}h_{2})(y)\mathrm{sgn}(x-y)\,dx\,dy,$

which is just the pullback of the standard symplectic $\omega$ under the gauge
transformation $u\mapsto v$ defined in (1.1). Recall that we will study (CM-
DNLS) as Hamiltonian PDE on the phase spaces $H^{s}_{+}(\mathbb{R})$
corresponding to chiral solutions. It is interesting to note that
$\omega_{u}^{\sharp}$ provides a non-standard symplectic form on the spaces
$H^{s}_{+}(\mathbb{R})$ with $s\geq 0$.

Next, we discuss the conservation laws exhibited by (CM-DNLS). Due to symmetry
by complex phase shifts, spatial translations and its Hamiltonian nature, we
easily obtain the following conserved quantities:

$M(u)=\int_{\mathbb{R}}|u|^{2}\,dx\quad\mbox{($L^{2}$-mass)},\qquad
P(u)=\int_{\mathbb{R}}(Du\overline{u}-\frac{1}{2}|u|^{4})\,dx\quad\mbox{(Momentum)},$
$E(u)=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}u-i\Pi_{+}(|u|^{2})u|^{2}\,dx\quad\mbox{(Energy)}.$

In the expression for the conserved momentum $P(u)$, the nonlinear term
$|u|^{4}$ arises due to the non-standard symplectic structure
$\omega_{u}^{\sharp}$. In fact, we will show below that $M(u),P(u)$ and $E(u)$
belong to an infinite hierarchy of conservation laws
$\\{I_{k}(u)\\}_{k=0}^{\infty}$ based on a Lax pair structure for (CM-DNLS);
see Section 2.

Finally, we briefly comment on the $L^{2}$-critical nature of (CM-DNLS). As
one may expect, there exists a special solution which separates solutions into
small and large data. Indeed, we will prove that the energy $E(u)$ has a
unique (up to symmetries) minimizer given by the rational function

$\mathcal{R}(x)=\frac{\sqrt{2}}{x+i}\in H^{1}_{+}(\mathbb{R}),$

which we refer to as the ground state for (CM-DNLS); see Section 4. An
elementary calculation shows that $u(t,x)=\mathcal{R}(x)$ provides a static
solution of (CM-DNLS) and its $L^{2}$-mass is found to be

$M(\mathcal{R})=\int_{\mathbb{R}}\frac{2}{1+x^{2}}\,dx=2\pi.$

As we will see below, this number $2\pi$ provides a threshold in the analysis
of (CM-DNLS). Consequently, we shall refer to solutions $u(t)\in
H^{s}_{+}(\mathbb{R})$ with

$M(u_{0})<M(\mathcal{R}),\quad M(u_{0})=M(\mathcal{R}),\quad
M(u_{0})>M(\mathcal{R})$

as having sub-critical, critical, and super-critical $L^{2}$-masses,
respectively. The main results of this paper will address these various
regimes.

###  1.2. Main Results

As a staring point, we first establish local well-posedness of (CM-DNLS) for
initial data in $H^{s}_{+}(\mathbb{R})$ with $s>\nicefrac{{1}}{{2}}$. For
sufficiently regular initial data in $H^{s}_{+}(\mathbb{R})$ with
$s>\nicefrac{{3}}{{2}}$, this follows from Kato’s classical iteration scheme
for quasilinear evolution equations. Extending the local well-posedness to
less regular data in $H^{s}_{+}(\mathbb{R})$ with
$\nicefrac{{1}}{{2}}<s\leq\nicefrac{{3}}{{2}}$ can then be achieved by
adapting arguments from [13], which in turn is inspired by Tao’s gauge trick
for the Benjamin-Ono equation [35]. However, for the rest of the paper, we
will be mainly be concerned with solutions of (CM-DNLS) such that $u(t)\in
H^{s}_{+}(\mathbb{R})$ with some integer $s\geq 1$.

The general question of global well-posedness for (CM-DNLS) seems to be rather
delicate because of the focusing $L^{2}$-criticality of the nonlinearity,
which might generate blowup of solutions in finite time. The following result
establishes global well-posedness for initial data with finite energy and
$L^{2}$-mass that is less or equal to the ground state mass.

######  Theorem 1.1 (Global Well-Posedness Result).

Let $s\geq 1$ be an integer. Then (CM-DNLS) is globally well-posed for initial
data in $u_{0}\in H^{s}_{+}(\mathbb{R})$ with $L^{2}$-mass

$M(u_{0})\leq M(\mathcal{R})=2\pi.$

Moreover, we have the a-priori bound

$\sup_{t\in\mathbb{R}}\|u(t)\|_{H^{s}}<+\infty,$

provided the strict inequality $M(u_{0})<M(\mathcal{R})$ holds.

######  Remarks 1.1.

1) In the case of sub-critical $L^{2}$-mass, the a-priori bounds on
$\|u(t)\|_{H^{s}}$ will follow from exploiting an infinite hierarchy of
conservation laws for (CM-DNLS). We refer to Section 5 for a detailed
discussion.

2) The case of critical $L^{2}$-mass when $M(u_{0})=M(\mathcal{R})$ is rather
delicate to handle and will follow from ruling out the so-called minimal mass
blowup solutions for (CM-DNLS) with finite energy. A key element in the proof
will be the slow algebraic decay of the ground states $\mathcal{R}$.

3) It is an interesting open question whether global-in-time existence holds
for large initial data in $H^{s}_{+}(\mathbb{R})$ with $s\geq 1$ and
$L^{2}$-mass $M(u_{0})>M(\mathcal{R})$. As a striking example below, there
exist smooth global-in-time solutions for (CM-DNLS) given by multi-solitons,
which always blowup in infinite time due to unbounded growth of all Sobolev
norms $\|u(t)\|_{H^{s}}$ for any $s>0$.

4) By applying the pseudo-conformal transformation to the static solution
$\mathcal{R}(x)$, we obtain the explicit solution

$u_{\mathrm{sing}}(t,x)=\frac{1}{t^{1/2}}e^{ix^{2}/4t}\mathcal{R}\left(\frac{x}{t}\right)\in
L^{2}(\mathbb{R})\quad\mbox{for all $t>0$},$

which solves (CM-DNLS) and becomes singular as $t\to 0^{-}$. Due to slow
algebraic decay of $\mathcal{R}(x)$, we find that $u_{\mathrm{sing}}(t)\not\in
H^{1}(\mathbb{R})$ has no finite energy111A closer inspection shows that
$u_{\mathrm{sing}}(t,\cdot)\in H^{s}(\mathbb{R})$ for all $0\leq
s<\nicefrac{{1}}{{2}}$. and, moreover, we see that the solution
$u_{\mathrm{sing}}(t)\not\in L^{2}_{+}(\mathbb{R})$ fails to be chiral. Still,
this explicit example shows that we cannot expect global well-posedness for
(CM-DNLS) with arbitrary initial data in the scaling-critical space
$L^{2}(\mathbb{R})$. It remains an intriguing open question if initial data in
$L^{2}_{+}(\mathbb{R})$ will always lead to global-in-time solutions for (CM-
DNLS).

Next, we turn our attention to sufficiently regular solutions $u(t)\in
H^{s}_{+}(\mathbb{R})$ of (CM-DNLS) with initial data having critical or
super-critical $L^{2}$-mass:

$M(u_{0})\geq M(\mathcal{R})=2\pi.$

In this regime of sufficiently large data, we expect (CM-DNLS) to possess
traveling ground state solitons as well as multi-soliton solutions. As a main
result in this setting, we completely classify all traveling solitary waves
for the Calogero–Moser DNLS with finite energy by showing that are given by
the ground state $\mathcal{R}(x)$ up to scaling, phase, translation, and
Galilean boosts prserving the chirality condition.

######  Theorem 1.2 (Classification of Traveling Solitary Waves).

Every traveling solitary wave for equation (CM-DNLS) in
$H^{1}_{+}(\mathbb{R})$ is of the form

$u(t,x)=e^{i\theta+i\eta x-i\eta^{2}t}\lambda^{1/2}\mathcal{R}(\lambda(x-2\eta
t)+y)$

with some $\theta\in[0,2\pi),y\in\mathbb{R},\lambda>0$, and $\eta\geq 0$.

In particular, every traveling solitary waves $u(t)\in H^{1}_{+}(\mathbb{R})$
for (CM-DNLS) have critical $L^{2}$-mass $M(u)=M(\mathcal{R})$.

######  Remarks 1.2.

1) The condition $\eta\geq 0$ enters through the chirality condition $u(t)\in
H^{1}_{+}(\mathbb{R})$ and thus traveling solitary waves can only move to
right. If we take negative values $\eta<0$ above, we obtain left-moving
traveling solitary waves $u(t)\in H^{1}(\mathbb{R})$ solving (CM-DNLS); see
Section 4 for the definition of traveling solitary waves.

2) A key step in the complete classification above is to establish uniqueness
of (non-trivial) minimizers of the energy $E(u)$, which is equivalent to
classifying all solutions $u\in H^{1}_{+}(\mathbb{R})$ of the nonlinear
equation

$Du-\Pi_{+}(|u|^{2})u=0.$

We refer to Section 4 below for details including a more general result
assuming only that $u\in H^{1}(\mathbb{R})$.

As our final main result, we study the dynamics of multi-soliton solutions for
(CM-DNLS); see Section 6 below for a precise definition using the Lax pair
structure. For the Calogero–Moser DNLS, it turns out that multi-solitons
$u=u(t,x)$ are rational functions of $x\in\mathbb{R}$ in the Hardy spaces
$L^{2}_{+}(\mathbb{R})$. As an interesting fact, we remark that they
necessarily have a quantized $L^{2}$-mass given by

$M(u)=2\pi N\quad\mbox{with $N=1,2,3,\ldots$}$

In the special case when $N=1$, the multi-solitons are given by the ground
state $\mathcal{R}(x)$ up to symmetries. For $N\geq 2$, we note that multi-
solitons have super-critical $L^{2}$-mass. As a consequence, the proof of
their global-in-time existence is far from trivial and will follow from the
analysis of a suitable inverse spectral formula based on the Lax structure. As
an outcome, we obtain a detailed dynamical description in the long-time limit.
Here, a surprising feature is the general ‘turbulent’ behavior of multi-
solitons with $N\geq 2$, leading to unbounded growth of higher Sobolev norms
(energy cascades) as follows.

######  Theorem 1.3 (Growth of Sobolev Norms).

For every $N\geq 2$, every multi-soliton $u=u(t,x)$ for (CM-DNLS) exists for
all times $t\in\mathbb{R}$ and it exhibits growth of Sobolev norms such that

$\|u(t)\|_{H^{s}}\sim_{s}|t|^{2s}\quad\mbox{as}\quad t\to\pm\infty,$

for any real number $s>0$.

######  Remarks.

1) In Section 6, we make a detailed analysis of the dynamics of multi-
solitons. After having established their global-in-time existence, we show
that there exists a sufficiently large time $T=T(u_{0})\gg 1$ such that a
multi-soliton reads

$u(t,x)=\sum_{k=1}^{N}\frac{a_{k}(t)}{x-z_{k}(t)}\quad\mbox{for $t\geq T$},$
(1.4)

with residues $a_{1}(t),\ldots,a_{N}(t)\in\mathbb{C}$ and pairwise distinct
poles $z_{1}(t),\ldots,z_{N}(t)\in\mathbb{C}_{-}$ that satisfy a complexified
version of the rational Calogero–Moser system for $N$ classical particles. A
detailed investigation (exploiting on the Lax pair structure) then yields that
the poles – except for $z_{1}(t)$ – will all approach the real axis
asymptotically, i. e.,

$\mathrm{Im}\,z_{k}(t)\to 0\quad\mbox{as}\quad t\to+\infty\quad\mbox{for
$2\leq k\leq N$}.$

A careful analysis of this fact then leads to the precise growth bound in
Theorem 1.3. The limit $t\to-\infty$ can be handled in the same way.

2) It is a subtle fact that multi-solitons $u(t,x)$ may fail to be of the form
(1.4) for all times $t\in\mathbb{R}$. That is, we can have collisions of poles
in finite time, which renders the form (1.4) invalid. To handle this collision
scenario (see explicit examples for $N=2$ in Section 6), we will make use of a
general representation formula of $u(t,x)$ in terms of an inverse spectral
formula. See Section 6 for details.

3) It is an interesting open question whether the growth phenomenon in Theorem
1.3 is stable under perturbations of multi-solitons.

###  1.3. Comments on the Lax Structure

A central feature of (CM-DNLS) is the fact that it admits a Lax pair. That is,
as detailed in Section 2, we can recast the dynamical evolution into
commutator form

$\frac{d}{dt}L_{u}=[B_{u},L_{u}],$ (1.5)

where the Lax operator $L_{u}$ is given by

$L_{u}=D-T_{u}T_{\bar{u}}.$ (1.6)

This defines an unbounded self-adjoint operator acting the Hardy space
$L^{2}_{+}(\mathbb{R})$ with a suitable operator domain, depending on the
regularity of $u$. Here $T_{b}(f)=\Pi_{+}(bf)$ denotes the Toeplitz operator
on $L^{2}_{+}(\mathbb{R})$ with symbol $b$.

As an important consequence of (1.5), we will find an infinite hierarchy of
conservation laws in terms of expressions

$I_{k}(u)=\langle L^{k}_{u}u,u\rangle\quad\mbox{with}\quad k=0,1,2,\ldots$

provided that $u(t)\in H^{s}_{+}(\mathbb{R})$ is a sufficiently regular
solution of (CM-DNLS). It is an intriguing feature, due to the
$L^{2}$-criticality of the problem, that the hierarchy
$\\{I_{k}(u)\\}_{k\in\mathbb{N}}$ will generally provide a-priori bounds on
solutions only if we have sub-critical $L^{2}$-mass; see Theorem 1.1. This is
in striking contrast to many other completely integrable PDEs (e. g. KdV,
Benjamin-Ono, cubic NLS etc.) where the corresponding hierarchy of
conservation laws yields control over any smooth solutions without any
assumption on its size.

In Section 5, we examine the spectral properties of the Lax operator $L_{u}$
in more detail. Based on a key commutator formula, we find a sharp bound on
the number of eigenvalues $N$ of the form

$N\leq\frac{\|u\|_{L^{2}}^{2}}{2\pi}.$ (1.7)

Furthermore, we prove that every eigenvalue of $L_{u}$ is simple. As
interesting aside, we remark that this bound not only applies to isolated
eigenvalues, but also to eigenvalues which are embedded in the essential
spectrum $\sigma_{\mathrm{ess}}(L)=[0,\infty)$. Moreover, we emphasize the
fact that we can easily generate embedded eigenvalues of $L_{u}$ by action of
the Beurling–Lax semigroup $\\{e^{i\eta x}\\}_{\eta\geq 0}$ acting on
$L^{2}_{+}(\mathbb{R})$; see Section 5 again.

In terms of spectral theory, it is a natural question to study which
potentials $u\in L^{2}_{+}(\mathbb{R})$ will lead to equality in the general
bound (1.7). Here we will find a distinguished class of potentials given by
rational functions of the form

$u(x)=\frac{P(x)}{Q(x)}\in H^{1}_{+}(\mathbb{R}),$

where $Q,P\in\mathbb{C}[x]$ are suitable polynomials with $\deg Q=N$ and $\deg
P\leq N-1$; see Proposition 5.2. We will refer to these $u(x)=P(x)/Q(x)$ as
above as multi-soliton potentials and the corresponding solution will called
multi-solitons for (CM-DNLS). As an immediate consequence of saturating the
bound (1.7), we obtain the multi-solitons $u(t,x)$ have quantized $L^{2}$-mass
with

$M(u)=2\pi N.$

Another noteworthy feature of any multi-soliton solution $u(t,x)$ is that it
is completely supported in the pure point spectrum of the Lax operator, i. e,
we have

$u(t)\in\mathcal{E}_{pp}(L_{u(t)}),$

where $\mathcal{E}_{pp}$ denotes the $N$-dimensional space spanned the
eigenfunctions of $L_{u}$. This fact will allow us to derive a very explicit
inverse spectral formula representing a multi-soliton. This will enable us to
prove global-in-time existence and, more strikingly, the growth bounds states
in Theorem 1.3. In the future, we plan to further refine the spectral analysis
of $L_{u}$ in order to study the long-time behavior of solutions of (CM-DNLS)
beyond the case of multi-solitons.

Finally, we remark that the Lax structure for (CM-DNLS) bears some resemblance
to the Lax structure for (BO), which is known to have the Lax operator

$L_{u}^{\mathrm{(BO)}}=D-T_{u}$

acting on the Hardy space $L^{2}_{+}(\mathbb{R})$; see e. g. [16]. Note that
the occurrence of $T_{u}$ in $L_{u}^{\mathrm{(BO)}}$ instead of
$T_{u}T_{\bar{u}}$ in $L_{u}$ is consistent with the different degrees of the
nonlinearity (quadratic vs. cubic).

###  1.4. Comparison to other PDEs

Let us comment on the energy cascade phenomenon in Theorem 1.3 in comparison
to other Hamiltonian PDEs on the line. Among the recently studied Hamiltonian
PDEs on the line, the closest one to (CM-DNLS) is certainly the derivative
nonlinear Schrödinger equation,

$i\partial_{t}u+\partial_{xx}u+i\partial_{x}(|u|^{2}u)=0,$ (DNLS)

which – like (CM-DNLS) – is $L^{2}$-mass critical with a Lax pair structure.
Using the Lax pair structure, global existence was first proved in [20] in the
space $H^{2}(\mathbb{R})\cap\hat{H}^{2}(\mathbb{R})$. Then global well-
posedness with uniform bounds in $H^{s}$, $s\geq 1/2$ was obtained in [5],
[4]. Quite recently, the flow map was extended to the whole of
$L^{2}(\mathbb{R})$ in [18], proving that all trajectories of (DNLS) are
uniformly equicontinuous with values in $L^{2}(\mathbb{R})$. All these results
prevent any kind of energy cascade and are therefore in strong contrast with
the dynamics of (CM-DNLS), which turns out to be much richer.

Another integrable Hamiltonian PDE on the line is the cubic Szegő equation,
see [31],

$i\partial_{t}u=\Pi_{+}(|u|^{2}u),$ (1.8)

where multi-solitons were recently studied in [17], and where energy cascades
were displayed under some degeneracy assumption of the spectrum of the
corresponding Lax operator. There is definitely some similarity in the
approaches to multi-solitons in (1.8) and (CM-DNLS), particularly in the
inverse spectral formulae. However, let us emphasize that the spectral
properties of the Lax operators are very different, and that energy cascades
for multi-solitons in (1.8) only occur under some degeneracy assumption, while
they always occur for multi-solitons in (CM-DNLS). This suggests that the
dynamics of (CM-DNLS) is particularly turbulent, even compared to the non-
dispersive equation (1.8). We hope to explore other aspects of this dynamics
in the near future.

###  1.5. Notation

We denote by $\langle f,g\rangle=\int_{\mathbb{R}}f\overline{g}$ the
$L^{2}$-inner product of functions $f,g$ on the line. We recall that $\Pi_{+}$
denotes the orthogonal projector from $L^{2}(\mathbb{R})$ onto the Hardy space
$L^{2}_{+}(\mathbb{R})$. Sometimes, we will also use the notation
$\Pi_{-}=1-\Pi_{+}$. Notice that, for every $L^{2}$ function $f$,

$\Pi_{-}(f)=\overline{\Pi_{+}(\overline{f})}\ .$

Finally, observe that the Hilbert transform $\mathsf{H}=-i\,\mathrm{sgn}(D)$
is related to $\Pi_{\pm}$ by the identities

$\Pi_{+}=\frac{1}{2}(1+i\mathsf{H})\ ,\ \Pi_{-}=\frac{1}{2}(1-i\mathsf{H})\ .$

### Acknowledgments

E. L. acknowledges financial support from the Swiss National Science
Foundation (SNSF) under Grant No. 204121.

##  2\. Well-Posedness, Lax Structure, and Conservation Laws

In this section, we study the Cauchy problem for (CM-DNLS) in
$H^{s}_{+}(\mathbb{R})$ with suitable $s$. As a key element for obtaining
global-in-time solutions, we will find a Lax pair structure on the Hardy-type
space $H^{s}_{+}$, which will generate an infinite hierarchy of conservation
laws. With this at hand, we will derive a-priori bounds for initial $u_{0}\in
H^{s}_{+}(\mathbb{R})$ with integer $s\geq 1$ and sub-critical $L^{2}$-mass
$M(u_{0})<M(\mathcal{R})$.

###  2.1. Local Well-Posedness

As starting point for local well-posedness, we consider the case of initial
data in $H^{s}_{+}(\mathbb{R})$ with $s>\nicefrac{{3}}{{2}}$, where Kato’s
classical iterative scheme for quasilinear evolution equations can be
utilized. We remark that the presence of the derivative term $D_{+}(|u|^{2})u$
raises some analytic challenges that need to be addressed.

######  Proposition 2.1.

Let $s>\nicefrac{{3}}{{2}}$. For any $R>0$, there is some $T(R)>0$ such that,
for every $u_{0}\in H^{s}_{+}(\mathbb{R})$ with $\|u_{0}\|_{H^{s}}\leq R$,
there exists a unique solution $u\in C([-T,T];H^{s}_{+}(\mathbb{R}))$ of (CM-
DNLS) with $u(0)=u_{0}$.

