10.1080/0003681YYxxxxxxxx 1563-504X 0003-6811 00 00 2009 July

# Traveling Waves of Discrete Nonlinear Schrödinger Equations with Nonlocal
Interactions

Michal Fečkan†∗ and Vassilis M. Rothos††
†Department of Mathematical Analysis and Numerical Mathematics, Comenius
University, Mlynská dolina, 842 48 Bratislava, Slovakia, and Mathematical
Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava,
Slovakia;
††School of Mathematics, Physics and Computational Sciences, Faculty of
Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
∗Corresponding author. Email<EMAIL_ADDRESS>
(v3.3 released July 2009)

###### Abstract

Existence and bifurcation results are derived for quasi periodic traveling
waves of discrete nonlinear Schrödinger equations with nonlocal interactions
and with polynomial type potentials. Variational tools are used. Several
concrete nonlocal interactions are studied as well.

###### keywords:

nonlocal interactions, discrete Schödinger equation, traveling wave, symmetry

34K14, 37K60, 37L60

##  1 Introduction

One of the most exciting areas in applied mathematics is the study of the
dynamics associated with the propagation of information. Coherent structures
like solitons, kinks, vortices, etc., play a central role, as carriers of
energy, in many nonlinear physical systems [17]. Solitons represent a rare
example of a (relatively) recently arisen mathematical object which has found
successful high-technology applications [24]. The nature of the system
dictates that the relevant and important effects occur along one axial
direction. Interplay between nonlinearity and periodicity is the focus of
recent studies in different branches of modern applied mathematics and
nonlinear physics. Applications range from nonlinear optics, in the dynamics
of guided waves in inhomogeneous optical structures and photonic crystal
lattices, to atomic physics, in the dynamics of Bose-Einstein condensate (BEC)
droplets in periodic potentials, and from condensed matter, in Josephson-
junction ladders, to biophysics, in various models of the DNA double strand.
Analysis and modeling of these physical situations are based on nonlinear
evolution equations derived from underlying physics equations, such as
nonlinear Maxwell equations with periodic coefficients [37]. In particular,
the systems of 2nd-order NLS equations, both continuous and discrete, were
applied in nonlinear physics to study a number of experimental and theoretical
problems. Spatial non-locality of the nonlinear response is also naturally
present in the description of BECs where it represents the finite range of the
bosonic interaction. Demands on the mathematics for techniques to analyze
these models may best be served by developing methods tailored to determining
the local behavior of solutions near these structures. The discreteness of
space i.e., the existence of an underlying spatial lattice is crucial to the
structural stability of these spatially localized nonlinear excitations.

During the early years, studies of intrinsic localized modes were mostly of a
mathematical nature, but the ideas of localized modes soon spread to
theoretical models of many different physical systems, and the discrete
breather concept has been recently applied to experiments in several different
physics subdisciplines. Most nonlinear lattice systems are not integrable even
if the partial differential equation (PDE) model in the continuum limit is.
While for many years spatially continuous nonlinear PDE’s and their localized
solutions have received a great deal of attention, there has been increasing
interest in spatially discrete nonlinear systems. Namely, the dynamical
properties of nonlinear systems based on the interplay between discreteness,
nonlinearity and dispersion (or diffraction) can find wide applications in
various physical, biological and technological problems. Examples are coupled
optical fibres (self-trapping of light) [18, 1, 19, 26], arrays of coupled
Josephson junctions [39], nonlinear charge and excitation transport in
biological macromolecules, charge transport in organic semiconductors [40].

Prototype models for such nonlinear lattices take the form of various
nonlinear lattices [6], a particularly important class of solutions of which
are so called discrete breathers which are homoclinic in space and oscillatory
in time. Other questions involve the existence and propagation of topological
defects or kinks which mathematically are heteroclinic connections between a
ground and an excited steady state. Prototype models here are discrete version
of sine-Gordon equations, also known as Frenkel-Kontorova (FK) models, e.g.
[4]. There are many outstanding issues for such systems relating to the global
existence and dynamics of localized modes for general nonlinearities, away
from either continuum or anti-continuum limits.

In the main part of the previous studies of the discrete NLS models the
dispersive interaction was assumed to be short-ranged and a nearest-neighbor
approximation was used. However, there exist physical situations that
definitely can not be described in the framework of this approximation. The
DNA molecule contains charged groups, with long-range Coulomb interaction
$1/r$ between them. The excitation transfer in molecular crystals [16] and the
vibron energy transport in biopolymers [35] are due to transition dipole-
dipole interaction with $1/r^{3}$ dependence on the distance, $r$. The
nonlocal (long-range) dispersive interaction in these systems provides the
existence of additional length-scale: the radius of the dispersive
interaction. We will show that it leads to the bifurcating properties of the
system due to both the competition between nonlinearity and dispersion, and
the interplay of long-range interactions and lattice discreteness.

In some approximation the equation of motion is the nonlocal discrete NLS

$\imath\dot{u}_{n}=\sum\limits_{m\neq
n}J_{n-m}(u_{n}-u_{m})+|u_{n}|^{2}u_{n},\quad n\in\mathbb{Z}\,,$ (1)

where the long-range dispersive coupling is taken to be either exponentially
$J_{n}=J{\rm e}^{-\beta|n|}$ with $\beta>0$, or algebraically
$J_{n}=J|n|^{-s}$ with $s>0$, decreasing with the distance $n$ between lattice
sites. In both cases the constant $J$ is normalized such that
$\sum_{n=1}^{\infty}J_{n}=1$, for all $\beta$ or $s$. The parameters $\beta$
and $s$ are introduced to cover different physical situations from the
nearest-neighbor approximation $(\beta\to\infty,s\to\infty)$ to the
quadrupole-quadrupole $(s=5)$ and dipole-dipole $(s=3)$ interactions. The
Hamiltonian $H$ and the number of excitations $N$

$H=\frac{1}{2}\sum_{n,m\in\mathbb{Z}}J_{n-m}|u_{n}-u_{m}|^{2}-\frac{1}{2}\sum_{n\in\mathbb{Z}}|u_{n}|^{4},\quad{\rm
and}\quad N=\sum_{n\in\mathbb{Z}}|u_{n}|^{2}$ (2)

are conserved quantities corresponding to the set of (1).

It should be also noted that the derivation of a discrete equation from the
Gross-Pitaevskii equation produces at the intermediate step a fully nonlocal
discrete NLS equation for the coefficients of the wave function expansion over
the complete set of the Wannier functions. Further reduction to the case of
the only band with the strong localization of the Wannier functions (the
tight-binding approximation) leads to the standard local DNLS equation.
Recently Abdullaev et al. [2] extended this approach to the case of periodic
nonlinearities and derived a number of nonintegrable lattices with different
nearest-neighbor nonlinearities.

In this paper, we study the discrete nonlinear Schrödinger equations on the
lattice $\mathbb{Z}$ (DNLS) with nonlocal interactions of forms

$\imath\dot{u}_{n}=\sum\limits_{j\in\mathbb{N}}a_{j}\Delta_{j}u_{n}+f(|u_{n}|^{2})u_{n},\quad
n\in\mathbb{Z}$ (3)

where $u_{n}\in\mathbb{C}$, $\Delta_{j}u_{n}:=u_{n+j}+u_{n-j}-2u_{n}$ are
$1$-dimensional discrete Laplacians and it holds

  * (H1)

$f\in C(\mathbb{R}_{+},\mathbb{R})$ for $\mathbb{R}_{+}:=[0,\infty)$, $f(0)=0$
and $a_{j}\in\mathbb{R}$ with $\sum\limits_{j\in\mathbb{N}}|a_{j}|<\infty$.
Moreover, there are constants $s>0$, $\mu>1$, $c_{1}>0$, $c_{2}>0$ and
$\bar{r}>0$ such that

$\begin{gathered}|f(w)|\leq c_{1}(w^{s}+1),\quad c_{2}(w^{s+1}-1)\leq
F(w),\quad\mu F(w)-\bar{r}<f(w)w\end{gathered}$

for any $w\geq 0$, where $F(w)=\int\limits_{0}^{w}f(z)dz$. Furthermore,
$\limsup_{w\to 0_{+}}f(w)/w^{\widetilde{s}}<\infty$ for a constant
$\widetilde{s}>0$.

Of course we suppose that not all $a_{j}$ are zero. Note any polynomial
$f(w)=p_{1}w+\cdots+p_{s}w^{s}$, $s\in\mathbb{N}$ with $p_{s}>0$ satisfies
(H1). Furthermore, (3) can be rewritten into a standard form

$\imath\dot{u}_{n}=\sum\limits_{m\neq
n}a_{|m-n|}\left(u_{m}-u_{n}\right)+f(|u_{n}|^{2})u_{n},\quad n\in\mathbb{Z}.$
(4)

It is well known that (4) conserves two dynamical invariants

$\begin{gathered}\sum\limits_{n\in\mathbb{Z}}|u_{n}|^{2}\quad-\textrm{the
norm},\\\ \sum\limits_{n\in\mathbb{Z}}\left[-\frac{1}{2}\sum\limits_{m\neq
n}a_{|m-n|}\left|u_{m}-u_{n}\right|^{2}+F(|u_{n}|^{2})\right]\quad-\textrm{the
energy}.\end{gathered}$

Differential equations with nonlocal interactions on lattices have been
studied in [3, 5, 7, 8, 9, 13, 14, 25, 30], while DNLS (discrete nonlinear
Schrödinger) in [10, 11, 13, 22, 28]. Nowadays it is clear that a large number
of important models of various fields of physics are based on DNLS type
equations with several forms of polynomial nonlinearities starting with the
simplest self-focusing cubic (Kerr) nonlinearity, then following with the
cubic onsite nonlinearity relevant for Bose-Einstein condensates, then with
more general discrete cubic nonlinearity in Salerno model up to cubic-quintic
ones (see [11] for more references).