Furthermore, the $H^{\sigma}$-regularity of $u_{0}$ for $\sigma>s$ is
propagated on the whole maximal interval of existence of $u$, and the flow map
$u_{0}\mapsto u(t)$ is continuous on $H^{s}$.

###### Proof.

As already mentioned above, we apply a Kato-type iterative scheme to obtain
this result. For concreteness, we shall consider the case

$s=2$

in what follows. The general case $s>\nicefrac{{3}}{{2}}$ can be handled in an
analogous way.

We first write (CM-DNLS) as

$\partial_{t}u=i\partial_{xx}u+2T_{u}T_{\overline{u}}\partial_{x}u+2uH_{u}\partial_{x}u,$
(2.1)

where

$T_{a}f:=\Pi_{+}(af),\quad H_{b}f:=\Pi_{+}(b\overline{f})$ (2.2)

denote the Toeplitz and Hankel operators acting on $L^{2}_{+}(\mathbb{R})$
with symbols $a$ and $b$, respectively. Our first observation is that the term
$H_{u}\partial_{x}u$ is of order 0 if $u$ is smooth enough.

######  Lemma 2.1.

If $u\in H^{\frac{3}{2}}_{+}(\mathbb{R})$, then
$H_{u}\partial_{x}:L^{2}_{+}(\mathbb{R})\to L_{+}^{2}(\mathbb{R})$ is bounded
with
$\|H_{u}(\partial_{x}f)\|_{L^{2}}\leq\frac{1}{\sqrt{2\pi}}\|u\|_{\dot{H}^{3/2}}\|f\|_{L^{2}}$.
If $u,v\in H^{2}_{+}(\mathbb{R})$, then $H_{u}\partial_{x}v\in
H^{2}_{+}(\mathbb{R})$ with

$\|H_{u}\partial_{x}v\|_{H^{2}}\leq C\|u\|_{H^{2}}\|v\|_{H^{2}}$

with some constant $C>0$.

###### Proof.

Let $f\in H^{1}_{+}(\mathbb{R})$. Then

$\widehat{H_{u}\partial_{x}f}(\xi)=-\int_{0}^{\infty}\widehat{u}(\xi+\eta)\overline{\widehat{f}(\eta)}\frac{d\eta}{2\pi}\quad\mbox{for}\quad\xi\geq
0.$

Consequently,

$\displaystyle|\widehat{H_{u}\partial_{x}f}(\xi)|^{2}$
$\displaystyle\leq\left|\int_{0}^{\infty}|\widehat{u}(\xi+\eta)||\xi+\eta||\widehat{f}(\eta)|\frac{d\eta}{2\pi}\right|^{2}$
$\displaystyle\leq\int_{0}^{\infty}|\widehat{u}(\xi+\eta)|^{2}(\xi+\eta)^{2}\frac{d\eta}{2\pi}\cdot\int_{0}^{\infty}|\widehat{f}(\eta)|^{2}\frac{d\eta}{2\pi}.$

Thus

$\displaystyle\|H_{u}\partial_{x}f\|_{L^{2}}^{2}$
$\displaystyle\leq\int_{0}^{\infty}\int_{0}^{\infty}|\widehat{u}(\xi+\eta)|^{2}(\xi+\eta)^{2}\frac{d\eta}{2\pi}\frac{d\xi}{2\pi}\|f\|_{L^{2}}$
$\displaystyle\leq\int_{0}^{\infty}|\widehat{u}(\zeta)|^{2}\zeta^{3}\frac{d\zeta}{4\pi^{2}}\|f\|_{L^{2}}^{2}=\frac{1}{2\pi}\|u\|_{\dot{H}^{3/2}}^{2}\|f\|_{L^{2}}^{2}.$

By density, this bound extends to all $f\in L^{2}_{+}(\mathbb{R})$. This
proves the first claim of the lemma.

For the second statement, we note that this follows from the first statement
combined with Sobolev embeddings and the identity

$\partial_{xx}(H_{u}\partial_{x}v)=H_{u}\partial_{x}(\partial_{xx}v)+2H_{\partial_{x}u}\partial_{x}v+H_{\partial_{xx}u}\partial_{x}v.$

This completes the proof of Lemma 2.1. ∎

In view of (2.1), we consider the following iteration scheme

$\partial_{t}u^{k+1}=i\partial_{xx}u^{k+1}+2T_{u^{k}}T_{\overline{u}^{k}}\partial_{x}u^{k+1}+2u^{k}H_{u^{k}}\partial_{x}u^{k}$
(2.3)

with initial datum $u^{k+1}(0,x)=u_{0}(x)\in H^{2}_{+}(\mathbb{R})$. Notice
that $T_{u}T_{\overline{u}}$ is a self-adjoint operator. Hence a standard
energy methods yields the following result.

######  Lemma 2.2.

Let $u\in C([-T,T],H^{2}_{+}(\mathbb{R}))$ with some $T>0$, $p\in\\{0,1,2\\}$,
and $w_{0}\in H^{p}_{+}(\mathbb{R})$, $f\in
L^{1}([-T,T];H^{p}_{+}(\mathbb{R}))$. Then there exists a unique $w\in
C([-T,T];H^{p}_{+}(\mathbb{R}))$ such that

$\partial_{t}w=i\partial_{xx}w+2T_{u}T_{\overline{u}}w+f,\quad
w(0,x)=w_{0}(x).$

Furthermore,

$\sup_{|t|\leq T}\|w(t)\|_{H^{p}}\leq
C\mathrm{e}^{C\int_{-T}^{T}\|u(t)\|_{H^{2}}^{2}\,dt}\left(\|w_{0}\|_{H^{p}}+\|f\|_{L^{1}_{t}H^{p}}\right).$

Coming back to the scheme (2.3), we see that Lemmas 2.1 and 2.2 allow us to
construct by induction a sequence $(u^{k})$ in
$C(\mathbb{R};H^{2}_{+}(\mathbb{R}))$ with $u^{0}(t,x)=u_{0}(x)$. We are now
going to prove that if $\|u_{0}\|_{H^{2}}\leq R$ and $T(R)>0$ suitably chosen,
then the sequence $(u^{k})$ is bounded in $H^{2}_{+}(\mathbb{R})$ and
uniformly convergent in $L^{2}_{+}(\mathbb{R})$ for $|t|\leq T(R)$.

Let us first prove that $(u^{k})$ is bounded in $H^{2}_{+}(\mathbb{R})$ for
$|t|\leq T(R)$ with suitably chosen $T(R)>0$. Indeed, by using the second
estimate in Lemma 2.1 together with the bound in Lemma 2.2 for $p=2$, we
obtain

$\sup_{|t|\leq T}\|u^{k+1}(t)\|_{H^{2}}\leq
C\mathrm{e}^{C\int_{-T}^{T}\|u^{k}(t)\|_{H^{2}}^{2}\,dt}\left(\|u_{0}\|_{H^{2}}+\int_{-T}^{T}\|u^{k}(t)\|_{H^{2}}^{3}\,dt\right).$

Assume $\|u_{0}\|_{H^{2}}\leq R$ and let $R_{1}=(1+C)R$. Since $R_{1}>CR$, we
can choose $T=T(R)>0$ such that

$C\mathrm{e}^{2CTR_{1}^{2}}(R+2TR_{1}^{3})\leq R_{1}.$

By an elementary induction argument, we find that $\sup_{|t|\leq
T(R)}\|u^{k}(t)\|_{H^{2}}\leq R_{1}$ for all $k$.

Next, we show that we have a contraction property of the sequence $(u^{k})$ in
$L^{2}_{+}(\mathbb{R})$ for $|t|\leq T(R)$ as follows. Observe that

$\displaystyle\partial_{t}(u^{k+1}-u^{k})=i\partial_{xx}(u^{k+1}-u^{k})+2T_{u^{k}}T_{\overline{u}^{k}}\partial_{x}(u^{k+1}-u^{k})+$
$\displaystyle+2(T_{u^{k}}T_{\overline{u}k}-2T_{u^{k-1}}T_{\overline{u}^{k-1}})\partial_{x}u^{k}+2u^{k}H_{u^{k}}\partial_{x}u^{k}-2u^{k-1}H_{u^{k-1}}\partial_{x}u^{k-1}.$

Using the estimate of Lemma 2.2 with $p=0$ and the bound on $u^{k}$ in
$H^{2}$, we infer

$\sup_{|t|\leq T}\|u^{k+1}(t)-u^{k}(t)\|_{L^{2}}\leq KT\sup_{|t|\leq
T}\|u^{k}(t)-u^{k-1}(t)\|_{L^{2}}$

with some constant $K>0$. If we choose $T=T(R)>0$ from above small enough to
ensure that $KT<1$, then the series $\sup_{|t|\leq
T(R)}\|u^{k+1}(t)-u^{k}(t)\|_{L^{2}}$ is geometrically convergent.

Finally, the sequence $(u^{k})$ is uniformly weakly convergent in
$C([-T,T];H^{2}_{+}(\mathbb{R}))$ and strongly convergent in
$C([-T,T];L^{2}_{+}(\mathbb{R}))$. Hence its limit $u(t)$ solves (2.1) – and
therefore (CM-DNLS). (To prove that the limit $u(t)$ actually belongs to
$C([-T,T];H^{2}_{+}(\mathbb{R}))$, we can invoke Tao’s frequency envelope
method [35] or adapt an argument due to Bona-Smith [8].)

Uniqueness follows along the same lines as the contraction property in
$L^{2}_{+}(\mathbb{R})$. The proof of Proposition 2.1 is now complete. ∎

Following the analysis in [13], we can further lower the regularity for local
well-posedness to initial data in $H^{s}_{+}(\mathbb{R})$ with
$s>\nicefrac{{1}}{{2}}$. In particular, we can reach the energy space
$H^{1}_{+}(\mathbb{R})$ for (CM-DNLS). In fact, the arguments adapt Tao’s
frequency localized gauge transform introduced to treat low regularity
solutions for the Benjamin-Ono equation. For (CM-DNLS), we obtain the
following local well-posedness result.

######  Theorem 2.1 (Local Well-Posedness in $H^{s}_{+}$ with
$s>\nicefrac{{1}}{{2}}$).

Let $u_{0}\in H^{s}_{+}(\mathbb{R})$ with some $s>\nicefrac{{1}}{{2}}$. Then
there exist a time $T=T(\|u_{0}\|_{H^{s}})>0$ and some Banach space
$Z_{s,T}\subset C([-T,T];H^{s}_{+}(\mathbb{R}))$ and a unique solution $u\in
Z_{s,T}$ of (CM-DNLS) with initial datum $u(0)=u_{0}$.

Furthermore, the $H^{\sigma}$-regularity of $u_{0}$ for $\sigma>s$ is
propagated on the whole maximal interval of existence of $u$, and the flow map
$u_{0}\mapsto u(t)$ is continuous on $H^{s}$.

######  Remark.

Recall that (CM-DNLS) is $L^{2}$-critical with respect to scaling. It remains
a fundamental open problem to understand the case $u\in H^{s}_{+}(\mathbb{R})$
when $0\leq s\leq\nicefrac{{1}}{{2}}$.

###### Proof.

We can adapt the estimates proven in [13] to our case. Suppose that $u_{0}\in
H^{s}_{+}(\mathbb{R})$ with some $s>\nicefrac{{1}}{{2}}$. For $\varepsilon>0$,
let
$\eta_{\varepsilon}(x)=\sqrt{\frac{1}{\varepsilon\pi}}\mathrm{e}^{-|x|^{2}/\varepsilon}$
be a Gaussian mollifier. Then $u_{0,\varepsilon}(x)=(\eta_{\varepsilon}\ast
u_{0})(x)$ satisfies $u_{0,\varepsilon}\in H^{\infty}_{+}(\mathbb{R})\subset
H^{2}_{+}(\mathbb{R})$. By Proposition 2.1, there exists a unique solution
$u_{\varepsilon}\in
C([-T_{\varepsilon},T_{\varepsilon}];H^{2}_{+}(\mathbb{R}))$ with
$u_{\varepsilon}(0)=u_{0,\varepsilon}$. We can now apply the arguments in the
proof of Theorem 1.1 in [13]. First, we can show that there exists
$T=T(\|u_{0}\|_{H^{s}})>0$ satisfying $T_{\varepsilon}\geq T$ for all
$\varepsilon>0$. Then following Proposition 3.2 in [13] we see that
$(u_{\varepsilon})$ is Cauchy in $Z_{s,T}$ as $\varepsilon\to 0$; we refer to
[13] for the definition of the Banach space $Z_{s,T}$. Finally, the uniqueness
of the limit of $(u_{\varepsilon})$ can be proven by the estimate (3.40) in
[13]. ∎

###  2.2. Lax Pair and Conservation Laws

In this subsection, we will show that (CM-DNLS) admits a Lax pair with certain
densely defined operators $L_{u}$ and $B_{u}$ acting on the Hardy space
$L^{2}_{+}(\mathbb{R})$. Here we will exploit this fact to derive an infinite
hierachy of conservation laws. For an analysis of the spectral properties of
$L_{u}$, we refer to Section 6 below.

For $u\in H^{s}_{+}(\mathbb{R})$ with some $s\geq 0$, we formally define the
operators $L_{u}$ and $B_{u}$ acting on $L^{2}_{+}(\mathbb{R})$ by setting

$\boxed{L_{u}=D-T_{u}T_{\bar{u}}\quad\mbox{and}\quad
B_{u}=T_{u}T_{\partial_{x}\bar{u}}-T_{\partial_{x}u}T_{\bar{u}}+i(T_{u}T_{\bar{u}})^{2}}$
(2.4)

Here $T_{b}(f)=\Pi_{+}(bf)$ denotes the Toeplitz operator on
$L^{2}_{+}(\mathbb{R})$ with symbol $b\in
L^{2}(\mathbb{R})+L^{\infty}(\mathbb{R})$. For $u\in
H^{1}_{+}(\mathbb{R})\subset L^{\infty}(\mathbb{R})$, we readily check that
$T_{u}$ and $T_{\bar{u}}$ are bounded operators on $L^{2}_{+}(\mathbb{R})$.
Thus, for $u\in H^{1}_{+}(\mathbb{R})$, it is straightforward to verify that
$L_{u}$ is semibounded and a self-adjoint operator, i. e.,

$L_{u}^{*}=L_{u}$

with operator domain $\mathrm{dom}(L_{u})=H^{1}_{+}(\mathbb{R})$. For $u\in
H^{2}_{+}(\mathbb{R})$, we readily check that $B_{u}=-B_{u}^{*}$ is a skew-
adjoint and bounded operator on $L^{2}_{+}(\mathbb{R})$.

######  Remark 2.1.

In Appendix A below, we detail how $L_{u}$ can be defined via quadratic forms
if we only assume that $u\in L^{2}_{+}(\mathbb{R})$, which is a natural class
in view of the $L^{2}$-criticality of (CM-DNLS).

Next, we see that $L_{u}$ and $B_{u}$ form indeed a Lax pair for the
Calogero–Moser DNLS.

######  Lemma 2.3 (Lax Equation).

If $u\in C([0,T];H^{s}_{+}(\mathbb{R}))$ solves (CM-DNLS) with $s\geq 0$
sufficiently large (e. g. with $s=2$), then it holds

$\frac{d}{dt}L_{u}=[B_{u},L_{u}].$

###### Proof.

We divide the proof into the following steps.

Step 1. We first calculate the commutators

$I:=[T_{u}T_{\partial_{x}\bar{u}},D],\quad
II:=[T_{\partial_{x}u}T_{\bar{u}},D],\quad III:=i[(T_{u}T_{\bar{u}})^{2},D].$

We find

$\displaystyle I$
$\displaystyle=T_{u}[T_{\partial_{x}\bar{u}},D]+[T_{u},D]T_{\partial_{x}\bar{u}}=T_{u}T_{i\partial_{xx}\bar{u}}+T_{i\partial_{x}u}T_{\partial_{x}\bar{u}},$
$\displaystyle II$
$\displaystyle=T_{\partial_{x}u}[T_{\bar{u}},D]+[T_{\partial_{x}u},D]T_{\bar{u}}=T_{\partial_{x}u}T_{i\partial_{x}\bar{u}}+T_{i\partial_{xx}u}T_{\bar{u}}.$

In addition, we see

$\displaystyle III$
$\displaystyle=iT_{u}T_{\bar{u}}[T_{u}T_{\bar{u}},D]+i[T_{u}T_{\bar{u}},D]T_{u}T_{\bar{u}}$
$\displaystyle=iT_{u}T_{\bar{u}}\left(T_{u}T_{i\partial_{x}\bar{u}}+T_{i\partial_{x}u}T_{\bar{u}}\right)+i\left(T_{u}T_{i\partial_{x}\bar{u}}+T_{i\partial_{x}u}T_{\bar{u}}\right)T_{u}T_{\bar{u}}.$

As a next step, we consider the terms

$IV:=[T_{u}T_{\partial_{x}\bar{u}},T_{u}T_{\bar{u}}],\quad
V:=[T_{\partial_{x}u}T_{\bar{u}},T_{u}T_{\bar{u}}],\quad
VI=i[(T_{u}T_{\bar{u}})^{2},T_{u}T_{\bar{u}}].$

We find

$IV=T_{u}T_{\partial_{x}\bar{u}}T_{u}T_{\bar{u}}-T_{u}T_{\bar{u}}T_{u}T_{\partial_{x}\bar{u}},\quad
V=T_{\partial_{x}u}T_{\bar{u}}T_{u}T_{\bar{u}}-T_{u}T_{\bar{u}}T_{\partial_{x}u}T_{\bar{u}},\\\
$

and, clearly, we have $VI=0$. If we combine all commutator terms, we see

$\displaystyle[B_{u},L_{u}]$ $\displaystyle=I-II+III-IV+V$
$\displaystyle=T_{u}T_{i\partial_{x}x\bar{u}}-T_{i\partial_{xx}u}T_{\bar{u}}-2T_{u}T_{\bar{u}}T_{\partial_{x}u}T_{\bar{u}}-2T_{u}T_{\partial_{x}\bar{u}}T_{u}T_{\bar{u}}$
$\displaystyle=T_{u}T_{i\partial_{xx}\bar{u}}-T_{i\partial_{xx}u}T_{\bar{u}}-2T_{u}T_{\partial_{x}|u|^{2}}T_{\bar{u}},$

where in the last step we used that $u\in H^{1}_{+}(\mathbb{R})$, which
implies that

$T_{\bar{u}}T_{\partial_{x}u}+T_{\partial_{x}\bar{u}}T_{u}=T_{\bar{u}\partial_{x}u+\partial_{x}\bar{u}u}=T_{\partial_{x}|u|^{2}}$

holds on $L^{2}_{+}(\mathbb{R})$.

Step 2. We now calculate

$\displaystyle\frac{d}{dt}L_{u}$
$\displaystyle=-T_{\dot{u}}T_{\bar{u}}-T_{u}T_{\dot{\bar{u}}}$
$\displaystyle=-T_{i\partial_{xx}u}T_{\bar{u}}-2T_{\Pi_{+}(\partial_{x}|u|^{2})u}T_{\bar{u}}-T_{u}T_{-i\partial_{xx}\bar{u}}-2T_{u}T_{\overline{\Pi_{+}(\partial_{x}|u|^{2})}\bar{u}}$

In view of the expression for $[B_{u},L_{u}]$ derived in Step 1, it remains to
show the identity

$T_{\Pi_{+}(\partial_{x}|u|^{2})u}T_{\bar{u}}+T_{u}T_{\overline{\Pi_{+}(\partial_{x}|u|^{2})}\bar{u}}=T_{u}T_{\partial_{x}|u|^{2}}T_{\bar{u}}$
(2.5)

Indeed, using that $u\in H^{1}_{+}(\mathbb{R})$, we find

$T_{\Pi_{+}(\partial_{x}|u|^{2})u}=T_{u}T_{\Pi_{+}(\partial_{x}|u|^{2})}\quad\mbox{and}\quad
T_{\overline{\Pi_{+}(\partial_{x}|u|^{2})}\bar{u}}=T_{\overline{\Pi_{+}(\partial_{x}|u|^{2})}}T_{\bar{u}}.$

Since
$\partial_{x}|u|^{2}=\Pi_{+}(\partial_{x}|u|^{2})+\overline{\Pi_{+}(\partial_{x}|u|^{2})}$,
we deduce that (2.5) holds true. This completes the proof of Lemma 2.3. ∎

######  Remark 2.2.

Since we have that $[L_{u},L_{u}^{2}]=0$, the skew-adjoint operator

$\widetilde{B}_{u}=B_{u}-iL_{u}^{2}$

also satisfies the Lax equation $\frac{d}{dt}L_{u}=[\widetilde{B}_{u},L_{u}]$.
A direct calculation shows that

$\widetilde{B}_{u}=-iD^{2}+2iT_{u}DT_{\bar{u}}$ (2.6)

Note that $\widetilde{B}_{u}$ is an unbounded skew-adjoint operator on
$L^{2}_{+}(\mathbb{R})$ with operator domain
$\mathrm{dom}(\widetilde{B}_{u})=H^{2}_{+}(\mathbb{R})$. Actually, we first
found the operator $\widetilde{B}_{u}$ in the analysis of the Lax structure of
(CM-DNLS). We also note that the relation between $B_{u}$ and
$\widetilde{B}_{u}$ is reminiscent to the Lax structure for the Benjamin-Ono
equation used in [16].