We are interested in the existence of traveling wave solutions
$u_{n}(t)=U(n-\nu t)$ of (3) with a quasi periodic function $U(z)$, $z=n-\nu
t$ and some $\nu\neq 0$.

First, we introduce a function

$\Phi(x):=\frac{4}{x}\sum\limits_{j\in\mathbb{N}}a_{j}\sin^{2}\left[\frac{x}{2}j\right]\,.$

######  Remark 1.1.

Clearly $\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$, $\Phi$ is odd,
$\Phi(2\pi k)=0$ for any $k\in\mathbb{Z}\setminus\\{0\\}$, and $\Phi(x)\to 0$
as $|x|\to\infty$. If $\sum\limits_{j\in\mathbb{N}}j|a_{j}|<\infty$ then
$\Phi\in C(\mathbb{R},\mathbb{R})$ and if
$\sum\limits_{j\in\mathbb{N}}j^{2}|a_{j}|<\infty$ then $\Phi\in
C^{1}(\mathbb{R},\mathbb{R})$. Consequently the range
${\mathcal{R}}\Phi:=\Phi(\mathbb{R}\setminus\\{0\\})$ is either an interval
$[-\bar{R},\bar{R}]$ or $(-\bar{R},\bar{R})$ here with possibility
$\bar{R}=\infty$ (see Section 2.4 for concrete examples).

Now we can state the following existence result.

######  Theorem 1.2.

Let (H1) hold and $T>0$. Then for almost each
$\nu\in\mathbb{R}\setminus\\{0\\}$ and any rational $r\in\mathbb{Q}\cap(0,1)$,
there is a nonzero periodic traveling wave solution $u_{n}(t)=U(n-\nu t)$ of
(3) with $U\in C^{1}(\mathbb{R},\mathbb{C})$ and such that

$U(z+T)={\,\textrm{\rm e}}^{2\pi r\imath}U(z),\,\forall z\in\mathbb{R}\,.$ (5)

Moreover, for any $\nu\in\mathbb{R}\setminus\\{0\\}$ there is at most a finite
number of $\bar{r}_{1},\bar{r}_{2},\cdots,\bar{r}_{m}\in(0,1)$ such that
equation

$-\nu=\Phi\left(\frac{2\pi}{T}(\bar{r}_{j}+k)\right)$

has a solution $k\in\mathbb{Z}$. Then for any
$r\in(0,1)\setminus\\{\bar{r}_{1},\bar{r}_{2},\cdots,\bar{r}_{m}\\}$ there is
a nonzero quasi periodic traveling wave solution $u_{n}(t)=U(n-\nu t)$ with
the above properties. In particular, for any $|\nu|>\bar{R}$ and $r\in(0,1)$,
there is such a nonzero quasi periodic traveling wave solution.

When a nonresonance condition of Theorem 1.2 fails, then we have the following
bifurcation results.

######  Theorem 1.3.

Suppose $f\in C^{2}(\mathbb{R}_{+},\mathbb{R})$ with $f(0)=0$. If there are
$\bar{r}_{1}\in(0,1)$, $\nu\in{\mathcal{R}}\Phi\setminus\\{0\\}$ and $T>0$
such that all solutions $k_{1},k_{2},\cdots,k_{m_{1}}\in\mathbb{Z}$ of
equation

$-\nu=\Phi\left(\frac{2\pi}{T}(\bar{r}_{1}+k)\right)$

are either nonnegative or negative, and $m_{1}>0$. Then for any
$\varepsilon>0$ small there are $m_{1}$ branches of nonzero quasi periodic
traveling wave solutions
$u_{n,j,\varepsilon}(t)=U_{j,\varepsilon}(n-\nu_{\varepsilon}t)$ of (3) with
$U_{j,\varepsilon}\in C^{1}(\mathbb{R},\mathbb{C})$, $j=1,2,\cdots,m_{1}$, and
nonzero velocity $\nu_{\varepsilon}$ satisfying
$U_{j,\varepsilon}(z+T)={\,\textrm{\rm
e}}^{2\pi\bar{r}_{1}\imath}U_{j,\varepsilon}(z)$, $\forall z\in\mathbb{R}$
along with $\nu_{\varepsilon}\to\nu$ and $U_{j,\varepsilon}\rightrightarrows
0$ uniformly on $\mathbb{R}$ as $\varepsilon\to 0$.

######  Remark 1.4.

If $a_{j}\geq 0$ for all $j\in\mathbb{N}$, then the assumptions of Theorem 1.3
are satisfies for any $\nu\in{\mathcal{R}}\Phi\setminus\\{0\\}$ such that
$\frac{T}{2\pi}\Phi^{-1}(-\nu)\setminus\mathbb{Z}\neq\emptyset$, and so there
are bifurcations of quasi periodic traveling waves in the generic resonant
cases. On the other hand, if $\nu\in{\mathcal{R}}\Phi\setminus\\{0\\}$ with
$\frac{T}{2\pi}\Phi^{-1}(-\nu)\subset\mathbb{Z}$ then Theorem 1.2 is
applicable for any $r\in(0,1)$.