As a consequence of the Lax equation, we obtain an infinite hierarchy of
conservation of laws for (CM-DNLS) as follows

######  Lemma 2.4 (Hierarchy of Conservation Laws).

Let $u\in C([0,T],H^{s}_{+}(\mathbb{R}))$ be a solution of (CM-DNLS) with
sufficiently large $s\geq 0$. Then, for every
$\lambda\not\in\sigma(L_{u(0)})$, we have the conserved quantity

$\mathcal{H}_{\lambda}(u):=\langle(L_{u}-\lambda I)^{-1}u,u\rangle.$

As a consequence, if $u\in C([0,T];H^{n/2}_{+}(\mathbb{R}))$ with some
$n\in\mathbb{N}$, the quantities

$I_{k}(u):=\langle L_{u}^{k}u,u\rangle\quad\mbox{with $k=0,\ldots,n$}$

are conserved, where $\langle\cdot,\cdot\rangle$ denotes the dual pairing of
$H^{-n/2}_{+}$ and $H^{n/2}_{+}$.

###### Proof.

Let $\lambda\not\in\sigma(L_{u_{0}})$, which by the Lax equation implies that
$\lambda\not\in\sigma(L_{u})$ for all $u=u(t)$ with $t\in[0,T]$. A quick
calculation reveals that (CM-DNLS) can be written as

$\partial_{t}u=\widetilde{B}_{u}u$

with the operator $\widetilde{B}_{u}=B_{u}-iL_{u}^{2}$; see also (2.6) above.
Using that $\frac{d}{dt}L_{u}=[\widetilde{B}_{u},L_{u}]$, it is elementary to
verify that

$\frac{d}{dt}\mathcal{H}_{\lambda}(u)=0.$

Finally, we note the expansion

$\mathcal{H}_{\lambda}(u)=\sum_{k=0}^{\infty}\lambda^{-(k+1)}\langle
L^{k}_{u}u,u\rangle$

for all real $\lambda<0$ sufficiently negative (using that $L_{u}$ is bounded
below) and with $u\in H^{\infty}_{+}(\mathbb{R})=\cap_{s\geq
0}H^{s}_{+}(\mathbb{R})$. Thus we deduce that $I_{k}(u)=\langle
L^{k}_{u}u,u\rangle$ are constant in time for solutions in
$H^{\infty}_{+}(\mathbb{R})$. The conservation laws $I_{0}(u),\ldots,I_{n}(u)$
for solutions $u\in C([0,T];H^{n/2}_{+}(\mathbb{R}))$ follow from an
approximation argument, which we omit. ∎

###  2.3. Global Existence for sub-critical $L^{2}$-Mass

As an application of Lemma 2.4, we deduce the following global-in-time
existence result.

######  Corollary 2.1 (Global Existence and Bounds for sub-critical
$L^{2}$-Mass).

Let $u_{0}\in H^{s}_{+}(\mathbb{R})$ with some integer $s\geq 1$ and suppose
that

$M(u_{0})<M(\mathcal{R})=2\pi.$

Then the corresponding solution $u(t)\in H^{s}_{+}(\mathbb{R})$ of (CM-DNLS)
exists for all times $t\in\mathbb{R}$ with the a-priori bound

$\sup_{t\in\mathbb{R}}\|u(t)\|_{H^{s}}<+\infty.$

######  Remarks.

1) The condition $s\geq 1$ arises from the current state of the local well-
posedness theory for (CM-DNLS). It is conceivable that, with some great effort
though, that this result extends to initial data in
$H^{0}_{+}(\mathbb{R})=L^{2}_{+}(\mathbb{R})$ satisfying
$\|u_{0}\|_{L^{2}}^{2}<2\pi$.

2) From Theorem 1.3 we deduce that the infinite hierarchy of conservation laws

$I_{k}(u(t))=I_{k}(u_{0})\quad\mbox{with $k=1,2,\ldots$}$

fail in general to produce a-priori bounds on $H^{s}$-norms for $s>0$ for
solutions with initial data with $L^{2}$-mass $M(u_{0})\geq
M(\mathcal{R})=2\pi$.

###### Proof.

We first consider the case $s=1$. By the local well-posedness theory in
$H^{1}_{+}(\mathbb{R})$, we need to find an a-priori bound on $\sup_{t\in
I}\|u(t)\|_{H^{1}}$, where $I\subset\mathbb{R}$ denotes the maximal time
interval of existence. Indeed, we have the conservation laws

$I_{0}(u)=\langle u,u\rangle=\|u\|_{L^{2}}^{2},$ $I_{1}(u)=\langle
L_{u}u,u\rangle=\langle Du,u\rangle-\|T_{\bar{u}}u\|_{L^{2}}^{2}.$

Now, we use the sharp inequality (see Lemma A.1):

$\|T_{\bar{u}}u\|_{L^{2}}^{2}\leq\frac{1}{2\pi}\|u\|_{L^{2}}^{2}\langle
Du,u\rangle.$

Therefore if we assume that $\|u_{0}\|_{L^{2}}^{2}<2\pi$, we deduce the
a-priori bound

$\sup_{t\in I}\|u(t)\|_{H^{1/2}}\leq C(u_{0}).$

Next, we use the conservation of energy together with a standard
Gagliardo–Nirenberg interpolation and Sobolev inequality:

$\displaystyle E(u)$ $\displaystyle=2I_{2}(u)=\langle
L^{2}_{u}u,u\rangle=\langle
Du,Du\rangle-\frac{1}{2}\langle|u|^{2},|D||u|^{2}\rangle+\frac{1}{12}\|u\|_{L^{6}}^{6}$
$\displaystyle\geq\|\partial_{x}u\|_{L^{2}}^{2}-C\|u\|_{L^{6}}^{3}\|\partial_{x}u\|_{L^{2}}\geq\|\partial_{x}u\|_{L^{2}}^{2}-C\|u\|_{H^{1/2}}^{3}\|\partial_{x}u\|_{L^{2}}.$

From the a-priori bound on $\|u(t)\|_{H^{1/2}}$, we readily infer that
$\sup_{t\in I}\|\partial_{x}u(t)\|_{L^{2}}\leq C(u_{0})$. This completes the
proof for $s=1$.

The remaining case of integer $s\geq 2$ follows by iteration and using the
conserved quantities

$I_{k}(u)=\langle L^{k}u,u\rangle=\|u\|_{\dot{H}^{k/2}}^{2}+\mbox{lower order
terms}$

with $k=0,\ldots,2s$. We omit the details. ∎

###  2.4. Proof of Theorem 1.1 for $M(u_{0})<M(\mathcal{R})$

The assertions in Theorem 1.1 in the case $M(u_{0})<M(\mathcal{R})$ follow
directly from Corollary 2.1. ∎

The critical case $M(u_{0})=M(\mathcal{R})$ will be discussed in the following
section.

##  3\. Nonexistence of Minimal Mass Blowup

The goal of this section is to rule out finite-time minimal mass blowup for
(CM-DNLS) with finite energy. As a consequence, we obtain that initial data
$u(0)\in H^{s}_{+}(\mathbb{R})$ with some $s\geq 1$ with critical $L^{2}$-mass

$M(u(0))=M(\mathcal{R})=2\pi$

will always lead to global-in-time solutions $u\in
C(\mathbb{R};H^{s}_{+}(\mathbb{R}))$, completing the proof of Theorem 1.1.

Notice that the absence of minimal mass blowup is in striking contrast to
focusing $L^{2}$-critical NLS, where the existence of minimal mass blowup is a
direct consequence of applying the pseudo-conformal transform to ground state
solitary waves. For (CM-DNLS) on the other hand, we will see below that the
mechanism that prevents the existence of minimal mass blowup is due to the
slow algebraic decay of ground states $\mathcal{R}\in H^{1}_{+}(\mathbb{R})$
with $|\mathcal{R}(x)|\sim\frac{1}{|x|}$ as $|x|\to+\infty$.

We begin with the $L^{2}$-tightness property for $H^{1}$-solutions on finite
time intervals.

######  Lemma 3.1.

Let $I\subset\mathbb{R}$ be an interval of finite length $|I|<\infty$ and
suppose that $u\in C(I,H^{1}(\mathbb{R}))$ solves (CM-DNLS). Then the family
$\\{u(t)\\}_{t\in I}$ is tight in $L^{2}(\mathbb{R})$, i. e., for every
$\varepsilon>0$ there exists $R=R(\varepsilon)>0$ such that

$\int_{|x|\geq R}|u(t,x)|^{2}\,dx\leq\varepsilon\quad\mbox{for all $t\in I$}.$

######  Remarks.

1) It should be noted that no assumption on the size of the solution $u(t)$,
i. e., we do not assume that $M(u_{0})$ is sufficiently small.

2) Notice that we do not assume that $u(t)$ belongs to the Hardy space
$H^{1}_{+}(\mathbb{R})$, but in fact we allow for general $H^{1}$-valued
solutions of (CM-DNLS).

###### Proof.

The idea is to adapt an elegant argument in [6] developed for studying minimal
mass blowup for $L^{2}$-critical NLS. Let $\psi\in C^{\infty}(\mathbb{R})$ be
a smooth real-valued function with bounded derivative $\partial_{x}\psi\in
L^{\infty}(\mathbb{R})$. For any $a\in\mathbb{R}$ and $u\in
H^{1}(\mathbb{R})$, the non-negativity of the energy implies that

$E(\mathrm{e}^{ia\psi}u)\geq 0.$

Expanding the right-hand side, we find

$\displaystyle E(\mathrm{e}^{ia\psi}u)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}(\mathrm{e}^{ia\psi}u)-i\Pi_{+}(|u|^{2})\mathrm{e}^{ia\psi}u|^{2}\,dx$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}|ia\partial_{x}\psi
u+\partial_{x}u-i\Pi_{+}(|u|^{2})u|^{2}\,dx$
$\displaystyle=\frac{a^{2}}{2}\int_{\mathbb{R}}|\partial_{x}\psi|^{2}|\partial_{x}u|^{2}\,dx+a\mathrm{Re}\,\langle
i\partial_{x}\psi u,\partial_{x}u-i\Pi_{+}(|u|^{2})u\rangle+E(u).$

Now we observe that

$\mathrm{Re}\,\langle i\partial_{x}\psi
u,\partial_{x}u\rangle=\int_{\mathbb{R}}\partial_{x}\psi\cdot\mathrm{Im}(\overline{u}\partial_{x}u)\,dx,$
$\mathrm{Re}\langle i\partial_{x}\psi
u,-i\Pi_{+}(|u|^{2})u\rangle=-\frac{1}{2}\mathrm{Re}\langle\partial_{x}\psi
u,(1+i\mathsf{H})(|u|^{2})u\rangle=-\frac{1}{2}\int_{\mathbb{R}}\partial_{x}\psi|u|^{4}\,dx,$

using that $\Pi_{+}=\frac{1}{2}(1+iH)$ and $\mathrm{Re}\langle
u,iH(|u|^{2})u\rangle=0$ since $H(|u|^{2})$ is real-valued. Therefore, the
quadratic expansion in $a$ together with $E(\mathrm{e}^{ia\psi}u)\geq 0$
implies that

$\left|\int_{\mathbb{R}}\partial_{x}\psi\left(\mathrm{Im}(\overline{u}\partial_{x}u)-\frac{1}{2}|u|^{4}\right)dx\right|\leq\sqrt{2E_{0}}\left(\int_{\mathbb{R}}|\partial_{x}\psi|^{2}|\partial_{x}u|^{2}\,dx\right)^{1/2},$
(3.1)

with the energy $E_{0}=E(u)\geq 0$.

Next, we apply (3.1) to obtain the claimed $L^{2}$-tightness bound. From (CM-
DNLS) we deduce

$\partial_{t}|u|^{2}=-2\partial_{x}\left(\mathrm{Im}(\overline{u}\partial_{x}u)-\frac{1}{2}|u|^{4}\right)$
(3.2)

in view of
$\mathrm{Re}(\overline{u}(2iD_{+}|u|^{2})u)=\mathrm{Re}((\partial_{x}|u|^{2})|u|^{2})=\frac{1}{2}\partial_{x}|u|^{4}$.
Now let $\chi$ be a smooth nonnegative function such that $\chi(x)\equiv 0$
for $|x|\leq 1/2$ and $\chi(x)\equiv 1$ for $|x|\geq 1$. For $R>0$, we set
$\chi_{R}(x)=\chi(x/R)$. Integrating by parts and using (3.2) and (3.1), we
infer that

$\displaystyle\left|\frac{d}{dt}\int_{\mathbb{R}}\chi_{R}|u|^{2}\,dx\right|$
$\displaystyle=2\left|\int_{\mathbb{R}}\partial_{x}\chi_{R}\left(\mathrm{Im}(\overline{u}\partial_{x}u)-\frac{1}{2}|u|^{4}\right)\,dx\right|$
(3.3)
$\displaystyle\lesssim\sqrt{E_{0}}\left(\int_{\mathbb{R}}|\partial_{x}\chi_{R}|^{2}|u|^{2}\,dx\right)^{1/2}\lesssim\frac{\sqrt{E_{0}M_{0}}}{R}$
(3.4)

with the conserved $L^{2}$-mass $M_{0}=\|u\|_{L^{2}}^{2}$. If we integrate
this bound over the finite time interval $I\subset\mathbb{R}$ with some
$t_{0}\in I$ fixed, we finally obtain

$\int_{\mathbb{R}}\chi_{R}(x)|u(t,x)|^{2}\,dx\leq\int_{\mathbb{R}}\chi_{R}(x)|u(t_{0},x)|^{2}\,dx+\frac{C|I|}{R},$

for all $t\in I$. This readily implies the claimed tightness bound. ∎

######  Theorem 3.1 (No Minimal Mass Blowup).

Let $I\subset\mathbb{R}$ with $0\in I$ and finite length $|I|<\infty$. Suppose
$u\in C(I;H^{1}(\mathbb{R}))$ solves (CM-DNLS) with
$M(u_{0})=M(\mathcal{R})=2\pi$. Then it holds

$\sup_{t\in I}\|u(t)\|_{H^{1}}<+\infty.$

######  Remark.

1) Note again that we allow for general $H^{1}$-valued solutions $u(t)$.

###### Proof.

We argue by contradiction. Without loss of generality, we can assume that
$I=[0,T)$ with some finite time $T\in(0,+\infty)$ and let $u\in
C([0,T);H^{1}(\mathbb{R}))$ satisfy

$\lim_{t\to T^{-}}\|u(t)\|_{H^{1}}=+\infty.$

Step 1. We first show that $u(t)$ must have finite variance, i. e., we have

$\int_{\mathbb{R}}|x|^{2}|u(t,x)|^{2}\,dx<+\infty\quad\mbox{for $t\in[0,T)$.}$
(3.5)

Here we adapt a strategy developed for classifying minimal-mass finite-time
blowup solutions for $L^{2}$-critical NLS; originally due to Merle in [25]. To
prove the claim (3.5), we follow the arguments laid out in [19, 6].

Let $t_{n}\to T^{-}$ be a sequence of times. We define

$\varepsilon_{n}:=\frac{1}{\|\partial_{x}u(t_{n})\|_{L^{2}}}\ ,\
v_{n}(x):=\varepsilon_{n}^{1/2}u(t_{n},\varepsilon_{n}x)\ .$

Applying the Minimal Mass Bubble Lemma B.1 from Appendix B to $v_{n}$, after
passing to a subsequence if necessary, there exist sequences
$x_{n}\in\mathbb{R}$ and $\lambda_{n}>0$ such that $\lambda_{n}\to 0$ and

$\lambda_{n}^{1/2}u(t_{n},\lambda_{n}(x+x_{n}))\to\mathrm{e}^{i\theta}\mathcal{R}(x)\quad\mbox{strongly
in $L^{2}(\mathbb{R})$}$

for some $\theta\in[0,2\pi[$, with the ground state $\mathcal{R}\in
H^{1}_{+}(\mathbb{R})$ minimizing the energy functional $E(u)$ on
$H^{1}(\mathbb{R})$. Thus we obtain

$|u(t_{n},x)|^{2}\,dx-\|\mathcal{R}\|_{L^{2}}^{2}\delta_{x=\lambda_{n}x_{n}}\rightharpoonup
0$

in the weak sense of measures. By the $L^{2}$-tightness property in Lemma 3.1,
we easily deduce that $\lambda_{n}|x_{n}|\leq C$ with some constant $C>0$.
From this fact (and passing to a subsequence) and by translational invariance
we can henceforth assume that $\lambda_{n}x_{n}\to 0$ holds.

Next, let $\psi\in C^{\infty}_{0}(\mathbb{R})$ be a non-negative function such
that $\psi(x)\equiv|x|^{2}$ for $|x|<1$ and $|\partial_{x}\psi(x)|^{2}\leq
C\psi(x)$ with some constant $C>0$. For $R>0$, we set
$\psi_{R}(x)=R^{2}\psi(x/R)$ and we define

$g_{R}(t)=\int_{\mathbb{R}}\psi_{R}(x)|u(t,x)|^{2}\,dx.$

In analogy to (3.3) based on (3.1) we find

$\left|\frac{d}{dt}{g}_{R}(t)\right|\lesssim\int_{\mathbb{R}}|\partial_{x}\psi_{R}|^{2}|u|^{2}\,dx\lesssim\sqrt{g_{R}(t)}$
(3.6)

where the last step used that $|\partial_{x}\psi_{R}|^{2}\lesssim\psi_{R}$ by
construction. By integrating this on $[t,t_{n}]$ and using that
$g_{R}(t_{n})\to 0$ as $n\to\infty$, we deduce

$g_{R}(t)=\int_{\mathbb{R}}\psi_{R}(x)|u(t,x)|^{2}\,dx\lesssim(T-t)^{2}\quad\mbox{for
$t\in[0,T)$}.$

Passing to the limit $R\to+\infty$, this yields

$\int_{\mathbb{R}}|x|^{2}|u(t,x)|^{2}\,dx\lesssim(T-t)^{2}\quad\mbox{for
$t\in[0,T)$}.$ (3.7)

In particular, this implies that (3.5) holds, showing that $u(t)$ has finite
variance.

Step 2. By Step 1, we have $u_{0}=u(0)\in\Sigma=H^{1}(\mathbb{R})\cap
L^{2}(\mathbb{R};|x|^{2}dx)$ and thus we can apply the pseudo-conformal
identity (see Lemma C.1) to conclude that

$8t^{2}E(\mathrm{e}^{i|x|^{2}/4t}u_{0})=\int_{\mathbb{R}}|x|^{2}|u(t,x)|^{2}\,dx\lesssim(T-t)^{2}.$
(3.8)

If we pass to the limit $t\to T^{-}$, we obtain

$E(\mathrm{e}^{i|x|^{2}/4T}u_{0})=0.$ (3.9)

Since $\|\mathrm{e}^{i|x|^{2}/4T}u_{0}\|_{L^{2}}^{2}=2\pi$, the uniqueness
result in Lemma 4.1 implies that

$\mathrm{e}^{i|x|^{2}/4T}u_{0}(x)=\mathcal{R}(x)$ (3.10)

up to translation, phase, and scaling. However, the fact that
$\mathrm{e}^{i|x|^{2}/4T}u_{0}\in\Sigma$ contradicts that $\mathcal{R}(x)$ has
infinite variance, i. e.,

$\int_{\mathbb{R}}|x|^{2}|\mathcal{R}(x)|^{2}\,dx=+\infty.$

This contradiction shows that any $u(t)\in H^{1}(\mathbb{R})$ solving (CM-
DNLS) with $M(u(0))=M(\mathcal{R})=2\pi$ cannot blowup in finite time. This
completes the proof of Theorem 3.1. Therefore, the proof of Theorem 1.1 is
also complete. ∎

##  4\. Ground States and Traveling Solitary Waves

In this section, we show that all ground states (minimizers) for the energy
functional $E(u)$ for (CM-DNLS) are given by the rational function

$\mathcal{R}(x)=\frac{\sqrt{2}}{x+i}\in H^{1}_{+}(\mathbb{R})$ (4.1)

modulo translation, phase, and scaling. As as a second main result of this
section, we prove that all traveling solitary waves in $H^{1}_{+}(\mathbb{R})$
for (CM-DNLS) are given by Galilean boosts (with positive velocity) of
$\mathcal{R}$ up to the symmetries just mentioned.

We start with the following key result, which shows uniqueness of non-trivial
solutions of the (first-order) Euler-Lagrange equation for minimizers of
$E(u)$.

######  Lemma 4.1 (Uniqueness of Ground States).

Suppose that $u\in H^{1}(\mathbb{R})$ with $u\not\equiv 0$ solves

$Du-\Pi_{+}(|u|^{2})u=0.$

Then it holds

$u(x)=\mathrm{e}^{i\theta}\lambda^{1/2}\mathcal{R}(\lambda x+y)$

with some $\theta\in[0,2\pi)$, $\lambda>0$, $y\in\mathbb{R}$, and
$\mathcal{R}\in H^{1}_{+}(\mathbb{R})$ is given by (4.1).

As a consequence, all minimizers $u\in H^{1}(\mathbb{R})\setminus\\{0\\}$ for
$E(u)$ are of the form $u=\mathcal{R}$ modulo phase, translation, and scaling.

######  Remarks.

1) Note we that only assume that $u\in H^{1}(\mathbb{R})$ and we obtain a
posteriori that $u\in H^{1}_{+}(\mathbb{R})$ due to its explicit form.

2) As an interesting aside, we remark that the equation for $u\in
H^{1}(\mathbb{R})$ above can be recast into the nonlocal Liouville equation on
the real line:

$|D|w=\mathrm{e}^{w}\quad\mbox{in $\mathbb{R}$}.$ (4.2)

To prove this claim (neglecting any technicalities of function spaces for
simplicity), we first apply the gauge transform introducing the function
$v=\mathrm{e}^{-i/2\int_{-\infty}^{x}|u|^{2}}u$, which leads to the equation
for $v$ given by

$\partial_{x}v+\frac{1}{2}\mathsf{H}(|v|^{2})v=0\quad\mbox{in $\mathbb{R}$},$
(4.3)

where we recall that $\mathsf{H}$ denotes the Hilbert transform. It is easy to
see non-trivial solutions $v$ are (up to a complex phase factor) strictly
positive $v>0$. Finally, if we define $w:\mathbb{R}\to\mathbb{R}$ by setting
$w=\log(v^{2})$, we readily check that $w$ solves (4.2). In [11, 37, 12] it is
proven that all solutions $w\in L^{1}(\mathbb{R};\frac{dx}{1+x^{2}})$ of (4.2)
are explicitly given by

$w(x)=\log\left(\frac{2\lambda}{1+\lambda^{2}(x-y)}\right)$ (4.4)

with some constants $\lambda>0$ and $y\in\mathbb{R}$. From this uniqueness
result for $w$, we could obtain the result of Lemma 4.1. However, we will
below give a self-contained (and short) uniqueness proof based on Hardy space
arguments and complex ODEs. This method also provides (yet) another proof of
the stated uniqueness result for $w$ solving (4.2). See also [2] for a recent
uniqueness result for solutions $w$ of the nonlocal Liouville equation
$|D|w=Ke^{w}$ in $\mathbb{R}$ with the prescribed function
$K:\mathbb{R}\to\mathbb{R}$.

###### Proof of Lemma 4.1.

We introduce the function

$w:=\Pi_{+}(|u|^{2})\in H^{1}_{+}(\mathbb{R}),$

so that $u^{\prime}=iuw$ and $|u|^{2}=w+\overline{w}$. We obtain the complex
ordinary differential equation

$w^{\prime}=\Pi_{+}(u^{\prime}\overline{u}+\overline{u}^{\prime}u)=\Pi_{+}(iw(w+\overline{w})-i\overline{w}(\overline{w}+w))=iw^{2},$

using that $\Pi_{+}(\overline{w}^{2})=0$ for $w\in H^{1}_{+}(\mathbb{R})$.
Since $w\not\equiv 0$, we deduce

$w(x)=\frac{i}{x-z}$

for some constant $z\in\mathbb{C}$ with $\mathrm{Im}\,z<0$ because $w\in
L^{2}_{+}(\mathbb{R})$. Consequently,

$u^{\prime}=iuw=-\frac{u}{x-z}.$

Thus we have $(x-z)u(x)=c$ with some constant $c\in\mathbb{C}$. To determine
$c$, we notice

$\frac{i}{x-z}=\Pi_{+}(|u|^{2})=\Pi_{+}\left(\frac{|c|^{2}}{|x-z|^{2}}\right)=\frac{|c|^{2}}{(\overline{z}-z)(x-z)}.$

This implies that $|c|^{2}=-2\mathrm{Im}\,z=2|\mathrm{Im}\,z|$. In summary, we
have found that

$u(x)=\mathrm{e}^{i\theta}\frac{\sqrt{2|\mathrm{Im}\,z|}}{x-z}$

with some $\theta\in[0,2\pi)$ and $z\in\mathbb{C}_{-}$. From this fact we
readily deduce that $u(x)=\mathrm{e}^{i\theta}\lambda^{1/2}\mathcal{R}(\lambda
x+y)$ with $\lambda=-(\mathrm{Im}z)^{-1}>0$ and $y=-\mathrm{Re}\,z$. ∎

Our next goal is to classify all traveling solitary wave solutions for (CM-
DNLS). By a traveling solitary waves (with finite energy), we mean solutions
of the form

$u(t,x)=\mathrm{e}^{i\omega t}\mathcal{R}_{v,\omega}(x-vt)$ (4.5)

where $\omega\in\mathbb{R}$ is a frequency parameter and $v\in\mathbb{R}$
denotes the velocity. Here the non-trivial profile $\mathcal{R}_{v,\omega}\in
H^{1}(\mathbb{R})$ is allowed to depend on $\omega$ and $v$. Note that
a-priori we allow also for $\mathcal{R}_{v,\omega}$ in $H^{1}(\mathbb{R})$ and
not just restricted to $H^{1}_{+}(\mathbb{R})$. We have the following complete
classification result, which shows that all traveling solitary waves for (CM-
DNLS) are generated by Galilean boosts, translations, scaling and phase
transformations of the ground state $\mathcal{R}(x)$.

######  Proposition 4.1.

Let $v,\omega\in\mathbb{R}$ and suppose $u(t,x)$ is a traveling solitary wave
for (CM-DNLS) with profile $\mathcal{R}_{v,\omega}\in
H^{1}(\mathbb{R})\setminus\\{0\\}$. Then it holds

$\omega=-\frac{v^{2}}{4}\quad\mbox{and}\quad\mathcal{R}_{v,\omega}(x)=\mathrm{e}^{\frac{i}{2}vx}\mathrm{e}^{i\theta}\lambda^{1/2}\mathcal{R}(\lambda
x+y)$

with some $\theta\in[0,2\pi),\lambda>0$, and $y\in\mathbb{R}$. Moreover, we
have $\mathcal{R}_{v,\omega}\in H^{1}_{+}(\mathbb{R})$ if and only if $v\geq
0$ holds.

###### Proof.

We divide the proof into the following steps.

Step 1. We easily check that $\mathcal{R}_{v,\omega}\in H^{1}(\mathbb{R})$
must solve

$-\partial_{xx}\mathcal{R}_{v,\omega}+iv\partial_{x}\mathcal{R}_{v,\omega}-2D_{+}(|\mathcal{R}_{v,\omega}|^{2})\mathcal{R}_{v,\omega}=\omega\mathcal{R}_{v,\omega}.$
(4.6)

The term $iv\partial_{x}\mathcal{R}_{v,\omega}$ can be removed by a Galilean
boost transform. That is, we write

$\mathcal{R}_{v,\omega}(x)=\mathrm{e}^{\frac{i}{2}vx}\mathcal{R}_{\tilde{\omega}}(x).$
(4.7)

An elementary calculation yields that $\mathcal{R}_{\tilde{\omega}}\in
H^{1}(\mathbb{R})$ satisfies

$-\partial_{xx}\mathcal{R}_{\tilde{\omega}}-2D_{+}(|\mathcal{R}_{\tilde{\omega}}|^{2})\mathcal{R}_{\tilde{\omega}}=\tilde{\omega}\mathcal{R}_{\tilde{\omega}}\quad\mbox{with}\quad\tilde{\omega}=\omega+\frac{v^{2}}{4}.$
(4.8)

Step 2. We claim that every solution $\mathcal{R}_{\tilde{\omega}}\in
H^{1}(\mathbb{R})$ of (4.8) has zero energy:

$E(\mathcal{R}_{\tilde{\omega}})=0.$ (4.9)

Since $\mathcal{R}_{\tilde{\omega}}\not\equiv 0$, we see from Lemma 4.1 that

$\mathcal{R}_{\tilde{\omega}}=\mathcal{R}$

modulo symmetries. As a consequence, we obtain $\tilde{\omega}=0$ and
therefore $\omega=-\frac{v^{2}}{4}$ as claimed.

The proof of (4.9) follows from applying the gauge transform together with a
Pohozaev-type argument. Since $D_{+}(|\mathcal{R}_{\tilde{\omega}}|^{2})\in
L^{2}(\mathbb{R})$, we notice that equation (4.8) tells us that
$\mathcal{R}_{\tilde{\omega}}\in H^{2}(\mathbb{R})$ holds. Next, we apply the
gauge transform $\Phi$ discussed in Appendix C to
$\mathcal{R}_{\tilde{\omega}}$. That is, we set

$S(x):=\Phi(\mathcal{R}_{\tilde{\omega}})(x)=\mathrm{e}^{-\frac{i}{2}\int_{-\infty}^{x}|\mathcal{R}_{\tilde{\omega}}(y)|^{2}\,dy}\mathcal{R}_{\tilde{\omega}}(x).$
(4.10)

We directly check that $S\in H^{2}(\mathbb{R})$ and
$|S|^{2}=|\mathcal{R}_{\tilde{\omega}}|^{2}$. Using that
$|D|=\mathsf{H}\partial_{x}$ and $\Pi_{+}=\frac{1}{2}(1+i\mathsf{H})$, a
calculation yields that

$-\partial_{xx}S-(|D||S|^{2})S+\frac{1}{4}|S|^{4}S=\tilde{\omega}S.$ (4.11)

If we integrate this equation against $\overline{S}$, we directly obtain

$\int_{\mathbb{R}}|\partial_{x}S|^{2}-\int_{\mathbb{R}}|S|^{2}(|D||S|^{2})+\frac{1}{4}\int_{\mathbb{R}}|S|^{6}=\tilde{\omega}\int_{\mathbb{R}}|S|^{2}.$
(4.12)

Next, we integrate (4.11) against $x\partial_{x}\overline{S}$ over the compact
interval $[-R,R]$. By taking the real part and taking the limit $R\to+\infty$,
we find the identity

$\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}S|^{2}-\frac{1}{24}\int_{\mathbb{R}}|S|^{6}=-\frac{\tilde{\omega}}{2}\int_{\mathbb{R}}|S|^{2}.$
(4.13)

For details of this step, we refer to Appendix C. The combination of (4.12)
and (4.13) yields

$\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}S|^{2}-\frac{1}{4}\int_{\mathbb{R}}|S|^{2}(|D||S|^{2})+\frac{1}{24}\int_{\mathbb{R}}|S|^{6}=0.$
(4.14)

Recall that the right side can be written as a complete square (see Appendix
C)

$\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}S+\frac{1}{2}\mathsf{H}(|S|^{2})S|^{2}\,dx=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}\mathcal{R}_{\tilde{\omega}}-\frac{i}{2}\Pi_{+}(|\mathcal{R}_{\tilde{\omega}}|^{2})\mathcal{R}_{\tilde{\omega}}|^{2}\,dx,$

where the last step follows by using (4.10). Thus we deduce that (4.9) holds,
which completes the proof. ∎

###  4.1. Proof of Theorem 1.2

This claim immediately follows from Proposition 4.1 by taking $\eta=v/2$. ∎

##  5\. Analysis of the Lax Operator

In this section, we will further study the Lax operator

$L_{u}=D-T_{u}T_{\bar{u}}$

introduced in Section 2 above, where $T_{b}(f)=\Pi_{+}(bf)$ denotes the
Toeplitz operator on $L^{2}_{+}(\mathbb{R})$ with symbol $b$. In particular,
we will derive important commutator identities and prove simplicity of
eigenvalues of $L_{u}$ for general potentials $u\in H^{1}_{+}(\mathbb{R})$
along with optimal bounds on the number of eigenvalues of $L_{u}$. Moreover,
we will define the notion of multi-soliton potentials below.

###  5.1. Simplicity of Eigenvalues and Sharp Bounds for $L_{u}$

Given $u\in H^{1}_{+}(\mathbb{R})$222Recall that the regularity assumption can
be relaxed to $u\in L^{2}_{+}(\mathbb{R})$ via an approach using quadratic
forms; see Appendix A below., we recall that the operator $L_{u}$ defined
above is an unbounded self-adjoint operator on the Hardy space
$L^{2}_{+}(\mathbb{R})$, with the operator domain $H^{1}_{+}(\mathbb{R})$.
First we show simplicity of eigenvalues for $L_{u}$ together with a bound on
the number of eigenvalues.

######  Proposition 5.1.

If $L_{u}\psi=\lambda\psi$, then

$|\langle u,\psi\rangle|^{2}=2\pi\|\psi\|_{L^{2}}^{2}\ .$ (5.1)

In particular, every eigenvalue of $L_{u}$ is simple, and the number $N$ of
eigenvalues of $L_{u}$ is finite with

$N\leq\frac{\|u\|_{L^{2}}^{2}}{2\pi}\ .$

In particular, the operator $L_{u}$ has no point spectrum if
$\|u\|_{L^{2}}^{2}<2\pi$.

######  Remark.

In [36], an identity reminiscent to (5.1) was derived for the Lax operator of
the Benjamin-Ono equation on the real line.

###### Proof.

The key is to introduce the Lax–Beurling semigroup of contractions defined on
$L^{2}_{+}(\mathbb{R})$ as

$S(\eta)f(x)={\rm e}^{i\eta x}f(x)\ ,\ \eta\geq 0\ .$

and the corresponding adjoint semigroup defined as

$S(\eta)^{*}f=\Pi_{+}\left({\rm e}^{-ix\eta}f\right)\ ,\ \eta\geq 0\ .$

Clearly, $S(\eta)$ and $S(\eta)^{*}$ act on the domain $H^{1}_{+}(\mathbb{R})$
of $L_{u}$. Also notice that, via the Fourier transform, we have

$\widehat{S(\eta)^{*}f}(\xi)=\hat{f}(\xi+\eta)\ ,\ \xi\geq 0,\eta\geq 0\ .$

To complete the proof of Proposition 5.1, we need the following identity.

######  Lemma 5.1.

For every $f\in H^{1}_{+}(\mathbb{R})$, we have the following limit in the
$L^{2}$-norm,

$\lim_{\eta\to
0}\left[\frac{S(\eta)^{*}}{\eta},L_{u}\right]f=f-\frac{1}{2\pi}\langle
f,u\rangle u\ .$

###### Proof of Lemma 5.1.

Notice that, for $\xi>0$ ,

$\widehat{L_{u}f}(\xi)=\xi\hat{f}(\xi)-\frac{1}{4\pi^{2}}\int_{0}^{\xi}\hat{u}(\xi-\zeta)\left[\int_{0}^{\infty}\overline{\hat{u}(\tau)}\hat{f}(\zeta+\tau)\,d\tau\right]\,d\zeta\
.$

Set

$A(\eta):=\left[\frac{S(\eta)^{*}}{\eta},L_{u}\right]\ ,\ \xi\geq 0\ .$

Then

$\widehat{A(\eta)f}(\xi)=\hat{f}(\xi+\eta)-\frac{1}{4\pi^{2}\eta}\int_{0}^{\eta}\hat{u}(\xi+\eta-\zeta)\left[\int_{0}^{\infty}\overline{\hat{u}(\tau)}\hat{f}(\zeta+\tau)\,d\tau\right]\,d\zeta\
.$

Passing to the limit in $L^{2}$, we get

$\lim_{\eta\to
0}\widehat{A(\eta)f}(\xi)=\hat{f}(\xi)-\frac{1}{4\pi^{2}}\hat{u}(\xi)\int_{0}^{\infty}\overline{\hat{u}(\tau)}\hat{f}(\tau)\,d\tau\
,$

which yields Lemma 5.1. ∎

For future reference, we state the following lemma, which can be proved
similarly to Lemma 5.1.

######  Lemma 5.2.

Let $a,b\in L^{2}_{+}(\mathbb{R})$. For every $f\in L^{2}_{+}(\mathbb{R})$, we
have the following limit in the $L^{2}$-norm,

$\lim_{\eta\to
0}\left[\frac{S(\eta)^{*}}{\eta},T_{a}T_{\overline{b}}\right]f=\frac{1}{2\pi}\langle
f,b\rangle a\ .$

Let us come back to the proof Proposition 5.1. If we apply Lemma 5.1 to
$f=\psi$, we obtain

$\lim_{\eta\to 0}\frac{(\lambda
I-L_{u})S(\eta)^{*}\psi}{\eta}=\psi-\frac{1}{2\pi}\langle\psi,u\rangle u\ .$

Taking inner product of both sides with $\psi$, we obtain

$0=\|\psi\|_{L^{2}}^{2}-\frac{1}{2\pi}|\langle\psi,u\rangle|^{2}\ $

which is (5.1). Now the next statements of Proposition 5.1 follow easily.
Indeed, identity (5.1) implies that the kernel of the linear form
$\langle.,u\rangle$ on any eigenspace is reduced to $\\{0\\}$, so that this
eigenspace is one dimensional. Finally, if $\psi_{1},\dots,\psi_{N}$ is an
orthonormal system of eigenvectors, we have

$\|u\|_{L^{2}}^{2}\geq\sum_{j=1}^{n}|\langle u,\psi_{j}\rangle|^{2}=2\pi N\ .$

This proof of Proposition 5.1 is now complete. ∎

###  5.2. Multi-Soliton Potentials

Next, we exhibit a class of potentials $u\in H^{1}_{+}(\mathbb{R})$ which are
given by rational functions and which optimize the general bound for the
number $N$ of eigenvalues of $L_{u}$ found in Proposition 5.1 above. To this
end, we use $\sigma_{pp}(L_{u})$ to denote the pure point spectrum of $L_{u}$
and correspondingly we define

$\mathcal{E}_{pp}(u)={\rm span}\\{\psi\in\mathrm{ker}(L_{u}-\lambda
I):\lambda\in\sigma_{pp}(L_{u})\\}$

to be the space spanned by the eigenfunctions of $L_{u}$. We have the
following spectral characterization result.

######  Proposition 5.2.

Let $u\in H^{1}_{+}(\mathbb{R})$ be given and let $N\geq 1$ be an integer.
Then the following properties are equivalent and preserved by the flow of (CM-
DNLS).

  1. (i)

The space $\mathcal{E}_{pp}(u)$ has dimension $N$ and is invariant under the
adjoint semigroup $\\{S(\eta)^{*}\\}_{\eta\geq 0}$ and it holds that
$u\in\mathcal{E}_{pp}(u)$.

  2. (ii)

There exist a polynomial $Q\in\mathbb{C}[x]$ of degree $N$, with all its zeros
in $\mathbb{C}_{-}$, and a polynomial $P\in\mathbb{C}[x]$ of degree at most
$N-1$ such that

$u(x)=\frac{P(x)}{Q(x)}\quad\mbox{and}\quad
P\overline{P}=i(Q^{\prime}\overline{Q}-\overline{Q}^{\prime}Q).$

Before we prove this result, let us make some general comments as follows. We
refer to the rational functions $u\in H^{1}_{+}(\mathbb{R})$ above as a multi-
soliton potential or $N$-soliton potential. In Section 6 below, we will study
the time evolution of multi-soliton potentials, which correspond to multi-
solitons. In view of Proposition 5.1, we see that multi-solitons must have a
quantized $L^{2}$-mass according to

$\|u\|_{L^{2}}^{2}=2\pi N$

with some integer $N\geq 1$. As a generic example for a multi-soliton
potential, we can take the polynomial $Q$ to have simple zeros, i. e.,

$Q(x)=\prod_{j=1}^{N}(x-z_{j}),$

where $z_{1},\ldots,z_{N}\in\mathbb{C}_{-}$ are pairwise distinct. Then the
condition in (ii) above yields that

$u(x)=\sum_{j=1}^{N}\frac{a_{j}}{x-z_{j}}\quad\mbox{with}\quad\sum_{j=1}^{N}\frac{a_{j}\overline{a}_{k}}{z_{j}-\overline{z}_{k}}=i\quad\mbox{with
$k=1,\dots,N$}.$

Let us also remark that, for a given polynomial $Q\in\mathbb{C}[x]$ of degree
$N\geq 1$, there exist only a finite number of polynomials $P\in\mathbb{C}[x]$
satisfying the constraint in (ii) up to a constant complex phase. We can find
them as follows. Consider the polynomial
$F:=i(Q^{\prime}\overline{Q}-\overline{Q}^{\prime}Q)\in\mathbb{C}[x]$ of
degree $2N-2$. Suppose that

$Q(x)=\prod_{j=1}^{N}(x-z_{j})$

with zeros $z_{1},\dots,z_{N}\in\mathbb{C}_{-}$, which are not necessarily
distinct. For $x\in\mathbb{R}$, we get

$\frac{F(x)}{|Q(x)|^{2}}=\sum_{j=1}^{N}\frac{-2{\rm
Im}z_{j}}{|x-z_{j}|^{2}}>0\ .$

In particular, the zeroes of $F$ come in pairs as
$(\alpha_{j},\overline{\alpha}_{j})$, $j=1,\dots,N-1$, with
$\alpha_{j}\notin\mathbb{R}$. Thus we can write

$P(x)=c\prod_{j=1}^{N-1}(x-\alpha_{j}),$

where the constant $c\in\mathbb{C}$ is adjusted so that $P\overline{P}=F$
holds. Of course, exchanging one $\alpha_{j}$ with $\overline{\alpha}_{j}$
leads to a different $P$ and hence to a different function $u$.

###### Proof of Proposition 5.2.

We first proof the equivalence of statements (i) and (ii). Finally, we address
the preservation by the flow of (CM-DNLS).