Theorem 1.3 is a Lyapunov center theorem for traveling wave solutions. Similar
results are derived in [23] for Fermi-Pasta-Ulam lattices.

We also discuss in Section 4 the extension of these results of (3) on the
lattices $\mathbb{Z}^{2}$ and $\mathbb{Z}^{3}$ [10, 11, 22, 28]. The final
Section 5 is devoted to traveling wave solutions of more general forms than
above [32].

##  2 Existence of Traveling Wave Solutions

In this section, we study the existence of traveling wave solutions of the
form $u_{n}(t)=U(n-\nu t)$, i.e. we are interested in the equation

$-\nu\imath
U^{\prime}(z)=\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\,,$
(6)

where $z=n-\nu t$, $\nu\neq 0$ and $\partial_{j}U(z):=U(z+j)+U(z-j)-2U(z)$. We
are interested in the existence of quasi periodic solutions $U(z)$ of (6)
stated in Theorem 1.2.

###  2.1 Preliminaries

In this subsection we recall some results from critical point theory of [27].
Let $H$ be a Hilbert space and let $J\in C^{1}(H,\mathbb{R})$. Suppose
$H=H_{1}\oplus H_{2}$ for closed linear subspaces, and let
$e_{1},e_{2},\cdots$ be the orthonormal basis of $H_{1}$. Let us put
$H_{n}^{1}:=\textrm{span}\,\\{e_{1},e_{2},\cdots,e_{n}\\}$ and
$H_{n}:=H_{n}^{1}\oplus H_{2}$. Let $P_{n}$ be the orthogonal projection of
$H$ onto $H_{n}$. Set $J_{n}:=J/H_{n}$ \- the restriction of functional $J$ on
subspace $H_{n}$ \- and so $\nabla J_{n}(x)=P_{n}\nabla J(x)$ if $x\in H_{n}$.

######  Definition 2.1.

If there are two positive constants $\alpha$ and $\beta$ such that

$\begin{gathered}J(x)\geq 0\quad\forall x\in\\{x\in
H_{1}\mid\|x\|\leq\beta\\}\,,\\\ J(x)\geq\alpha\quad\forall x\in\\{x\in
H_{1}\mid\|x\|=\beta\\}\,,\\\ J(x)\leq 0\quad\forall x\in\\{x\in
H_{2}\mid\|x\|\leq\beta\\}\,,\\\ J(x)\leq-\alpha\quad\forall x\in\\{x\in
H_{2}\mid\|x\|=\beta\\}\,,\end{gathered}$

then $J$ is said to satisfy the local linking condition at $0$.

######  Definition 2.2.

We shall say that $J$ satisfies the Palais-Smale (PS)∗-condition if any
sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$ in $H$ such that $x_{n}\in H_{n}$,
$J(x_{n})\leq c<\infty$ and $P_{n}\nabla J(x_{n})=\nabla J_{n}(x_{n})\to 0$ as
$n\to\infty$, possesses a convergent subsequence.

Now we can state the following theorem of [27] which we apply.

######  Theorem 2.3.

Suppose

  * $(I_{1})$

$J\in C^{1}(H,\mathbb{R})$ satisfies (PS)∗-condition.

  * $(I_{2})$

$J$ satisfies the local linking condition at $0$.

  * $(I_{3})$

$\forall n$, $J_{n}(x)\to-\infty$ as $\|x\|\to\infty$ and $x\in H_{n}$.

  * $(I_{4})$

$\nabla J=A+C$ for a bounded linear self-adjoint operator $A$ such that
$AH_{n}\subset H_{n}$, $\forall n\in\mathbb{N}$ and $C$ is a compact mapping.

Then $J$ possesses a critical point $\bar{x}$ with $|J(\bar{x})|\geq\alpha$.

######  Remark 2.4.

If $0$ is an indefinite nondegenerate critical point of $J$, then $J$
satisfies the local linking condition at $0$.

###  2.2 Proof of Theorem 1.2

In this section, we use Theorem 2.3 to prove Theorem 1.2. Without loss of
generality, we set $T=2\pi$. We suppose $\nu>0$, the case $\nu<0$ can be
handled similarly. First, we identify $\mathbb{C}$ with $\mathbb{R}^{2}$ in
this section. Let $r\in(0,1)$ be fixed. Next, we consider real Banach spaces

$L_{r}^{\widetilde{s}}:=\left\\{U\in
L^{\widetilde{s}}_{loc}(\mathbb{R},\mathbb{C})\mid U(z+2\pi)={\,\textrm{\rm
e}}^{2\pi r\imath}U(z),\,\forall z\in\mathbb{R}\right\\}$

for $\widetilde{s}\geq 1$. Clearly $U\in L_{r}^{\widetilde{s}}$ if and only if
$U(z)={\,\textrm{\rm e}}^{rz\imath}V(z)$ for some $V\in
L^{\widetilde{s}}:=L^{\widetilde{s}}(S^{2\pi},\mathbb{C})$. Consequently
$U_{1}(z+c_{1})\overline{U_{2}(z+c_{2})}$ is $2\pi$-periodic for any
$c_{1},c_{2}\in\mathbb{R}$ and $U_{1},U_{2}\in L_{r}^{\widetilde{s}}$, hence
$|U(z)|$ is $2\pi$-periodic. So we consider the norm on
$L_{r}^{\widetilde{s}}$ like on $L^{\widetilde{s}}$. In particular, we have