Step 1: (ii) $\Rightarrow$ (i). For $u(x)=\frac{P(x)}{Q(x)}\in
H^{1}_{+}(\mathbb{R})$ as in (ii), we define

$\theta(x)=\frac{\overline{Q}(x)}{Q(x)}\quad\mbox{and}\quad
K_{\theta}:=\frac{\mathbb{C}_{N-1}[x]}{Q(x)}\ ,$

so that we have the orthogonal decomposition

$L^{2}_{+}(\mathbb{R})=K_{\theta}\oplus\theta L^{2}_{+}(\mathbb{R})\ .$

We claim that $K_{\theta}$ is an invariant subspace of $L_{u}$. Indeed, if
$f=A/Q\in K_{\theta}$, then

$\displaystyle L_{u}(f)$ $\displaystyle=$
$\displaystyle\frac{-iA^{\prime}}{Q}+\frac{iQ^{\prime}A}{Q^{2}}-\frac{P}{Q}\Pi_{+}\left(\frac{\overline{P}}{\overline{Q}}\frac{A}{Q}\right)$
$\displaystyle=$
$\displaystyle\frac{-iA^{\prime}}{Q}+\Pi_{+}\left[\frac{P\overline{P}}{Q\overline{Q}}\frac{A}{Q}+i\frac{\overline{Q}^{\prime}}{\overline{Q}}\frac{A}{Q}\right]-\frac{P}{Q}\Pi_{+}\left(\frac{\overline{P}}{\overline{Q}}\frac{A}{Q}\right)$
$\displaystyle=$
$\displaystyle\frac{-iA^{\prime}}{Q}+i\Pi_{+}\left(\frac{\overline{Q}^{\prime}}{\overline{Q}}\frac{A}{Q}\right)+\Pi_{+}\left(\frac{P}{Q}\Pi_{-}\left(\frac{\overline{P}A}{\overline{Q}Q}\right)\right)\
.$

Observe that, if $R\in L^{2}(\mathbb{R})$ is a rational function, and if the
denominator of $R$ reads $Q_{+}Q_{-}$, where $Q_{\pm}$ is a polynomial with
zeroes in $\mathbb{C}_{\pm}$, then $\Pi_{+}(R)$ is of the form $P/Q_{-}$,
where $P$ is a polynomial of degree less than the degree of $Q_{-}$. From this
observation and the above identity, we conclude that $L_{u}(f)\in K_{\theta}$.
Therefore $L_{u}$ is a self-adjoint endomorphism on the finite dimensional
space $K_{\theta}$, which has dimension $N$. This implies that $L_{u}$ has at
least $N$ eigenvalues. Since $u\in K_{\theta}$, it follows that $u$ is a
linear combination of an orthonormal basis of eigenfunctions
$\psi_{1},\ldots,\psi_{N}\in K_{\theta}$. From Proposition 5.1 we conclude
that

$\|u\|_{L^{2}}^{2}=\sum_{k=1}^{N}|\langle u,\psi_{k}\rangle|^{2}=2\pi N.$

By invoking Proposition 5.1 again, we deduce that $L_{u}$ has exactly $N$
eigenvalues. Thus we have shown $\mathcal{E}_{pp}(u)=K_{\theta}$ and therefore
we conclude $\dim\mathcal{E}_{pp}(u)=N$ as well as $u\in\mathcal{E}_{pp}(u)$.
Since the semigroup $\\{S(\eta)\\}_{\eta\geq 0}$ leaves the space $\theta
L_{+}^{2}(\mathbb{R})$ invariant, we obtain that its adjoint semigroup
$\\{S(\eta)^{*}\\}_{\eta\geq 0}$ leaves its orthogonal complement
$\mathcal{E}_{pp}(u)=K_{\theta}$ invariant.

Step 2: (i) $\Rightarrow$ (ii). Suppose that $\mathcal{E}_{pp}(u)$ has
dimension $N\geq 1$ and is invariant under the adjoint semigroup
$\\{S(\eta)^{*}\\}_{\eta\geq 0}$. Thus the orthogonal complement
$(\mathcal{E}_{pp}(u))^{\perp}$ is invariant preserved by the action of the
semigroup $S(\eta)$ with $\eta\geq 0$. By the Lax–Beurling theorem [23], we
conclude

$(\mathcal{E}_{pp}(u))^{\perp}=\theta L^{2}_{+}(\mathbb{R})$

with some inner function $\theta$ defined on the upper complex halfplane
$\mathbb{C}_{+}$. Furthermore, since $\mathcal{E}_{pp}(u)$ is $N$–dimensional,
one can choose $\theta$ of the form

$\theta(x)=\frac{\overline{Q}(x)}{Q(x)}\ ,\ Q(x)=\prod_{j=1}^{N}(x-z_{j})\ ,\
\mathrm{Im}z_{j}<0\ .$

Consequently,

$\mathcal{E}_{pp}(u)=(\theta
L^{2}_{+}(\mathbb{R}))^{\perp}=K_{\theta}=\frac{\mathbb{C}_{N-1}[x]}{Q(x)}\ .$

Since $u\in\mathcal{E}_{pp}(u)$, there exists $P\in\mathbb{C}_{N-1}[x]$ such
that $u=P/Q$. Since $L_{u}$ is self-adjoint, we have

$L_{u}[(\mathcal{E}_{pp}(u))^{\perp}\cap
H^{1}_{+}(\mathbb{R})]\subset(\mathcal{E}_{pp}(u))^{\perp}\ ,$

alternatively $L_{u}(\theta H^{1}_{+}(\mathbb{R}))\subset\theta
L^{2}_{+}(\mathbb{R})$. Let $h\in H^{1}_{+}(\mathbb{R})$. We have

$L_{u}(\theta h)=\theta Dh+(D\theta)h-u\Pi_{+}(\overline{u}\theta h)=\theta
Dh+(D\theta)h-|u|^{2}\theta h\ ,$

because

$\overline{u}\theta=\frac{\overline{P}}{\overline{Q}}\frac{\overline{Q}}{Q}=\frac{\overline{P}}{Q}\in
L^{2}_{+}(\mathbb{R})\ .$

We infer $(D\theta)h-\theta|u|^{2}h\in\theta L^{2}_{+}(\mathbb{R})$ for every
$h\in H^{1}_{+}(\mathbb{R})$, or

$\frac{D\theta}{\theta}-|u|^{2}\in L^{2}_{+}(\mathbb{R})\ .$

Notice that, for every $x\in\mathbb{R}$,

$\frac{D\theta}{\theta}=\sum_{j=1}^{N}\frac{2\mathrm{Im}z_{j}}{|x-z_{j}|^{2}}$

hence $D\theta/\theta-|u|^{2}$ is real valued, therefore it belongs to
$L^{2}_{+}(\mathbb{R})\cap\overline{L^{2}_{+}(\mathbb{R})}=\\{0\\}$.
Reformulating this identity in terms of $P$ and $Q$, we obtain
$P\overline{P}=i(Q^{\prime}\overline{Q}-\overline{Q}^{\prime}Q)\ .$

Step 3: Preservation by the Flow. In order to prove the last part of
Proposition 5.2, we introduce the infinitesimal generator of the adjoint
Lax–Beurling semi–group, namely the operator $G$ such that

$S(\eta)^{*}={\rm e}^{-i\eta G}\ .$

Notice that its operator domain is given by

${\rm dom}(G)=\\{f\in L^{2}_{+}(\mathbb{R}):\hat{f}_{|]0,+\infty[}\in
H^{1}(]0,+\infty[)\\}$

and that

$\widehat{(Gf)}(\xi)=i\frac{d\hat{f}}{d\xi}\quad\mbox{for}\quad\xi>0.$

In particular, the operator $G$ acts on $K_{\theta}$ for every finite Blaschke
product $\theta$. We claim that properties (i) and (ii) are equivalent to the
following statement:
(iii) The space $\mathcal{E}_{pp}(u)$ has dimension $N$, contains $u$, and
moreover $u\in\mathrm{dom}(G)$ with $Gu\in\mathcal{E}_{pp}(u)$.

Indeed, as we just observed, if $u$ is a $N$-soliton, then it satisfies (iii).
Conversely, assume that $u\in H^{1}_{+}$ satisfies (iii). We appeal to a
corollary of Lemma 5.1.

######  Lemma 5.3.

Let $f\in{\rm dom}(G)\cap H^{1}_{+}$ such that $L_{u}f\in{\rm dom}(G)$. Then
$Gf\in H^{1}_{+}$ and

$GL_{u}f-L_{u}Gf=if-i\frac{\langle f,u\rangle}{2\pi}u\ .$

###### Proof.

For every $h\in{\rm dom}(G)$, we have

$Gh=i\lim_{\eta\to 0^{+}}\frac{S(\eta)^{*}h-h}{\eta}\ .$

Rewriting Lemma 5.1 as

$\lim_{\eta\to
0}\left[\frac{S(\eta)^{*}-I}{\eta},L_{u}\right]f=f-\frac{1}{2\pi}\langle
f,u\rangle u.$

Using that $L_{u}$ is a closed operator, the lemma follows. ∎

Applying Lemma 5.3 to $f=u$, we infer that $L_{u}(u)\in\mathrm{dom}(G)$ and
that

$GL_{u}(u)=L_{u}(Gu)+iu-i\frac{\|u\|^{2}_{L^{2}}}{2\pi}u\ .$

In particular, $GL_{u}(u)\in\mathcal{E}_{pp}(u)$. Iterating this process, we
conclude by an easy induction that, for every integer $k$,
$L_{u}^{k}u\in\mathrm{dom}(G)$ and that $GL_{u}^{k}u\in\mathcal{E}_{pp}(u)$.
Now recall from Proposition 5.1 that $L_{u}$ has $N$ simple eigenvalues on
$\mathcal{E}_{pp}(u)$, and that the component of $u$ on any eigenvector is
different from $0$. Consequently, $u$ is a cyclic vector for $L_{u}$ in
$\mathcal{E}_{pp}(u)$, namely the $N$ vectors $u,L_{u},\cdots,L_{u}^{N-1}u$
form a basis of $\mathcal{E}_{pp}(u)$. From this we infer that $G$ acts on
$\mathcal{E}_{pp}(u)$, and finally that $S(\eta)^{*}$ acts on
$\mathcal{E}_{pp}(u)$, whence (i).
Let us prove that property (iii) is preserved by the flow of (CM-DNLS).
Consider $u_{0}$ satisfying (iii), and denote by $u$ the solution of (CM-DNLS)
with $u(0)=u_{0}$, on its maximal time interval. Notice that we know from (ii)
that $u_{0}$ belongs to every $H^{s}$. Therefore, by the well-posedness result
Proposition 2.1, $u(t)$ belongs to every $H^{s}$, hence we do not have to
worry about its regularity. We are going to use the Lax equation provided by
Lemma 2.3. Denote by $U(t)$ the one-parameter family of unitary operators on
$L^{2}_{+}(\mathbb{R})$ defined as

$\frac{d}{dt}U(t)=B_{u(t)}U(t)\ ,\ U(0)=I\ .$

Then Lemma 2.3 implies

$L_{u(t)}=U(t)L_{u_{0}}U(t)^{*}\ .$ (5.2)

Consequently, $\mathcal{E}_{pp}(u(t))=U(t)[\mathcal{E}_{pp}(u_{0})]$ has
dimension $N$. Furthermore, in view of Lemma 2.4, the spectral measure of
$L_{u(t)}$ associated to the vector $u(t)$ is the same as the spectral measure
of $L_{u_{0}}$ associated to the vector $u_{0}$. This implies that
$\mathcal{E}_{pp}(u(t))$ contains $u(t)$. It remains to prove that
$u(t)\in\mathrm{dom}(G)$ and that $Gu(t)\in\mathcal{E}_{pp}(u(t))$. Let us
appeal to the reformulation of the dynamics as

$\partial_{t}u=\tilde{B}_{u}u\ ,$

where $\tilde{B}_{u}=B_{u}-iL_{u}^{2}$ according to (2.6). Given $\eta>0$,
define

$v(t,\eta):=i\frac{S(\eta)^{*}u(t)-u(t)}{\eta}$

and observe that

$\partial_{t}v(t,\eta)=\tilde{B}_{u}v(t,\eta)+g(t,\eta)\ ,\
g(t,\eta):=i\left[\frac{S(\eta)^{*}}{\eta},\tilde{B}_{u(t)}\right]u(t)\ ,$

which, in view of (5.2), can be solved as

$U(t)^{*}v(t,\eta)={\rm e}^{-itL_{u_{0}}^{2}}v(0,\eta)+\int_{0}^{t}{\rm
e}^{i(\tau-t)L_{u_{0}}^{2}}U(\tau)^{*}g(\tau,\eta)\,d\tau\ .$

Using Lemma 5.2 and the expression of $B_{u}$, we obtain

######  Lemma 5.4.

If $u\in H^{1}_{+}(\mathbb{R})$, we have, for every $f\in
L^{2}_{+}(\mathbb{R})$,

$\lim_{\eta\to
0}i\left[\frac{S(\eta)^{*}}{\eta},B_{u}\right]f=\frac{1}{2\pi}(\langle
f,L_{u}u\rangle u+\langle f,u\rangle L_{u}u)\ .$

Combining Lemma 5.4 with Lemma 5.1, we infer, locally uniformly in $t$,

$\lim_{\eta\to 0}g(t,\eta)=2L_{u(t)}u(t)\ .$

This shows that $v(t,\eta)$ has a limit $Gu(t)$ in $L^{2}_{+}$ as $\eta\to 0$,
characterized by

$\displaystyle U(t)^{*}Gu(t)$ $\displaystyle=$ $\displaystyle{\rm
e}^{-itL_{u_{0}}^{2}}Gu_{0}+2\int_{0}^{t}{\rm
e}^{i(\tau-t)L_{u_{0}}^{2}}L_{u_{0}}U(\tau)^{*}u(\tau)\,d\tau$
$\displaystyle=$ $\displaystyle{\rm e}^{-itL_{u_{0}}^{2}}Gu_{0}+2t\,{\rm
e}^{-itL_{u_{0}}^{2}}L_{u_{0}}u_{0}\ .$

Notice that, by (iii), the right hand side of the above equation belongs to
$\mathcal{E}_{pp}(u_{0})$. Consequently, $Gu(t)\in
U(t)[\mathcal{E}_{pp}(u_{0})]=\mathcal{E}_{pp}(u(t))$. This completes the
proof. ∎

######  Remark.

In fact, one can easily check that the operator $L_{u}$ restricted to the
invariant subspace $\theta L^{2}_{+}=(K_{\theta})^{\perp}$ has absolutely
continuous simple spectrum with

$L_{u}(\theta h)=\theta Dh\quad\mbox{for all $h\in H^{1}_{+}(\mathbb{R})$.}$

For any $u\in H^{1}_{+}(\mathbb{R})$, we notice that the operator $L_{u}$ has
the essential spectrum $\sigma_{ess}(L_{u})=[0,\infty)$. In the case of multi-
solitons, we find that $0$ is always an embedded eigenvalue.

######  Proposition 5.3.

For any multi-soliton potential $u\in H^{1}_{+}(\mathbb{R})$, we have that

$L_{u}(1-\theta)=0,$

where $\theta(x)=\frac{\overline{Q}(x)}{Q(x)}$ with the notation from
Proposition 5.2 (ii) above.

###### Proof.

We observe that

$L_{u}(1-\theta)=-D\theta-u\Pi_{+}(\overline{u}(1-\theta))=-D\theta+u\Pi_{+}(\overline{u}\theta)\
.$

Notice that the function

$\overline{u}\theta=\frac{\overline{P}}{\overline{Q}}\frac{\overline{Q}}{Q}=\frac{\overline{P}}{Q}$

belongs to $L^{2}_{+}(\mathbb{R})$. Hence we can deduce

$L_{u}(1-\theta)=-D\theta+u\overline{u}\theta=0,$

because of the constraint in Proposition 5.2 (ii). ∎

##  6\. Dynamics of Multi-Solitons

This section is devoted to the study of multi-solitons, i. e., solutions
$u(t,x)$ with initial datum giving by a multi-soliton potential $u_{0}\in
H^{1}_{+}(\mathbb{R})$ (see Proposition 5.2 above). By means of an inverse
spectral formula, we will be able to prove global-in-time existence for all
multi-solitons. This is a large data result which is beyond the scope of
a-priori bounds. Second, we prove that all multi-solitons with $N\geq 2$
exhibit an energy cascade (growth of Sobolev norms) as $t\to\pm\infty$.

###  6.1. Preliminary Discussion

Let us first consider the following pole ansatz of the form

$u(t,x)=\sum_{j=1}^{N}\frac{a_{j}(t)}{x-z_{j}(t)}\in H^{1}_{+}(\mathbb{R}),$
(6.1)

where $a_{1}(t),\ldots,a_{N}(t)\in\mathbb{C}\setminus\\{0\\}$ and pairwise
distinct poles $z_{1}(t),\ldots,z_{N}(t)$ in the complex lower halfplane
$\mathbb{C}_{-}$. If we plug this ansatz into (CM-DNLS), then a
straightforward calculation shows that the self-consistency of (6.1) leads to
the set of nonlinear constraints given by

$\sum_{j=1}^{N}\frac{a_{j}(t)\overline{a}_{k}(t)}{z_{j}(t)-\overline{z}_{k}(t)}=i\quad\mbox{for}\quad
k=1,\ldots,N.$ (6.2)

Note that these conditions have already appeared in the discussion of multi-
soliton potentials (see Proposition 5.2 above). Furthermore, the equations of
motions which govern the parameters $\\{a_{j}(t),z_{j}(t)\\}_{j=1}^{N}$ are
found to be

$\dot{a}_{k}=2i\sum_{\ell\neq
k}^{N}\frac{a_{\ell}-a_{k}}{(z_{k}-z_{\ell})^{2}}\quad\mbox{and}\quad
a_{k}\dot{z}_{k}=-2i\sum_{\ell\neq k}^{N}\frac{a_{\ell}}{z_{k}-z_{\ell}}$
(6.3)

with $k=1,\ldots,N$. A tedious calculation shows that the constraints (6.2)
are indeed preserved by the time evolution determined by (6.3). Finally, we
remark that the first-order system (6.3) can be used to derive that

$\ddot{z}_{k}=\sum_{\ell\neq
k}^{N}\frac{8}{(z_{k}-z_{\ell})^{3}}\quad\mbox{for}\quad k=1,\ldots,N,$ (6.4)

which is again confirmed by a lengthy calculation that we omit here. We note
that (6.4) can be regarded as a complexified version333That is, we formally
generalize the positions $x_{j}\in\mathbb{R}$ to complex numbers
$z_{j}\in\mathbb{C}_{-}$. of the Calogero–Moser (CM) system on the real line,
whose complete integrability was proven by J. Moser in [27] (see also [28, 21,
29]). Let us mention that the pole dynamics of rational solutions of
completely integrable PDEs are governed by (complexified versions) of CM
systems have also been found for the Benjamin-Ono, KdV and Half-Wave Maps
equations in [3, 9, 7].

However, we emphasize that working with the pole ansatz in (6.1) leads to the
following caveats that ultimately need to be addressed.

  1. (1)

Collision of poles: It may happen that $z_{j}(t)\to z_{k}(t)$ for some pair
$j\neq k$ as $t\to T$ with some finite time $T>0$. From (6.4) we expect that
the solution $z_{k}(t)$ blows up in $C^{2}$ as $t\to T$. But the solution
$u(t,x)$ itself may stay smooth as $t\to T$, whereas the pole ansatz (6.1)
becomes invalid only. Explicit examples of pole collisions will be given in
Subsection 6.4 below.

  2. (2)

Showing that $z_{k}(t)\in\mathbb{C}_{-}$: The major step in showing global-in-
existence for multi-solitons consists in proving that the poles $z_{k}(t)$
stay in the lower complex half-plane $\mathbb{C}_{-}$.

To systematically tackle the problems (1) and (2), we will derive an inverse
spectral formula for multi-solitons, which entails the pole ansatz (6.1) as a
special case. Furthermore, the dynamical evolution of multi-solitons $u(t,x)$
will be encoded by the linear flow of a suitable matrix
$\mathsf{M}(t)\in\mathbb{C}^{N\times N}$ such that

$\mathsf{M}(t)=2\mathsf{V}t+\mathsf{W}$ (6.5)

with some constant matrices $\mathsf{V}$ and $\mathsf{W}$ in
$\mathbb{C}^{N\times N}$; see Proposition 6.2 and (6.12)–(6.13) below.
Moreover, we remark that solving classical CM systems on the real line by
means of linear flows of $N\times N$-matrices was successfully used in [28].
Finally, notice that similar inverse formulae for multi-soliton solutions were
derived in the case of the Benjamin–Ono equation in [34] and in the case of
the cubic Szegő equation in [17]. However, in these two examples, the global
well-posedness result was established directly by other methods, while it
seems to be the first time that such inverse formulae provide global
existence.

###  6.2. Inverse Spectral Formula and Time Evolution

Let

$u(x)=\frac{P(x)}{Q(x)}\in H^{1}_{+}(\mathbb{R})$

be a multi-soliton potential. We use the notation introduced in the proof of
Proposition 5.2 with

$\theta(x)=\frac{\overline{Q}(x)}{Q(x)}\quad\mbox{and}\quad
K_{\theta}=\frac{\mathbb{C}_{N-1}[x]}{Q(x)}.$

Recall that the Lax–Beurling semigroup $S(\eta)$ leaves the space $\theta
L^{2}_{+}$ invariant, and hence its adjoint semigroup $S(\eta)^{*}$ leaves the
subspace $K_{\theta}$ invariant. This observation leads to the following
formula, where $G$ is the operator introduced in subsection 5.2.

######  Proposition 6.1.

For every $f\in K_{\theta}$,

$f(x)=\frac{1}{2\pi
i}\langle(G-xI)^{-1}f,1-\theta\rangle\quad\mbox{for}\quad{\rm Im}(x)>0\ .$

###### Proof.

We start from the inversion Fourier formula for every element of
$L^{2}_{+}(\mathbb{R})$,

$f(x)=\frac{1}{2\pi}\int_{0}^{\infty}\mathrm{e}^{ix\xi}\hat{f}(\xi)\,d\xi\ .$

We claim that if $f\in K_{\theta}$ we have

$\hat{f}(\xi)=\langle S(\xi)^{*}f,1-\theta\rangle\ ,\ \xi>0\ .$ (6.6)

Indeed, by the Plancherel theorem,

$\hat{f}(\xi)=\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}}{\rm
e}^{-ix\xi}\frac{f(x)}{1+i\varepsilon x}\,dx=\lim_{\varepsilon\to
0^{+}}\bigg{\langle}S(\xi)^{*}f,\frac{1}{1-i\varepsilon x}\bigg{\rangle}\ ,$

since $(1-i\varepsilon x)^{-1}\in L^{2}_{+}$. For the same reason
$\theta(1-i\varepsilon x)^{-1}\in\theta L^{2}_{+}$ which is orthogonal to
$K_{\theta}$ and in particular to $S(\xi)^{*}f$. Therefore

$\hat{f}(\xi)=\lim_{\varepsilon\to
0^{+}}\bigg{\langle}S(\xi)^{*}f,\frac{1-\theta}{1-i\varepsilon
x}\bigg{\rangle}\ ,$

which yields (6.6). It remains to plug (6.6) into the inversion Fourier
formula, reminding that $S(\xi)^{*}={\rm e}^{-i\xi G}$, and the proposition
follows. ∎

Next, we are going to use Proposition 6.1 in the particular case $f=u$. The
interesting feature is that the inner product in this formula takes place in
$K_{\theta}$, of which we can choose an orthonormal basis made of eigenvectors
$(\psi_{1},\ldots,\psi_{N})$ of $L_{u}$ so that

$L_{u}\psi_{j}=\lambda_{j}\psi_{j}\quad\mbox{for}\quad j=1,\dots,N.$

Here and throughout the following we label the eigenvalues such that

$\lambda_{1}=0,\lambda_{2},\ldots,\lambda_{N},$

where we recall that $0$ is always an eigenvalue of $L_{u}$ by Proposition
5.3. In view of Proposition 5.1, we can choose the normalisation

$\langle u,\psi_{j}\rangle=\sqrt{2\pi}\quad\mbox{for}\quad j=1,\dots,N.$

Note that, because of Proposition 5.3, it holds

$\langle 1-\theta,\psi_{j}\rangle=0\quad\mbox{for}\quad j=2,\dots,N.$

Let us first discuss the case of $\langle 1-\theta,\psi_{1}\rangle$. Notice
that every element $f$ of $K_{\theta}$ is smooth on $]0,+\infty[$ with limits
at $0^{+}$. In particular, we can pass to the limit in formula (6.6) as
$\xi\to 0^{+}$ to obtain

$\hat{f}(0^{+})=\langle f,1-\theta\rangle\ .$

On the other hand, the identity $L_{u}\psi_{1}=0$ reads

$\xi\hat{\psi}_{1}(\xi)=\frac{1}{4\pi^{2}}\int_{0}^{\xi}\hat{u}(\xi-\zeta)\left(\int_{0}^{\infty}\overline{\hat{u}(\tau)}\hat{\psi}_{1}(\tau+\zeta)\,d\tau\right)\,d\zeta$

so that

$\langle\psi_{1},1-\theta\rangle=\hat{\psi}_{1}(0^{+})=\frac{\hat{u}(0^{+})}{2\pi}\langle
u,\psi_{1}\rangle=\frac{\hat{u}(0^{+})}{\sqrt{2\pi}}\ .$ (6.7)

Coming back to the equation satisfied by $u$,

$i\partial_{t}u+\partial_{x}^{2}u+2D_{+}(|u|^{2})u=0\ ,$ (6.8)

and taking the Fourier transform, we observe that

$i\partial_{t}\hat{u}(t,\xi)-\xi^{2}\hat{u}(t,\xi)+2\int_{0}^{\infty}{\rm
e}^{-ix\xi}D_{+}(|u|^{2})(t,x)u(t,x)\,dx=0\ .$

Passing to the limit as $\xi\to 0^{+}$ and observing that $D_{+}(|u|^{2})$ and
$u$ both belong to $L^{2}_{+}$, we infer

$\partial_{t}\hat{u}(t,0^{+})=0.$

Hence $\hat{u}(0^{+})$ is a conserved quantity and, consequently, the inner
product product $\langle 1-\theta,\psi_{1}\rangle$ is conserved as well. Let
us come back to the inverse formula

$u(x)=\frac{1}{2i\pi}\langle(G-xI)^{-1}u,1-\theta\rangle\ ,\ {\rm Im}(x)>0\ .$
(6.9)

In the orthonormal basis $(\psi_{1},\dots,\psi_{N})$ of $K_{\theta}$, we have
just observed that the components of $u$ and of $1-\theta$ are conserved
quantities. Next, let us discuss the matrix of $G$ in this basis, which we
denote as

$\mathsf{M}:=(\langle G\psi_{k},\psi_{j}\rangle)_{1\leq j,k\leq N}\ .$

Of course, the matrix $\mathsf{M}=\mathsf{M}(t)$ will depend on $t$ as well
through the evolution of the eigenfunctions $\psi_{j}$ as given by the Lax
structure in order to keep $\langle u,\psi_{j}\rangle$ constant, i. e., we
have $\dot{\psi}_{j}=\tilde{B}_{u}\psi_{j}$. Indeed, we will later use this to
derive explicitly formulas for $\mathsf{M}$. For notational ease, we will
sometimes omit the $t$-dependence of $\mathsf{M}(t)$.

We first consider the case $j\neq k$. For this We recall that every element of
$K_{\theta}$ belongs to ${\rm dom}(G)$. Then, for $j\neq k$, we observe that

$(\lambda_{k}-\lambda_{j})\langle
G\psi_{k},\psi_{j}\rangle=\langle(GL_{u}-L_{u}G)\psi_{k},\psi_{j}\rangle=i\langle\psi_{k},\psi_{j}\rangle-i\frac{\langle\psi_{k},u\rangle\langle
u,\psi_{j}\rangle}{2\pi}=-i\ $

so that

$\langle
G\psi_{k},\psi_{j}\rangle=\frac{i}{\lambda_{j}-\lambda_{k}}\quad\mbox{if}\quad
j\neq k.$ (6.10)

Finally, let us discuss the diagonal elements $\langle
G\psi_{j},\psi_{j}\rangle$. Notice that their imaginary parts are easy to
calculate, since

${\rm Im}\,\langle G\psi_{j},\psi_{j}\rangle=\frac{1}{2\pi}{\rm
Re}\,\bigg{\langle}\frac{d\hat{\psi}_{j}}{d\xi},\hat{\psi}_{j}\bigg{\rangle}=-\frac{|\hat{\psi}_{j}(0^{+})|^{2}}{4\pi}=-\frac{|\langle\psi_{j},1-\theta\rangle|^{2}}{4\pi}$

which is $0$ whenever $j\neq 1$. For $j=1$, we use (6.7) to conclude

${\rm Im}\langle
G\psi_{1},\psi_{1}\rangle=-\frac{|\hat{u}(0^{+})|^{2}}{8\pi^{2}}\ .$

As for the real part of the diagonal elements, we are going to compute their
time derivatives if $u$ is a solution of (6.8). From the Lax pair formula, we
may assume that

$\dot{\psi}_{j}=B_{u}\psi_{j}\ ,\
B_{u}=T_{u}T_{\partial_{x}\overline{u}}-T_{\partial_{x}u}T_{\overline{u}}+i(T_{u}T_{\overline{u}})^{2}\
.$

Then

$\frac{d}{dt}\langle
G\psi_{j},\psi_{j}\rangle=\langle[G,\tilde{B}_{u}]\psi_{j},\psi_{j}\rangle=\langle[G,B_{u}]\psi_{j},\psi_{j}\rangle\
.$

Passing to the limit in Lemma 5.4, we have

$[G,B_{u}]f=\frac{1}{2\pi}(\langle f,L_{u}u\rangle u+\langle f,u\rangle
L_{u}u\ ).$

Consequently, we get

$\displaystyle\langle[G,B_{u}]\psi_{j},\psi_{j}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\langle L_{u}\psi_{j},u\rangle\langle
u,\psi_{j}\rangle+\frac{1}{2\pi}\langle\psi_{j},u\rangle\langle
L_{u}u,\psi_{j}\rangle$ $\displaystyle=$ $\displaystyle 2\lambda_{j}\ .$

Summing up, we have proved that

$\frac{d}{dt}\langle G\psi_{j},\psi_{j}\rangle=2\lambda_{j}\ .$ (6.11)

The inverse spectral formula therefore reads, setting

$\gamma_{j}:={\rm Re}(\mathsf{M}_{jj})\ ,\
\sqrt{2\varrho}\,\mathrm{e}^{i\varphi}:=\frac{\hat{u}(0^{+})}{2\pi i}\ ,\
\varrho>0.$

The discussion above shows that the following results holds.

######  Proposition 6.2.

If $u(t)\in H^{1}_{+}(\mathbb{R})$ is a multi-soliton potential such that
$L_{u}$ has eigenvalues $\lambda_{1}=0,\lambda_{2},\dots,\lambda_{N}$, then
$u$ can be recovered as

$u(t,x)=\sqrt{2\varrho}\,{\rm
e}^{i\varphi}\langle(\mathsf{M}(t)-xI)^{-1}X,Y\rangle_{\mathbb{C}^{N}}\quad\mbox{for}\quad{\rm
Im}\,x\geq 0,$

where

$\displaystyle X:=(1,\dots,1)^{T}\ ,\ Y:=(1,0,\dots,0)^{T}\ ,$
$\displaystyle\mathsf{M}_{jk}=\frac{i}{\lambda_{j}-\lambda_{k}}\ \ (1\leq
j\neq k\leq N),\quad\mathsf{M}_{jj}=\gamma_{j}-i\varrho\delta_{j1}\ \
(j=1,\dots,N).$

Furthermore, the following evolution laws hold:

$\frac{d}{dt}\varphi=0\ ,\frac{d}{dt}\varrho=0\ ,\
\frac{d}{dt}\gamma_{j}=2\lambda_{j},\quad(j=1,\dots,N).$

###  6.3. Global-in-Time Existence

In order to prove that $N$-soliton solutions $u(t,x)$ can be extended to all
times $t\in\mathbb{R}$, we are going to study the eigenvalues of the matrix
$\mathsf{M}(t)\in\mathbb{C}^{N\times N}$ introduced above. To this end, we
note that the time evolution of $\mathsf{M}(t)$ stated in Proposition 6.2 can
be written as

$\mathsf{M}(t)=2\mathsf{V}t+\mathsf{W}$ (6.12)

with the constant complex $N\times N$-matrices
$\mathsf{V}=(\mathsf{V}_{jk})_{1\leq j,k\leq N}$ and
$\mathsf{W}=(\mathsf{W}_{jk})_{1\leq j,k\leq N}$ having the entries:

$\mathsf{V}_{jk}=\lambda_{j}\delta_{jk},\quad\mathsf{W}_{jk}=\begin{dcases*}\gamma_{j}-i\varrho\delta_{j1}&if
$j=k$,\\\ \frac{i}{\lambda_{j}-\lambda_{k}}&if $j\neq k$.\end{dcases*}$ (6.13)

Recall that $\lambda_{1}=0,\lambda_{2},\ldots,\lambda_{N}\in\mathbb{R}$ are
real and pairwise distinct and $\varrho>0$ is a strictly positive real number,
whereas $\gamma_{1},\ldots,\gamma_{N}\in\mathbb{R}$ are real numbers (which
are not necessarily pairwise distinct).

To prove that multi-soliton solutions extend to all times $t\in\mathbb{R}$, we
show that all the eigenvalues of $\mathsf{M}(t)$ are always in the lower
complex plane $\mathbb{C}_{-}=\\{z\in\mathbb{C}:\mathrm{Im}\,z<0\\}$.

######  Lemma 6.1.

For any $t\in\mathbb{R}$, all eigenvalues of the matrix $\mathsf{M}(t)$ have
strictly negative imaginary parts, i. e., it holds

$\sigma(\mathsf{M}(t))\subset\mathbb{C}_{-}\quad\mbox{for $t\in\mathbb{R}$}.$

######  Remark.

Below we will prove the remarkable fact that all the eigenvalues of
$\mathsf{M}(t)$ except for one will asymptotically converge to the real axis
as $t\to\pm\infty$. As a consequence, this implies that all $N$-solitons with
$N\geq 2$ will have an algebraic growth of their Sobolev norms according to
$\|u(t)\|_{H^{s}}\sim|t|^{2s}$ for any $s>0$.

###### Proof.

Let $t\in\mathbb{R}$ be given. For notational convenience we write
$\mathsf{M}=\mathsf{M}(t)$ in what follows. We readily verify the identities

$\frac{(\mathsf{M}-\mathsf{M}^{*})_{jk}}{2i}=-2\varrho\delta_{1j}\delta_{1k},\quad[\mathsf{M},\mathsf{V}]=[\mathsf{W},\mathsf{V}]=iI-i\langle\cdot,X\rangle
X,$ (6.14)

where $\langle\cdot,\cdot\rangle_{\mathbb{C}^{N}}$ denotes the standard inner
product on $\mathbb{C}^{N}$ and $X=(1,\ldots,1)^{T}\in\mathbb{C}^{N}$. Because
of the first identity above, any eigenvalue $z\in\sigma(\mathsf{M})$ must
satisfy

$\mathrm{Im}\,z\leq 0.$

Let us prove that $z\not\in\mathbb{R}$ holds. We argue by contradiction as
follows. Suppose that $z\in\mathbb{R}$ and let
$v=(v_{1},\ldots,v_{N})^{T}\in\mathbb{C}^{N}\setminus\\{0\\}$ be a
corresponding eigenvector, i. e.,

$\mathsf{M}v=zv.$

Since we assume that $z$ is real, we can use the first identity in (6.14) to
deduce that

$0=\frac{1}{2i}\langle(\mathsf{M}-\mathsf{M}^{*})v,v\rangle_{\mathbb{C}^{N}}=-2\varrho|v_{1}|^{2}$

and therefore $v_{1}=0$. Projecting the equation $\mathsf{M}v=zv$ onto the 1st
mode and recalling that $\lambda_{1}=0$ by assumption, we infer

$\sum_{j=2}^{N}\frac{v_{j}}{\lambda_{j}}=0.$

This can be written as

$\langle w,X\rangle=0\quad\mbox{with}\quad
w:=\left(0,\frac{v_{2}}{\lambda_{2}},\ldots,\frac{v_{N}}{\lambda_{N}}\right)^{T}.$

Noticing that $\mathsf{V}w=v$ and applying the second identity in (6.14), we
find

$\mathsf{V}(\mathsf{M}-zI)w=i(w-\langle w,X\rangle_{\mathbb{C}^{N}}X)=iw.$
(6.15)

If we take inner product of the left side with $w$ and using that
$\mathsf{V}=\mathsf{V}^{*}$, we conclude

$\displaystyle\langle w,\mathsf{V}(\mathsf{M}-zI)w\rangle_{\mathbb{C}^{N}}$
$\displaystyle=\langle\mathsf{V}w,\mathsf{M}w\rangle_{\mathbb{C}^{N}}-z\langle
w,\mathsf{V}w\rangle_{\mathbb{C}^{N}}=\langle\mathsf{M}^{*}v,w\rangle_{\mathbb{C}^{N}}-z\langle
w,v\rangle_{\mathbb{C}^{N}}$ $\displaystyle=z\langle
v,w\rangle_{\mathbb{C}^{N}}-z\langle
w,v\rangle_{\mathbb{C}^{N}}=z\sum_{j=2}^{N}\frac{|v_{j}|^{2}}{\lambda_{j}}-z\sum_{j=2}^{N}\frac{|v_{j}|^{2}}{\lambda_{j}}=0,$

where we used that $z\in\mathbb{R}$ and that $\mathsf{M}^{*}v=\mathsf{M}v=zv$
holds thanks to the form of $\mathsf{M}$ and $v_{1}=0$. Thus from (6.15) we
deduce that $0=i|w|^{2}$. This shows that $w=0$ and consequently we find
$v=0$, which is a contradiction. Therefore $z\in\mathbb{R}$ cannot be an
eigenvalue of $\mathsf{M}$. ∎

As a direct consequence of Proposition 6.2 and Lemma 6.1, we deduce the
following result.

######  Theorem 6.1 (Global-in-Time Existence).

Suppose $u_{0}\in H^{1}_{+}(\mathbb{R})$ is an $N$-soliton potential with some
$N\geq 1$. Then the corresponding solution $u(t)$ of (CM-DNLS) with
$u(0)=u_{0}$ extends to all times $t\in\mathbb{R}$.

######  Remark 6.1.

The previous study shows that, to every $N$–soliton $u$, one can associate

$\Lambda(u):=(\varphi,\varrho,\lambda_{2},\dots,\lambda_{N},\gamma_{1},\dots,\gamma_{N})\in\mathbb{T}\times(0,+\infty)\times\mathbb{R}^{2N-1}$

with the condition that $\lambda_{1}=0,\lambda_{2},\dots,\lambda_{N}$ are
pairwise distinct, so that

$\sqrt{2\varrho}\mathrm{e}^{i\varphi}=\frac{\hat{u}(0^{+})}{2i\pi}$

and $\lambda_{1}=0,\lambda_{2},\dots,\lambda_{N}$ are the eigenvalues of
$L_{u}$, while $\gamma_{j}=\langle
G\psi_{j},\psi_{j}\rangle-i\varrho\delta_{j1}$ if
$L_{u}\psi_{j}=\lambda_{j}\psi_{j},\|\psi_{j}\|_{L^{2}}=1$.
We claim that the spectral mapping $u\mapsto\Lambda(u)$ is surjective. Indeed,
assume we are given
$(\varphi,\varrho,\lambda_{2},\dots,\lambda_{N},\gamma_{1},\dots,\gamma_{N})\in\mathbb{T}\times(0,+\infty)\times\mathbb{R}^{2N-1}$
with the condition that $\lambda_{1}=0,\lambda_{2},\dots,\lambda_{N}$ are
pairwise distinct, and consider the associated matrix $\mathsf{M}$. According
to Lemma 6.1, we already know that the eigenvalues of $\mathsf{M}$ belong to
the lower half plane, so that we may define $u$ as

$u(x)=\sqrt{2\varrho}\,{\rm
e}^{i\varphi}\langle(\mathsf{M}-xI)^{-1}X,Y\rangle_{\mathbb{C}^{N}}\quad\mbox{for}\quad{\rm
Im}\,x\geq 0.$

We claim that $u$ is a $N$–soliton with $Q(x)=\det(xI-\mathsf{M})$. Indeed,
this is equivalent to the identity

$|u(x)|^{2}=\frac{1}{i}\left(\frac{\overline{Q}^{\prime}(x)}{\overline{Q}(x)}-\frac{Q^{\prime}(x)}{Q(x)}\right)\
,\ x\in\mathbb{R}\ .$ (6.16)

Using the expressions of $u(x)$ and $Q(x)$, (6.16) is equivalent to

$|\langle Z(x),X\rangle_{\mathbb{C}^{N}}|^{2}=\|Z(x)\|^{2}$

where $Z(x):=(xI-\mathsf{M}^{*})^{-1}Y\ .$ The latter identity can be proved
as follows. From $(xI-\mathsf{M}^{*})Z(x)=Y$, we infer
$\mathsf{V}(xI-\mathsf{M}^{*})Z(x)=0\ .$ Taking the imaginary part of the
inner product of both sides with $Z(x)$, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\langle(\mathsf{V}(xI-\mathsf{M}^{*})-(xI-\mathsf{M})\mathsf{V})Z(x),Z(x)\rangle_{\mathbb{C}^{N}}$
$\displaystyle=$
$\displaystyle\langle[\mathsf{V},(xI-\mathsf{M}^{*})]Z(x),Z(x)\rangle_{\mathbb{C}^{N}}=i(|\langle
Z(x),X\rangle_{\mathbb{C}^{N}}|^{2}-\|Z(x)\|^{2})\ .$

Furthermore, from Proposition 6.2, the value of $\hat{u}(0^{+})$ can be
obtained by identifying the coefficient of $1/x$ in the expansion of $u(x)$ as
$x\to\infty$. In order to complete the proof of the surjectivity, we just have
to check the following identities,

$L_{u}\psi_{j}=\lambda_{j}\psi_{j}\ ,\ \langle
G\psi_{j},\psi_{j}\rangle=(\gamma_{j}-i\varrho\delta-{j1})\|\psi_{j}\|_{L^{2}}^{2}\
,$

if we define $\psi_{j}$ as
$\psi_{j}(x)=\langle(\mathsf{M}-xI)^{-1}Y_{j},Y\rangle_{\mathbb{C}^{N}}\ ,$
where $Y_{j}$ denotes the column with $1$ on the line $j$ and $0$ on the other
lines. This can be done by direct calculations. For instance, since
$\mathsf{V}Y_{j}=\lambda_{j}Y_{j}$ and $\mathsf{V}Y=0$,

$\displaystyle\lambda_{j}\psi_{j}(x)$ $\displaystyle=$
$\displaystyle\langle(\mathsf{M}-xI)^{-1}\mathsf{V}Y_{j},Y\rangle_{\mathbb{C}^{N}}=\langle(\mathsf{M}-xI)^{-1}[\mathsf{V},\mathsf{M}](\mathsf{M}-xI)^{-1}Y_{j},Y\rangle_{\mathbb{C}^{N}}$
$\displaystyle=$ $\displaystyle
D\psi_{j}(x)+i\langle(\mathsf{M}-xI)^{-1}Y_{j},X\rangle_{\mathbb{C}^{N}}\langle(\mathsf{M}-xI)^{-1}X,Y\rangle_{\mathbb{C}^{N}}$

while

$\displaystyle\langle(\overline{\mathsf{M}}-xI)^{-1}X,Y\rangle_{\mathbb{C}^{N}}\langle(\mathsf{M}-xI)^{-1}Y_{j},Y\rangle_{\mathbb{C}^{N}}=\langle(\mathsf{M}^{*}-xI)^{-1}YY^{T}(\mathsf{M}-xI)^{-1}Y_{j},X\rangle_{\mathbb{C}^{N}}$
$\displaystyle=(2i\varrho)^{-1}[\langle(\mathsf{M}-xI)^{-1}X,Y\rangle_{\mathbb{C}^{N}}-\langle(\mathsf{M}^{*}-xI)^{-1}Y_{j},X\rangle_{\mathbb{C}^{N}}],$

so that
$i\langle(\mathsf{M}-xI)^{-1}Y_{j},X\rangle_{\mathbb{C}^{N}}\langle(\mathsf{M}-xI)^{-1}X,Y\rangle_{\mathbb{C}^{N}}=-u(x)\Pi_{+}(\overline{u}\psi_{j})(x)$.