$V\in L_{r}^{2}\Leftrightarrow
V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath},\,V_{k}\in\mathbb{C},\,\sum\limits_{k\in\mathbb{Z}}|V_{k}|^{2}<\infty\,.$

Let

$\begin{gathered}X_{r}:=W_{r}^{1/2,2}(S^{2\pi},\mathbb{C})=\left\\{V\in
L^{2}_{r}\mid V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath},\,\sum\limits_{k\in\mathbb{Z}}|V_{k}|^{2}|r+k|<\infty\right\\}\,,\\\
Y_{r}:=W_{r}^{1,2}(S^{2\pi},\mathbb{C})=\left\\{V\in L^{2}_{r}\mid
V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath},\,\sum\limits_{k\in\mathbb{Z}}|V_{k}|^{2}(r+k)^{2}<\infty\right\\}\,.\end{gathered}$

Note $r+k\neq 0$ for any $k\in\mathbb{Z}$. Clearly $Y_{r}\subset X_{r}\subset
L_{r}^{2}$. We consider $L_{r}^{2}$, $X_{r}$ and $Y_{r}$ as real Hilbert
spaces with inner products

$\begin{gathered}\langle
V,W\rangle_{L_{r}^{2}}:=2\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}=\Re\int_{0}^{2\pi}V(z)\overline{W(z)}dz\,,\\\
\langle
V,W\rangle_{X_{r}}:=2\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}|r+k|\\\
\langle
V,W\rangle_{Y_{r}}:=2\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}(r+k)^{2}\end{gathered}$

for $V(z)=\sum\limits_{k\in\mathbb{Z}}V_{k}{\,\textrm{\rm e}}^{(r+k)z\imath}$
and $W(z)=\sum\limits_{k\in\mathbb{Z}}W_{k}{\,\textrm{\rm e}}^{(r+k)z\imath}$.

Clearly $\|U\|_{L^{2}}=\|U\|_{L_{r}^{2}}\leq r_{1}\|U\|_{X_{r}}$, $\forall
U\in X_{r}$ and $\|U\|_{X_{r}}\leq
r_{1}\|U\|_{Y_{r}}=r_{1}\|U^{\prime}\|_{L^{2}}$, $\forall U\in Y_{r}$ for
$r_{1}:=\min\left\\{\sqrt{r},\sqrt{1-r}\right\\}$. The following result is
well-known [27, 34].

######  Lemma 2.5.

For each $\widetilde{s}\geq 1$, $X_{r}$ is compactly embedded into
$L_{r}^{\widetilde{s}}$.

On $X_{r}$, we consider a continuous symmetric bilinear form

$B_{r}(U,V):=4\pi\Re\sum\limits_{k\in\mathbb{Z}}V_{k}\overline{W_{k}}(r+k)\,.$

Note, if $U\in X_{r}$ and $V\in Y_{r}$, then

$2\Re\int_{0}^{2\pi}\imath U(z)\overline{V(z)}\,^{\prime}dz=B_{r}(U,V)\,.$

Now we consider a real functional

$\begin{gathered}I_{r}(U):=\frac{\nu}{2}B_{r}(U,U)+\int_{0}^{2\pi}\left\\{\sum\limits_{j\in\mathbb{Z}}\frac{a_{|j|}}{2}|U(z+j)-U(z)|^{2}-F(|U(z)|^{2})\right\\}dz\\\
=\frac{\nu}{2}B_{r}(U,U)+\int_{0}^{2\pi}\left\\{\sum\limits_{j\in\mathbb{N}}a_{j}|U(z+j)-U(z)|^{2}-F(|U(z)|^{2})\right\\}dz\end{gathered}$

on $X_{r}$. Then $I_{r}\in C^{1}(X_{r},\mathbb{R})$ and for $U\in X_{r}$,
$V\in Y_{r}$, we derive

$\begin{gathered}DI_{r}(U)V=\\\ 2\Re\left\\{\int_{0}^{2\pi}\left(\nu\imath
U(z)\overline{V(z)}\,^{\prime}-\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\overline{V(z)}\right)dz\right\\}.\end{gathered}$

If $U\in X_{r}$ is a critical point of $I_{r}$ then

$\Re\left\\{\int_{0}^{2\pi}\left(\nu\imath
U(z)\overline{V(z)}\,^{\prime}-\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\overline{V(z)}\right)dz\right\\}=0$
(7)

for any $V\in Y_{r}$. Replacing $V$ with $\imath V$ in (7), we obtain

$\int_{0}^{2\pi}\left(\nu\imath
U(z)\overline{V(z)}\,^{\prime}-\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\overline{V(z)}\right)dz=0$

for any $V\in Y_{r}$. This means that $U$ is a weak solution of (6). Then a
standard regularity method shows [34] that $U$ is a $C^{1}$-smooth solution of
(6).