Below we will show that all multi-soliton solutions with $N\geq 2$ exhibit
growth of Sobolev norms such that
$(\phi,\varrho,\lambda_{2},\dots,\lambda_{N},\gamma_{1},\dots,\gamma_{N})\in\mathbb{T}\times(0,+\infty)\times\mathbb{R}^{2N-1}$
is given so that

$\|u(t)\|_{H^{s}}\sim|t|^{2s}\quad\mbox{as $t\to\pm\infty$ for any $s>0$}.$

This demonstrates that global-in-time existence for such $u(t,x)$ cannot be
inferred from a-priori bounds in $H^{1}$. Furthermore, it shows that the
infinite hierarchy of conversation laws $\\{I_{k}(u)\\}_{k=0}^{\infty}$
generally fail to produce a-priori bounds on solutions of (CM-DNLS) in the
large data regime with $\|u(0)\|_{L^{2}}^{2}>2\pi$.

###  6.4. Explicit Example: Two-Soliton Solutions

Before we study the case of $N$-solitons with arbitrary $N\geq 1$, it is
instructive to first consider the case $N=2$ in detail. In this case, the
constraints in Proposition 5.2 can be solved explicitly. Suppose that

$u(x)=\frac{P(x)}{Q(x)}\in H^{1}_{+}(\mathbb{R})$ (6.17)

is a two-soliton potential, i. e., $Q\in\mathbb{C}[x]$ is a polynomial of
degree 2 with zeros in $\mathbb{C}_{-}$ and $P\in\mathbb{C}[x]$ is a
polynomial of degree at most 1 satisfying the condition stated in Proposition
5.2 above.

We start with generic case by assuming that $Q(x)$ has two different zeros
$z_{1}\neq z_{2}$ in $\mathbb{C}_{-}$. Then condition (ii) in Proposition 5.2
implies that the polynomial $P(x)=\mbox{const}$ is constant and we find that

$u(x)=\frac{a_{1}}{x-x_{1}}+\frac{a_{2}}{x-z_{2}},\quad\sum_{j=1}^{2}\frac{a_{j}\overline{a}_{k}}{z_{j}-\overline{z}_{k}}=i\quad\mbox{with
$k=1,2$}.$

The constraints on $a_{1},a_{2}\in\mathbb{C}$ with given
$z_{1},z_{2}\in\mathbb{C}_{-}$ can be solved explicitly as follows. Writing
for convenience $y_{j}=-{\rm Im}z_{j}>0$, we find

$\frac{|a_{1}|^{2}}{2y_{1}}-i\frac{a_{2}\overline{a}_{1}}{z_{2}-\overline{z}_{1}}=1\
,\
-i\frac{a_{1}\overline{a}_{2}}{z_{1}-\overline{z}_{2}}+\frac{|a_{2}|^{2}}{2y_{2}}=1\
.$

Let us set

$\xi:=\frac{|a_{1}|^{2}}{2y_{1}}=\frac{|a_{2}|^{2}}{2y_{2}}\ ,\
\eta:=-i\frac{a_{2}\overline{a}_{1}}{z_{2}-\overline{z}_{1}}=-i\frac{a_{1}\overline{a}_{2}}{z_{1}-\overline{z}_{2}}=1-\xi\
.$

Then

$a_{j}=\sqrt{2\xi y_{j}}\mathrm{e}^{i\theta_{j}}$

and

$1-\xi=\eta=-2i\sqrt{y_{1}y_{2}}\,\xi\,\frac{\mathrm{e}^{i(\theta_{1}-\theta_{2})}}{z_{1}-\overline{z}_{2}}\
.$

Now we discuss the two solutions, according to $\xi<1$ or $\xi>1$. If $\xi<1$,

$1-\xi=\xi\frac{2\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}\ ,$

hence

$\xi=\frac{1}{1+\frac{2\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}}.$

Thus we get

$a_{1}=i\left(\frac{2y_{1}}{1+2\frac{\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}}\right)^{\frac{1}{2}}\frac{z_{1}-\overline{z}_{2}}{|z_{1}-\overline{z}_{2}|}\mathrm{e}^{i\theta}\
,\quad
a_{2}=\left(\frac{2y_{2}}{1+2\frac{\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}}\right)^{\frac{1}{2}}{\rm
e}^{i\theta}\ .$

If $\xi>1$,

$\xi-1=\xi\frac{2\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}\ ,$

hence

$\xi=\frac{1}{1-\frac{2\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}}.$

Therefore,

$a_{1}=-i\left(\frac{2y_{1}}{1-2\frac{\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}}\right)^{\frac{1}{2}}\frac{z_{1}-\overline{z}_{2}}{|z_{1}-\overline{z}_{2}|}\mathrm{e}^{i\theta}\
,\quad
a_{2}=\left(\frac{2y_{2}}{1-2\frac{\sqrt{y_{1}y_{2}}}{|z_{1}-\overline{z}_{2}|}}\right)^{\frac{1}{2}}\mathrm{e}^{i\theta}\
.$

Let us compute the remaining eigenvalue $\lambda$ of $L_{u}$ in both cases. It
is enough which is the trace of $L_{u}$ on $\mathcal{E}_{\rm pp}=K_{\theta}$
in view of Proposition 5.3. If $\\{j,k\\}=\\{1,2\\}$, one checks that

$L_{u}\left(\frac{1}{x-z_{k}}\right)=-i\frac{a_{j}}{a_{k}(z_{k}-z_{j})}\frac{1}{x-z_{k}}+i\frac{a_{j}}{a_{k}(z_{k}-z_{j})}\frac{1}{x-z_{j}}\
.$

This implies

$\lambda=-i\frac{a_{2}}{a_{1}(z_{1}-z_{2})}-i\frac{a_{1}}{a_{2}(z_{2}-z_{1})}.$

After some calculations we obtain

$\lambda=\frac{-(y_{1}+y_{2})}{\sqrt{y_{1}y_{2}}|z_{1}-\overline{z}_{2}|}<0\quad\mbox{and}\quad\lambda=\frac{(y_{1}+y_{2})}{\sqrt{y_{1}y_{2}}|z_{1}-\overline{z}_{2}|}>0$

in the first and in the second case, respectively.

Finally, we consider the non-generic case when $z_{1}=z_{2}=z$ and
$Q(x)=(x-z)^{2}$. Then

$F(x):=i(Q^{\prime}(x)\overline{Q}(x)-Q(x)\overline{Q}^{\prime}(x))=4y(x-z)(x-\overline{z}).$

Writing $y=-\mathrm{Im}\,z>0$ again, we have two possibilities:

$P(x)=\sqrt{4y}(x-z)\ ,\ u(x)=\frac{\sqrt{4y}}{x-z}$

and

$P(x)=\sqrt{4y}(x-\overline{z})\ ,\
u(x)=\frac{\sqrt{4y}(x-\overline{z})}{(x-z)^{2}}\ .$

Notice that the first case should not be confused with the case $N=1$, where

$Q(x)=x-z\ ,\ u(x)=\frac{\sqrt{2y}}{x-z}\ .$

In the first case, we have

$L_{u}\left(\frac{1}{(x-z)^{2}}\right)=-\frac{1}{y}\frac{1}{(x-z)^{2}},$

so that $\lambda=-y^{-1}$, which can obtained from the first formula by
letting $z_{1}\to z$ and $z_{2}\to z$. Similarly, in the second case, one
obtains that $\lambda=y^{-1}$.

As a next step, we study the time evolution of two-solitons. For $N=2$, the
matrix $\mathsf{M}(t)$ is given by

$\mathsf{M}(t)=\left(\begin{array}[]{cc}\gamma_{1}-i\varrho&-i\lambda^{-1}\\\
i\lambda^{-1}&\gamma_{2}+2\lambda t\end{array}\right)$ (6.18)

with some positive number $\varrho>0$ and real numbers
$\gamma_{1},\gamma_{2}\in\mathbb{R}$ and $\lambda\neq 0$ denotes the non-zero
eigenvalue of $L_{u}$. From Proposition 6.2 we deduce

$u(t,x)=\frac{\mathrm{e}^{i\varphi}\sqrt{2\varrho}(\gamma_{2}+2\lambda
t+i\lambda^{-1}-x)}{x^{2}-(\gamma_{1}-i\varrho+\gamma_{2}+2\lambda
t)x+(\gamma_{1}-i\varrho)(\gamma_{2}+2\lambda t)-\lambda^{-2}}$ (6.19)

with some constant $\varphi\in[0,2\pi)$. The discriminant of the denominator
of $u(t,x)$ is

$\Delta(t)=(\gamma_{1}-i\varrho-\gamma_{2}-2\lambda t)^{2}+4\lambda^{-2}.$
(6.20)

Note that $\Delta(t)=0$ if and only if $\gamma_{1}=\gamma_{2}+2\lambda t$ and
$\varrho|\lambda|=2$. This corresponds to the degenerate two-solitons, the
cases 1 and 2 occurring according to the sign of $\lambda$. Notice that, if
$\varrho|\lambda|=2$, the two-soliton solution of (6.8) will be degenerate at
exactly one time $t\in\mathbb{R}$ characterized by
$\gamma_{1}=\gamma_{2}+2\lambda t$.

Finally, let us study the large-time behaviour of a two-soliton. Let
$z_{+}(t)$ and $z_{-}(t)$ denote the poles of $u(t,x)$ at time $t$. We see
that

$z_{\pm}(t)=\frac{1}{2}\left(\gamma_{0}-i\varrho+\gamma_{1}+2\lambda
t\pm\sqrt{(\gamma_{0}-i\varrho-\gamma_{1}-2\lambda
t)^{2}+4\lambda^{-2}}\right).$ (6.21)

As $t\to\pm\infty$, we obtain

$z_{+}(t)\to\gamma_{0}-i\varrho,\quad{\rm Re}\,z_{-}(t)=2\lambda
t+O(1),\quad{\rm Im}\,z_{-}(t)=\frac{-\varrho}{4\lambda^{4}t^{2}}+O(t^{-3}).$
(6.22)

The vanishing of ${\rm Im}\,z_{-}(t)$ as $t\to\pm\infty$ implies growth of the
Sobolev norms for the two-soliton solution. More precisely, we claim that

$\|u(t)\|_{H^{s}}\sim c_{s}|t|^{2s}\quad\mbox{as $t\to\pm\infty$ for any
$s>0$}.$ (6.23)

To prove this, we note that $z_{+}(t)\neq z_{-}(t)$ for all $t\in\mathbb{R}$
(except for one time $t$ at most) and we have that

$u(t,x)=\frac{a_{+}(t)}{x-z_{+}(t)}+\frac{a_{-}(t)}{x-z_{-}(t)},\quad
a_{\pm}(t)=\sqrt{2\varrho}\mathrm{e}^{i\varphi}\frac{\gamma_{1}+2\lambda
t+i\lambda^{-1}-z_{\pm}(t)}{\sqrt{\Delta(t)}}.$

From (6.22), we infer as $|t|\to\infty$ that

$a_{+}(t)=O(1)\quad\mbox{and}\quad a_{-}(t)\sim\frac{1}{|t|}.$

Since

$\hat{u}(t,\xi)=-2\pi
i\left(a_{+}(t)\mathrm{e}^{-iz_{+}(t)\xi}+a_{-}(t)\mathrm{e}^{-iz_{-}(t)\xi}\right)\quad\mbox{for}\quad\xi>0,$

we deduce the bound (6.23) from (6.22) and by direct calculation.

###  6.5. Long-Time Asymptotics

We now study the long-time behavior for $N$-solitons with general $N\geq 2$.
The key ingredient for the general understanding is the following result about
the long-time asymptotics for the eigenvalues of the matrix $\mathsf{M}(t)$ in
(6.12).

######  Lemma 6.2.

There exists $T_{0}>0$ sufficiently large such that all eigenvalues

$\\{z_{1}(t),\ldots,z_{N}(t)\\}\subset\mathbb{C}_{-}$

of $\mathsf{M}(t)$ are simple for $|t|\geq T_{0}$. As $|t|\to+\infty$, we have
the asymptotic expansions:

$\mathrm{Re}\,z_{k}(t)=2\lambda_{k}t+\gamma_{k}+O(t^{-1})\quad\mbox{for
$k=1,\ldots,N$}$
$\mathrm{Im}\,z_{1}(t)=-\varrho+O(t^{-1}),\quad\mathrm{Im}\,z_{k}(t)=-\frac{\varrho}{4\lambda_{k}^{4}t^{2}}+O(t^{-3})\quad\mbox{for
$k=2,\ldots,N$}.$

###### Proof.

We use standard eigenvalue perturbation theory for

$\mathsf{A}(\varepsilon)=\mathsf{A}+\varepsilon\mathsf{B}$

with a small parameter $|\varepsilon|\ll 1$. Here
$\mathsf{A}=\mathsf{A}^{*}\in\mathbb{C}^{N\times N}$ is a Hermitian matrix and
$\mathsf{B}$ denotes an arbitrary matrix in $\mathbb{C}^{N\times N}$.

Since $\mathsf{M}(t)=t(2\mathsf{V}+t^{-1}\mathsf{W})$ and by taking
$\varepsilon=t^{-1}$, it suffices to study the eigenvalues of

$\mathsf{A}(\varepsilon)=\mathsf{A}+\varepsilon\mathsf{B}\quad\mbox{with}\quad\mbox{$\mathsf{A}=2\mathsf{V}$
and $\mathsf{B}=\mathsf{W}$},$

where the constant matrices $\mathsf{V}$ and $\mathsf{W}$ are displayed in
(6.13). Because $\mathsf{A}=2\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$
has $N$ simple eigenvalues, standard perturbation theory yields that
$\mathsf{A}(\varepsilon)$ has $N$ simple eigenvalue provided that
$|\varepsilon|\ll 1$ is sufficiently small. Thus for $|t|\geq T_{0}$ with
$T_{0}>0$ sufficiently large, we see that $\mathsf{M}(t)=t\mathsf{A}(t^{-1})$
has $N$ simple eigenvalues $\\{z_{1}(t),\ldots,z_{N}(t)\\}$ which all belong
to $\mathbb{C}_{-}$ by Lemma 6.1.

The derivation of the claimed asymptotics for the eigenvalues requires an
expansion up to order $\varepsilon^{3}=t^{-3}$. Let
$\\{\mu_{1}(\varepsilon),\ldots,\mu_{N}(\varepsilon)\\}$ denote the
eigenvalues of $\mathsf{A}+\varepsilon\mathsf{B}$, which are simple for
$|\varepsilon|\ll 1$ sufficiently small. From [32][Section XII.1] we recall
that

$\mu_{k}(\varepsilon)=\mu_{k}+\mu_{k}^{(1)}\varepsilon+\mu_{k}^{(2)}\varepsilon^{2}+\mu_{k}^{(3)}\varepsilon^{3}+O(\varepsilon^{4}),\quad\mbox{with
$\mu_{k}=2\lambda_{k}$ }.$

The first-order coefficient is the well-known expression

$\mu_{k}^{(1)}=\mathsf{B}_{kk}=\gamma_{k}-i\varrho\delta_{k1}.$

Next, the second-order contribution is purely real and given by

$\mu_{k}^{(2)}=-\sum_{j=1,j\neq
k}^{N}\frac{1}{\mu_{j}-\mu_{k}}\mathsf{B}_{kj}\mathsf{B}_{jk}=-\frac{1}{2}\sum_{j=1,j\neq
k}^{N}\frac{1}{(\mu_{j}-\mu_{k})^{3}}\in\mathbb{R}.$

Finally, the third-order term reads

$\mu_{k}^{(3)}=\sum_{j\neq k,\ell\neq
k}^{N}\frac{1}{(\mu_{j}-\mu_{k})(\mu_{\ell}-\mu_{k})}\mathsf{B}_{kj}\mathsf{B}_{j\ell}\mathsf{B}_{\ell
k}-\sum_{j\neq
k}^{N}\frac{1}{(\mu_{j}-\mu_{k})^{2}}\mathsf{B}_{kj}\mathsf{B}_{jk}\mathsf{B}_{kk}.$

For the proof of the lemma, it suffices to determine
$\mathrm{Im}\,\mu_{k}^{(3)}$ for $k=2,\ldots,N$. Since
$\mathsf{B}_{kk}\in\mathbb{R}$ for $k\geq 2$ and
$\mathsf{B}_{kj}\mathsf{B}_{jk}\in\mathbb{R}$ if $j\neq k$, we see

$\mathrm{Im}\sum_{j\neq
k}^{N}\frac{1}{(\mu_{j}-\mu_{k})^{2}}\mathsf{B}_{kj}\mathsf{B}_{jk}\mathsf{B}_{kk}=0\quad\mbox{for
$k=2,\ldots,N$}.$

As for the first sum in the expression for $\mu_{k}^{(3)}$, we notice the
symmetry property

$\mathsf{B}_{kj}\mathsf{B}_{j\ell}\mathsf{B}_{\ell k}=-\mathsf{B}_{\ell
j}\mathsf{B}_{\ell j}\mathsf{B}_{jk}\quad\mbox{for $j\neq\ell$}.$

Thus we only need to consider the diagonal case when $j=\ell$. This leads to

$\mathrm{Im}\,\mu_{k}^{(3)}=\mathrm{Im}\sum_{j\neq
k}^{N}\frac{1}{(\mu_{j}-\mu_{k})^{2}}\mathsf{B}_{kj}\mathsf{B}_{jj}\mathsf{B}_{jk}.$

Recalling that $\mathsf{B}_{kj}\mathsf{B}_{jk}\in\mathbb{R}$ for $j\neq k$ and
$\mathsf{B}_{jj}\in\mathbb{R}$ if $j\geq 2$, we deduce

$\mathrm{Im}\,\mu_{k}^{(3)}=\mathrm{Im}\frac{1}{(\mu_{1}-\mu_{k})^{2}}\mathsf{B}_{1k}\mathsf{B}_{11}\mathsf{B}_{k1}=-\frac{\varrho}{4\lambda_{k}^{4}}\quad\mbox{for
$k=2,\ldots,N$},$

using that $\mu_{k}=2\lambda_{k}$ and $\mu_{1}=2\lambda_{1}=0$. Since
$\mathsf{M}(t)=t^{-1}\mathsf{A}(t^{-1})$, we obtain the claimed asymptotic
formulae. ∎

As a consequence of the preceding lemma, we obtain the following result.

######  Proposition 6.3.

Let $u(t,x)$ be an $N$-soliton solution. Then there exists $T_{0}>0$
sufficiently large such that

$u(t,x)=\sum_{j=1}^{N}\frac{a_{j}(t)}{x-z_{j}(t)}\quad\mbox{for $|t|\geq
T_{0}$},$

where $\\{z_{1}(t),\ldots,z_{N}(t)\\}\subset\mathbb{C}_{-}$ denote the simple
eigenvalues of $\mathsf{M}(t)$ and coefficients
$a_{1}(t),\ldots,a_{N}(t)\in\mathbb{C}\setminus\\{0\\}$.

###### Proof.

By Lemma 6.2, the eigenvalues
$\\{z_{1}(t),\ldots,z_{N}(t)\\}\subset\mathbb{C}_{-}$ of $\mathsf{M}(t)$ are
simple whenever $|t|\geq T_{0}$, where $T_{0}>0$ is some sufficiently large
constant.