Now we split $X_{r}=X_{+}\oplus X_{-}$ for

$X_{-}:=\left\\{V(z)=\sum\limits_{k=-\infty}^{-1}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath}\right\\},\quad
X_{+}:=\left\\{V(z)=\sum\limits_{k=0}^{\infty}V_{k}{\,\textrm{\rm
e}}^{(r+k)z\imath}\right\\}\,.$

Clearly if $U=U_{+}+U_{-}$ then
$B_{r}(U,U)=2\left(\|U_{+}\|^{2}_{X_{r}}-\|U_{-}\|^{2}_{X_{r}}\right)$.

Next, let us define $\widetilde{K}_{r}:L^{2}_{r}\to X_{r}$ as

$\langle\widetilde{K}_{r}H,V\rangle_{X_{r}}:=2\Re\int_{0}^{2\pi}H(z)\overline{V(z)}dz,\,\forall
V\in X_{r}\,.$ (8)

Then

$\widetilde{K}_{r}H=\sum_{k\in\mathbb{Z}}\frac{2H_{k}}{|r+k|}{\,\textrm{\rm
e}}^{(r+k)\imath z}$

and so $\widetilde{K}_{r}$ is compact. To study $\nabla I_{r}(u)$, we
introduce the mapping $\Psi_{r}:X_{r}\to X_{r}$ defined by

$\langle\Psi_{r}(U),V\rangle_{X_{r}}:=2\Re\int_{0}^{2\pi}f(|U(z)|^{2})U(z)\overline{V(z)}dz$

for any $V\in X_{r}$. By Lemma 2.5, the Nemytskij operator $U\to
f(|U(z)|^{2})U(z)$ from $X_{r}$ to $L_{r}^{2}$ is continuous. Using (8), we
get

$\Psi_{r}(U)=\widetilde{K}_{r}f(|U|^{2})U\,.$

Hence $\Psi_{r}:X_{r}\to X_{r}$ is compact and continuous.

######  Lemma 2.6.

Under (H1) it hods $D\Psi_{r}(0)=0$.

######  Proof 2.7.

There is a constant $c_{3}$ such that

$|f(w)|\leq c_{3}(w+w^{s})$

for any $w\geq 0$. Then by Lemma 2.5, we derive

$\begin{gathered}|f(|U|^{2})U|_{L_{r}^{2}}^{2}=\int_{0}^{2\pi}f(|U(z)|^{2})^{2}|U(z)|^{2}dz\\\
\leq 2c^{2}_{3}\int_{0}^{2\pi}\left(|U(z)|^{6}+|U(z)|^{2(2s+1)}\right)dz\leq
c^{2}_{4}\left(\|U\|^{3}_{X_{r}}+\|U\|^{2s+1}_{X_{r}}\right)^{2}\end{gathered}$

for a constant $c_{4}>0$. Hence

$\left|\langle\Psi_{r}(U),V\rangle_{X_{r}}\right|\leq
2\|f(|U|^{2})U\|_{L_{r}^{2}}\|V\|_{L_{r}^{2}}\leq
c_{5}\left(\|U\|^{3}_{X_{r}}+\|U\|^{2s+1}_{X_{r}}\right)\|V\|_{X_{r}}$

for a constant $c_{5}>0$. This implies

$\|\Psi_{r}(U)\|_{X_{r}}\leq
c_{5}\left(\|U\|^{3}_{X_{r}}+\|U\|^{2s+1}_{X_{r}}\right),\,\forall U\in
X_{r}\,.$

Since $\Psi_{r}(0)=0$ and $s>0$, we get $D\Psi_{r}(0)=0$. The proof is
finished.

Finally, define ${\mathcal{L}}_{r}:L_{r}^{2}\to L_{r}^{2}$ as

${\mathcal{L}}_{r}U:=\sum_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)\,.$

Then

$\nabla
I_{r}(U)=\left(2\nu\bm{I}_{+}-2\nu\bm{I}_{-}-\widetilde{K}_{r}{\mathcal{L}}_{r}-\Psi_{r}\right)(U)$
(9)

for the identities $\bm{I}_{\pm}:X_{\pm}\to X_{\pm}$. Clearly

$A_{r}:=2\nu\bm{I}_{+}-2\nu\bm{I}_{-}-\widetilde{K}_{r}{\mathcal{L}}_{r}$

is a self-adjoint bounded operator $A_{r}:X_{r}\to X_{r}$ satisfying

$A_{r}U=2\sum_{k\in\mathbb{Z}}\left(\nu\,\textrm{sgn}\,(r+k)+\frac{4}{|r+k|}\sum_{j\in\mathbb{N}}a_{j}\sin^{2}\left[\frac{r+k}{2}j\right]\right)U_{k}{\,\textrm{\rm
e}}^{(r+k)\imath z}\,.$

Consequently, the spectrum $\sigma(A_{r})$ of $A_{r}$ is given by

$\sigma(A_{r})=\left\\{2\,\textrm{sgn}\,(r+k)\left(\nu+\Phi(r+k)\right)\mid
k\in\mathbb{Z}\right\\}\,.$

By Lemma 2.6, we get that under the assumption

$-\nu\neq\Phi(r+k)\,\forall k\in\mathbb{Z}\,,$ (10)

$0$ is an indefinite nondegenerate critical point of $I_{r}$: $\nabla
I_{r}(0)=0$ and $\textrm{Hess}\,I_{r}(0)=A_{r}$ with $0\notin\sigma(A_{r})$
and $X_{r}=X_{1,r}\oplus X_{2,r}$ with
$\sigma(A_{r}/X_{1,r})\subset(0,\infty)$ and
$\sigma(A_{r}/X_{2,r})\subset(-\infty,0)$ where $X_{1,r}$, $X_{2,r}$ are
suitable closed linear subspaces of $X_{r}$. Note $X_{1,r}$ and $X_{2,r}$ are
infinite dimensional, since $\Phi(r+k)\to 0$ as $|k|\to\infty$. Consequently
by Remark 2.4, under (10), $I_{r}$ satisfies the local linking condition at
$0$ in the sense of Definition 2.1, i.e. condition $(I_{2})$ of Theorem 2.3 is
verified.

We consider an equivalent scalar product $\langle\cdot,\cdot\rangle_{r}$ on
$X_{r}$ such that

$\langle A_{r}U,U\rangle_{r}=\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2},\quad
U_{1}\in X_{1,r},\,U_{2}\in X_{2,r}\,.$

Note there is a linear isomorphism $K_{r}:X_{r}\to X_{r}$ such that

$\langle U,V\rangle_{X_{r}}=\langle K_{r}U,V\rangle_{r},\quad\forall U,\forall
V\in X_{r}\,.$

Clearly $K_{r}$ is self-adjoint and positive definite. Then

$\begin{gathered}I_{r}(U)=\frac{\nu}{2}\|U_{1}\|_{r}^{2}-\frac{\nu}{2}\|U_{2}\|_{r}^{2}-\int_{0}^{2\pi}F(|U(z)|^{2})dz\,,\\\
\nabla I_{r}(U)=\nu\bm{I}_{1}-\nu\bm{I}_{2}-K_{r}\Psi_{r}\,,\\\ \langle\nabla
I_{r}(U),V\rangle_{r}=DI_{r}(U)V=\nu\|V_{1}\|_{r}^{2}-\nu\|V_{2}\|_{r}^{2}\\\
-2\Re\int_{0}^{2\pi}f(|U(z)|^{2})U(z)\overline{V(z)}dz\,.\end{gathered}$

Let $X_{2,r}=\textrm{span}\,\\{e_{1},e_{2},\cdots\\}$ and $e_{i}$ are
eigenvectors of $A_{r}$. Then we take
$X_{n}=\textrm{span}\,\\{e_{1},e_{2},\cdots,e_{n}\\}\oplus X_{1,r}$ for $n\geq
3$. So clearly $AX_{n}\subset X_{n}$, i.e. condition $(I_{4})$ of Theorem 2.3
is verified. Let $P_{n}:X_{r}\to X_{n}$ be the orthogonal projection with
respect $\langle\cdot,\cdot\rangle_{r}$.

We suppose there is a sequence $\\{U_{m}\\}_{m\in\mathbb{N}}\subset X_{r}$,
$U_{m}\in X_{m}$ and a constant $c$ such that

$I_{r}(U_{m})\leq c\quad\textrm{and}\quad P_{m}\nabla I_{r}(U_{m})\to 0\,.$

Then for $m$ large we get,

$\begin{gathered}c+\|U_{m}\|_{r}\geq I_{r}(U_{m})-\frac{1}{2}\langle
P_{m}\nabla I_{r}(U_{m}),U_{m}\rangle_{r}\\\
=\int_{0}^{2\pi}\left[f(|U_{m}(z)|^{2})|U_{m}(z)|^{2}-F(|U_{m}(z)|^{2})\right]dz\\\
\geq\int_{0}^{2\pi}(\mu-1)F(|U_{m}(z)|^{2})dz-2\pi\bar{r}\\\
\geq(\mu-1)c_{2}\int_{0}^{2\pi}\left(|U_{m}(z)|^{2(s+1)}-1\right)dz-2\pi\bar{r}\\\
\geq(\mu-1)c_{2}\left(\|U_{m}\|^{2(s+1)}_{L^{2(s+1)}}-c_{6}\right)\end{gathered}$
(11)

for a constant $c_{6}>0$.