Fix some time $t\in\mathbb{R}$ with $|t|\geq T_{0}$. For notational
convenience, we will omit the dependence of $u(t,x)$ and $\mathsf{M}(t)$ for
the rest of the proof. Since $u\in H^{1}_{+}(\mathbb{R})$ is a multi-soliton
potential, we have

$u(x)=\frac{P(x)}{Q(x)}$

with some polynomial $Q\in\mathbb{C}[x]$ of degree $N$ having all its zeros in
$\mathbb{C}_{-}$ and some polynomial $P\in\mathbb{C}_{N-1}[x]$ satisfying the
condition in Proposition 5.2. Recall that $\mathsf{M}$ denotes the matrix
(with respect to a suitable orthonormalbasis) of the operator $G$ acting on
the invariant space $K_{\theta}=\frac{\mathbb{C}_{N-1}[x]}{Q(x)}$. From
[34][Lemma 3.3] we observe that $Q(x)$ is the characteristic polynomial of $G$
acting on $K_{\theta}$. Hence we conclude

$Q(x)=\det(xI-\mathsf{M}).$

Since $\mathsf{M}$ has only simple eigenvalues, we find
$Q(x)=\prod_{j=1}^{N}(x-z_{j})$ with pairwise distinct zeros
$z_{j}\in\mathbb{C}_{-}$. Since
$P\overline{P}=i(Q^{\prime}\overline{Q}-\overline{Q}^{\prime}Q)$, we see that
$P\in\mathbb{C}_{N-1}[x]$ has no common zeros with $Q(x)$. Thus, by partial
fraction expansion, we conclude that

$u(x)=\frac{P(x)}{Q(x)}=\sum_{j=1}^{N}\frac{a_{j}}{x-z_{j}}$

with some $a_{1},\ldots,a_{N}\in\mathbb{C}\setminus\\{0\\}$. ∎

###  6.6. Growth of Sobolev Norms: Proof of Theorem 1.3

Let $N\geq 2$ and suppose $u(t,x)$ is an $N$-soliton, which by Theorem 6.1
exists for all times $t\in\mathbb{R}$. We claim that

$\|u(t)\|_{H^{s}}\sim_{s}t^{2s}\quad\mbox{as}|t|\to+\infty$ (6.24)

for any $s>0$.

For convenience, we discuss the limit $t\to+\infty$. (The case $t\to-\infty$
follows by the same reasoning.) We divide the proof into following steps.

Step 1. We first derive the asymptotics for $a_{1}(t),\ldots,a_{N}(t)$. From
Proposition 6.3 we deduce that

$u(t,x)=\sum_{j=1}^{N}\frac{a_{j}(t)}{x-z_{j}(t)}\quad\mbox{for}\quad t\geq
T_{0}$ (6.25)

with some sufficiently large constant $T_{0}>0$. Now we are in the position to
use the system of differential equations in (6.3) for the time evolution of
$a_{1}(t),\ldots,a_{N}(t)$. We claim that

$a_{k}(t)=\frac{\alpha}{\lambda_{k}^{2}t}+O(t^{-2})\quad\mbox{for
$k=2,\ldots,N$}$ (6.26)

with some non-zero constant $\alpha\in\mathbb{C}\setminus\\{0\\}$. Indeed,
using that $\dot{z}_{k}=2\lambda_{k}+o(1)$ as $t\to+\infty$ by Lemma 6.2, we
deduce from (6.3) that

$\vec{a}(t)=A(t)\vec{a}(t)+\vec{b}(t)$

where $\vec{a}(t)=(a_{2}(t),\ldots,a_{N}(t))\in\mathbb{C}^{N-1}$ and
$A(t)=(A(t))_{2\leq k,\ell\leq N}\in\mathbb{C}^{N-1}\times\mathbb{C}^{N-1}$,
$\vec{b}(t)=(b_{2}(t),\ldots,b_{N}(t))\in\mathbb{C}^{N-1}$ are given by

$(A(t))_{k\ell}=\frac{-i}{(\lambda_{k}+o(1))(z_{k}-z_{\ell})}=O(t^{-1})\quad\mbox{for
$k\neq\ell$},\quad(A(t))_{kk}=0,$
$b_{k}(t)=\frac{-ia_{0}(t)}{(\lambda_{k}+o(1))(z_{k}-z_{1})}=O(t^{-1})a_{1}(t).$

Here we also used that $|z_{k}(t)-z_{\ell}(t)|\sim t$ as $|t|\to\infty$ which
follows from Lemma 6.2 together with the fact that all
$\lambda_{k}\neq\lambda_{\ell}$ for $k\neq\ell$. Next, as a direct consequence
of (6.3), we infer the conservation law

$A=\sum_{j=1}^{N}a_{j}(t)=\frac{\hat{u}(t,0^{+})}{-2\pi i}\quad\mbox{for
$t\geq T_{0}$},$ (6.27)

with some constant $A\in\mathbb{C}$, where the last equation follows from
taking the Fourier transform of the right-hand side in (6.25). Since we must
have $\hat{u}(t,0^{+})\neq 0$ by the discussion above, we conclude that $A\neq
0$ as well. Hence we deduce that

$\vec{a}(t)=O(t^{-1})\vec{a}(t)+\vec{f}(t)$ (6.28)

where $\vec{f}(t)=(f_{2}(t),\ldots,f_{N}(t))$ is given by

$f_{k}(t)=\frac{-iA}{(\lambda_{k}+o(1))(z_{k}(t)-z_{1}(t))}=\frac{-iA}{2\lambda_{k}^{2}t}+O(t^{-2}),$

thanks to the fact that $z_{k}(t)-z_{1}(t)=2\lambda_{k}t+O(1)$ by Lemma 6.2
and $\lambda_{1}=0$. In view of (6.28), we conclude that (6.26) holds with the
constant $\alpha=\frac{-iA}{2}\neq 0$.

Finally, by the conservation law (6.27) together with (6.26) we immediately
find

$\lim_{t\to+\infty}a_{1}(t)=A\neq 0.$ (6.29)

Step 2. For $t\geq T_{0}$, we write

$u(t,x)=\sum_{j=1}^{N}\phi_{j}(t,x)\quad\mbox{with}\quad\phi_{j}(t,x)=\frac{a_{j}(t)}{x-z_{j}(t)}.$
(6.30)

We note that

$\hat{\phi}_{j}(t,\xi)=-2\pi
ia_{j}(t)\mathrm{e}^{-iz_{j}\xi}\mathds{1}_{\xi\geq 0}.$

By Plancherel, we find

$\displaystyle\langle\phi_{j}(t),\phi_{k}(t)\rangle_{H^{s}}$
$\displaystyle=-ia_{j}(t)\overline{a}_{k}(t)\int_{0}^{\infty}\mathrm{e}^{-i(z_{j}(t)-\overline{z}_{k}(t))\xi}\langle\xi\rangle^{2s}\,d\xi$

To estimate this expression for $j\neq k$, we observe, by Lemma 6.2, that
$\omega_{jk}(t):=z_{j}(t)-\overline{z}_{k}(t)$ satisfies
$\mathrm{Im}\,\omega_{jk}(t)<0$ as well as

$\omega_{jk}(t)=2(\lambda_{j}-\lambda_{k})t+O(1)\sim t\quad\mbox{as}\quad
t\to+\infty,$

since $\lambda_{j}\neq\lambda_{j}$ when $j\neq k$. Integrating by parts
sufficiently many times depending on $s>0$, we deduce that

$\left|\int_{0}^{\infty}\mathrm{e}^{-i\omega_{jk}(t)\xi}\langle\xi\rangle^{2s}\,d\xi\right|\lesssim_{s}\frac{1}{|\omega_{jk}(t)|}\sim\frac{1}{t}\quad\mbox{for
$t\geq T_{0}$},$

provided that $j\neq k$. If recall the bounds for $a_{1}(t),\ldots,a_{N}(t)$
derived in Step 1 above, we can conclude

$\left|\langle\phi_{j}(t),\phi_{k}(t)\rangle_{H^{s}}\right|\lesssim_{s}\frac{1}{t^{2}}\quad\mbox{for
$t\geq T_{0}$ and $j\neq k$}.$

Next, we consider the case $j=k$. This yields

$\displaystyle\langle\phi_{j}(t),\phi_{j}(t)\rangle_{H^{s}}$
$\displaystyle\sim|a_{j}(t)|^{2}\int_{0}^{\infty}\mathrm{e}^{2\mathrm{Im}(z_{j}(t))\xi}(1+|\xi|^{2s})d\xi$
$\displaystyle=|a_{j}(t)|^{2}\left(\frac{1}{2|\mathrm{Im}\,z_{j}(t)|}+\frac{C_{s}}{2|\mathrm{Im}\,z_{j}(t)|^{1+2s}}\right)$

with the constant $C_{s}=\int_{0}^{\infty}\mathrm{e}^{-y}y^{2s}\,dy>0$. By
combining the estimates for the coefficients $\\{a_{1}(t),\ldots,a_{N}(t)\\}$
from Step 1 and the poles $\\{z_{1}(t),\ldots,z_{N}(t)\\}$ from Lemma 6.2 we
finally obtain

$\langle\phi_{j}(t),\phi_{j}(t)\rangle_{H^{s}}\simeq_{s}\begin{dcases*}1&for
$j=1$,\\\ t^{4s}&for $j=2,\ldots,N$,\end{dcases*}$

for all times $t\geq T_{0}$. In summary, we conclude

$t^{4s}+t^{-2}\lesssim_{s}\|u(t)\|_{H^{s}}^{2}=\sum_{j=1}^{N}\langle\phi_{j}(t),\phi_{j}(t)\rangle+\sum_{j\neq
k}^{N}\langle\phi_{j}(t),\phi_{k}(t)\rangle\lesssim_{s}t^{4s}+t^{-2}.$

This proves (6.24) and completes the proof of Theorem 1.3. ∎

##  Appendix A Definition of the Lax Operator for $u\in L^{2}_{+}(\mathbb{R})$

In view of the $L^{2}$-criticality of (CM-DNLS), it is worthwhile giving a
definition of $L_{u}$ via quadratic forms if we only assume that $u\in
L^{2}_{+}(\mathbb{R})$ holds. We start with the following basic estimate.

######  Lemma A.1.

For every $u\in L^{2}_{+}(\mathbb{R})$ and $f\in
H^{\frac{1}{2}}_{+}(\mathbb{R})$, we have $T_{\overline{u}}f\in
L^{2}_{+}(\mathbb{R})$ with

$\|T_{\overline{u}}f\|_{L^{2}}^{2}\leq\frac{1}{2\pi}\|u\|_{L^{2}}^{2}\langle
Df,f\rangle\ .$

###### Proof.

Applying the Fourier transformation, we have

$\widehat{T_{\overline{u}}f}(\xi)=\int_{0}^{+\infty}\hat{f}(\xi+\eta)\overline{\hat{u}(\eta)}\,\frac{d\eta}{2\pi}\
.$

Thus, by the Cauchy–Schwarz inequality,

$\displaystyle\int_{0}^{\infty}|\widehat{T_{\overline{u}}f}(\xi)|^{2}\,d\xi$
$\displaystyle\leq$
$\displaystyle\int_{0}^{+\infty}\int_{0}^{+\infty}|\hat{f}(\xi+\eta)|^{2}\,\frac{d\xi\,d\eta}{2\pi}\
\int_{0}^{+\infty}|\hat{u}(\eta)|^{2}\,\frac{d\eta}{2\pi}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{+\infty}\zeta|\hat{f}(\zeta)|^{2}\,\frac{d\zeta}{2\pi}\
\int_{0}^{+\infty}|\hat{u}(\eta)|^{2}\,\frac{d\eta}{2\pi}\ ,$

and the claim follows from Plancherel’s theorem. ∎

Next, we define $L_{u}$ for given $u\in L^{2}_{+}(\mathbb{R})$ via a densely
defined quadratic form on $L_{+}^{2}(\mathbb{R})$ as follows. For $f,g\in
H^{1/2}_{+}(\mathbb{R})$, we set

$\mathcal{Q}_{u}(f,g)=\langle Df,g\rangle-\langle
T_{\overline{u}}f,T_{\overline{u}}g\rangle\ .$

We claim that, for every $\varepsilon>0$, there exists a constant
$C_{\varepsilon}(u)>0$ such that

$\forall h\in H^{1/2}_{+}(\mathbb{R})\ ,\
\|T_{\overline{u}}h\|_{L^{2}}\leq\varepsilon{\langle
Dh,h\rangle}^{1/2}+C_{\varepsilon}(u)\|h\|_{L^{2}}\ .$ (A.1)

Indeed, for every $\lambda>0$, set

$u_{<\lambda}:={\bf 1}_{[0,\lambda[}(D)u\ ,\ u_{\geq\lambda}:={\bf
1}_{[\lambda,+\infty[}(D)u\ .$

Then $\|u_{\geq\lambda}\|_{L^{2}}\to 0$ as $\lambda\to+\infty$, while

$\|u_{<\lambda}\|_{L^{\infty}}\leq\left(\frac{\lambda}{2\pi}\right)^{1/2}\|u\|_{L^{2}}\
.$

Choose $\lambda=\lambda(\varepsilon,u)$ such that
$(2\pi)^{-1/2}\|u_{\geq\lambda}\|_{L^{2}}\leq\varepsilon$. Then, by Lemma A.1,

$\|T_{\overline{u}}h\|_{L^{2}}\leq\|T_{\overline{u}_{\geq\lambda}}h\|_{L^{2}}+\|T_{\overline{u}_{<\lambda}}h\|_{L^{2}}\leq\varepsilon{\langle
Dh,h\rangle}^{1/2}+\|u_{<\lambda}\|_{L^{\infty}}\|h\|_{L^{2}}\ ,$

and (A.1) follows. Applying (A.1), we obtain

$\langle Df,f\rangle\geq\mathcal{Q}_{u}(f,f)\geq(1-2\varepsilon^{2})\langle
Df,f\rangle-2C_{\varepsilon}(u)^{2}\|f\|_{L^{2}}^{2}\ .$ (A.2)

Choosing $\varepsilon$ small enough, we find a constant $K=K(u)>0$ such that

$\tilde{\mathcal{Q}}_{u}(f,g):=\mathcal{Q}_{u}(f,g)+K(u)\langle f,g\rangle$

is an inner product on $H^{1/2}_{+}(\mathbb{R})$, defining a norm which is
equivalent to the standard one. Then we just define

${\rm dom}(L_{u})=\\{f\in H^{1/2}_{+}(\mathbb{R}):\mbox{$\exists C>0$
s.\,t.~{}$|\mathcal{Q}_{u}(f,g)|\leq C\|g\|_{L^{2}}$ for $g\in
H^{1/2}_{+}(\mathbb{R})$}\\}$

and

$\langle L_{u}(f),g\rangle=\mathcal{Q}_{u}(f,g)\ \quad\mbox{for $f\in{\rm
dom}(L_{u})$ and $g\in H^{1/2}_{+}(\mathbb{R})$}\ .$

Then the standard theory of quadratic forms (see [33]) implies that ${\rm
dom}(L_{u})$ is dense in $H^{1/2}_{+}(\mathbb{R})$, hence in
$L^{2}_{+}(\mathbb{R})$, and that $L_{u}$ is self-adjoint and bounded below.
Furthermore, using the quadratic form $\mathcal{Q}_{u}$, Lemma 5.1 and
Proposition 5.1 extend easily to the case $u\in L^{2}_{+}(\mathbb{R})$.

##  Appendix B Variational Properties of $E(u)$

We recall the energy functional

$E(u)=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}u-i\Pi_{+}(|u|^{2})u|^{2}\,dx,$

where we allow for general $u\in H^{1}(\mathbb{R})$, which are not necessarily
in the Hardy-Sobolev space $H^{1}_{+}(\mathbb{R})$. It is elementary to show
that $E:H^{1}(\mathbb{R})\to\mathbb{R}_{\geq 0}$ is weakly lower semi-
continuous, i. e., if $u_{k}\rightharpoonup u$ weakly in $H^{1}(\mathbb{R})$
then

$\liminf_{n\to\infty}E(u_{n})\geq E(u).$ (B.1)

######  Lemma B.1 (Minimal Mass Bubble Lemma).

Suppose $(v_{n})_{n\in\mathbb{N}}\subset H^{1}(\mathbb{R})$ is a sequence such
that

$\sup_{n\geq
1}\|v_{n}\|_{L^{2}}<+\infty\quad\mbox{and}\quad\|\partial_{x}v_{n}\|_{L^{2}}=\mu\quad\mbox{for
all $n\in\mathbb{N}$}.$

with some constant $\mu>0$. In addition, we assume that

$\lim_{n\to\infty}E(v_{n})=0.$

Then it holds that

$\liminf_{n\to\infty}\|v_{n}\|_{L^{2}}^{2}\geq\|\mathcal{R}\|_{L^{2}}^{2}=2\pi.$

Here equality holds if and only if, after possibly passing to a subsequence,

$v_{n}(x+x_{n})\to\mathrm{e}^{i\theta}\lambda^{1/2}\mathcal{R}(\lambda
x)\quad\mbox{strongly in $L^{2}(\mathbb{R})$}$

with some constants $\theta\in[0,2\pi)$, $\lambda>0$, and some sequence
$x_{n}\in\mathbb{R}$.

###### Proof.

We will give a proof that is based on a compactness lemma in [24].
Alternatively, we could use more refined analysis with a profile decomposition
[15].

By rescaling $v_{n}\to\mu^{-1/2}v_{n}(\mu^{-1}\cdot)$, we can assume that
$\|\partial_{x}v_{n}\|_{L^{2}}=1$ for all $n\in\mathbb{N}$. From the triangle
inequality and the form of $E(u)$, we find that

$\displaystyle\|\Pi_{+}(|v_{n}|^{2})v_{n}\|_{L^{2}}\geq\|\partial_{x}v_{n}\|_{L^{2}}-\|\partial_{x}v_{n}-i\Pi_{+}(|v_{n}|^{2})v_{n}\|_{L^{2}}=1-\sqrt{2E(v_{n})}.$

Since $E(v_{n})\to 0$ by assumption, we deduce that

$1\lesssim\|\Pi_{+}(|v_{n}|^{2})v_{n}\|_{L^{2}}\lesssim\|\Pi_{+}(|v_{n}|^{2})\|_{L^{3}}\|v_{n}\|_{L^{6}}\lesssim\||v_{n}|^{2}\|_{L^{3}}\|v_{n}\|_{L^{6}}\lesssim\|v_{n}\|_{L^{6}}^{3},$

by Hölder’s inequality and the classical fact that
$\Pi_{+}:L^{3}(\mathbb{R})\to L^{3}(\mathbb{R})$ is bounded. Thus we have
found that

$\|v_{n}\|_{L^{6}}\geq C>0$

with some constant $C>0$. On the other hand, by the fact
$\sup_{n}\|v_{n}\|_{H^{1}}<+\infty$ and by Sobolev embeddings, we deduce that

$\|v_{n}\|_{L^{2}}\leq C_{1}\quad\mbox{and}\quad\|v_{n}\|_{L^{8}}\leq C_{2}$

with some constants $C_{1},C_{2}>0$. Thus, by applying the $pqr$-Lemma in
[14], we deduce that there exist constants $\varepsilon>0$ and $\delta>0$ such
that

$\mu(\\{x\in\mathbb{R}:|v_{n}(x)|>\varepsilon\\})\geq\delta\quad\mbox{for all
$n\in\mathbb{N}$},$

where $\mu$ denotes the Lebesgue measure on $\mathbb{R}$. Hence we can apply
Lieb’s compactness lemma in [24] to deduce that, after passing to a
subsequence, there exists a sequence of translations
$(y_{n})_{n\in\mathbb{N}}\subset\mathbb{R}$ such that

$v_{n}(\cdot+y_{n})\rightharpoonup v\quad\mbox{weakly in $H^{1}(\mathbb{R})$}$

for some $v\in H^{1}(\mathbb{R})$ with $v\not\equiv 0$.

By translation invariance of the energy $E$, we can henceforth assume that
$y_{n}=0$ for all $n$. Furthermore, the weak lower semi–continuity of $E$
implies that

$0=\lim_{n\to\infty}E(v_{n})\geq E(v).$

On the other hand, we have $E(v)\geq 0$ in general and hence we conclude
$E(v)=0$. By Lemma 4.1, the equality $E(v)=0$ for $v\not\equiv 0$ holds if and
only if

$v(x)=\mathrm{e}^{i\theta}\lambda^{1/2}\mathcal{R}(\lambda x+y)$

with some constants $\theta\in[0,2\pi)$, $\lambda>0$ and $y\in\mathbb{R}$.
Since $v_{n}\rightharpoonup v$ in $L^{2}$, the weak lower semi–continuity of
the $L^{2}$-norm implies that

$\liminf_{n\to\infty}\|v_{n}\|_{L^{2}}^{2}\geq\|v\|_{L^{2}}^{2}=\|\mathcal{R}\|_{L^{2}}^{2}=2\pi.$

Finally, we have equality (after passing to a subsequence if necessary) if and
only if $v_{n}\to v$ strongly in $L^{2}(\mathbb{R})$, which completes the
proof. ∎

##  Appendix C Useful Identities and Gauge Transformation

For sufficiently regular and decaying functions $v:\mathbb{R}\to\mathbb{C}$
and the Hilbert transform $\mathsf{H}$, we have the following identities:

$\mathrm{Re}\langle
xv,\mathsf{H}(v)\rangle=\frac{1}{2\pi}\left|\int_{\mathbb{R}}v\,dx\right|^{2}\
,$ (C.1)
$\mathrm{Re}\,\langle\partial_{x}v,\mathsf{H}(|v|^{2})v\rangle=-\frac{1}{2}\langle|v|^{2},|D||v|^{2}\rangle,$
(C.2) $\langle
v\mathsf{H}(|v|^{2}),v\mathsf{H}(|v|^{2})\rangle=\frac{1}{3}\int_{\mathbb{R}}|v|^{6}\,dx.$
(C.3)

Let us check these identities. Using the Plancherel theorem, we have

$\mathrm{Re}\langle