By following the same arguments, we derive

$\begin{gathered}\nu\|U_{1,m}\|^{2}_{r}\leq\|P_{m}\nabla
I_{m}(U_{m})\|\cdot\|U_{1,m}\|_{r}+2\int_{0}^{2\pi}f(|U_{m}(z)|^{2})|U_{m}(z)||U_{1,m}(z)|dz\\\
\leq\|U_{1,m}\|_{r}+2c_{7}\int_{0}^{2\pi}\left(|U_{m}(z)|^{2s+1}+1\right)|U_{1,m}(z)|dz\\\
\leq\|U_{1,m}\|_{r}+2c_{7}\left\||U_{m}|^{2s+1}+1\right\|_{L^{\frac{2(s+1)}{2s+1}}}\|U_{1,m}\|_{L^{2(s+1)}}\\\
\leq\|U_{1,m}\|_{r}+2c_{7}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)\|U_{1,m}\|_{r}\end{gathered}$

and hence

$\|U_{1,m}\|_{r}\leq c_{8}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)\,.$

Similarly we obtain

$\|U_{2,m}\|_{r}\leq c_{8}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)$

and consequently by (11), we obtain

$\|U_{m}\|_{r}\leq 2c_{8}\left(\|U_{m}\|^{2s+1}_{L^{2(s+1)}}+1\right)\leq
c_{9}\left(\|U_{m}\|^{\frac{2s+1}{2(s+1)}}_{r}+1\right)$

for positive constants $c_{7}$, $c_{8}$ and $c_{9}$. Thus
$\\{U_{m}\\}_{m\in\mathbb{N}}\subset X_{r}$ is bounded. Since

$P_{m}\nabla I_{r}(U_{m})=\nu U_{1,m}-\nu U_{2,m}-K_{r}\Psi_{r}(U_{m})\to 0$

and $K_{r}\Psi_{r}$ is compact, there is a convergent subsequence of
$\\{U_{m}\\}_{m\in\mathbb{N}}$ in $X_{r}$. Summarizing, (PS)∗-condition is
verified for $I_{r}$, i.e. condition $(I_{1})$ of Theorem 2.3 is verified.

Next, let $U\in X_{n}$. Then using
$U_{1}\in\textrm{span}\,\\{e_{1},e_{2},\cdots,e_{n}\\}$, we derive

$\begin{gathered}I_{r}(U)=\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-\int_{0}^{2\pi}F(|U(z)|^{2})dz\\\
\leq\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-c_{2}\int_{0}^{2\pi}\left(|U(z)|^{2(s+1)}-1\right)dz\\\
\leq\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-c_{2}\|U\|^{2(s+1)}_{L^{2(s+1)}}+c_{10}\\\
\leq\frac{\nu}{2}\left(\|U_{1}\|_{r}^{2}-\|U_{2}\|_{r}^{2}\right)-c_{11}\left(\|U_{1}\|^{2(s+1)}_{L^{2}}+\|U_{2}\|^{2(s+1)}_{L^{2}}\right)+c_{10}\\\
\leq\frac{\nu}{2}\|U_{1}\|_{r}^{2}\left(1-c_{12}\|U_{1}\|^{2s}_{r}\right)-\frac{\nu}{2}\|U_{2}\|_{r}^{2}+c_{10}\end{gathered}$

for positive constants $c_{10}$, $c_{11}$ and $c_{12}$. Now it is clear that
$I_{r}(U)\to-\infty$ as $\|U\|_{r}\to\infty$, i.e. condition $(I_{3})$ of
Theorem 2.3 is verified.

Summarizing, under assumptions (H1) and (10), all conditions
$(I_{1})$-$(I_{4})$ of Theorem 2.3 are verified for $I_{r}$. Hence there is a
nonzero critical point $U_{r}\in X_{r}$ of $I_{r}$, which we already know to
be a $C^{1}$-smooth solution of (6) satisfying (5). Note (10) certainly holds
for any $|\nu|>\bar{R}$ and $r\in(0,1)$. Hence the proof of the second part of
Theorem 1.2 is finished. To prove the first part, it enough to observe that
the set

$\left\\{\Phi(r+k)\mid r\in\mathbb{Q}\cap(0,1),\quad k\in\mathbb{Z}\right\\}$

is countable, and thus for almost each $\nu\in\mathbb{R}\setminus\\{0\\}$ and
any $r\in\mathbb{Q}\cap(0,1)$, condition (10) holds.

###  2.3 Remarks

######  Remark 2.8.

When $r$ is rational in Theorem 1.2 then we get periodic $U(z)$ with
arbitrarily large minimal periods. If $r$ is irrational then clearly
$U(z)={\,\textrm{\rm e}}^{\frac{2\pi}{T}rz\imath}V(z)$ for a $T$-periodic
$V(z)=U(z){\,\textrm{\rm e}}^{-\frac{2\pi}{T}rz\imath}$. So $U(z)$ is quasi
periodic and its orbit in $\mathbb{C}\simeq\mathbb{R}^{2}$ is dense either in
a compact annulus or in a compact disc. But $|U(z)|$ is $T$-periodic in the
both cases.

######  Remark 2.9.

Changing $t\leftrightarrow-t$, we can also handle DNLS

$-\imath\dot{u}_{n}=\sum\limits_{j\in\mathbb{N}}a_{j}\Delta_{j}u_{n}+f(|u_{n}|^{2})u_{n},\quad
n\in\mathbb{Z}$ (12)

under (H1) and (10) becomes

$\nu\neq\Phi(r+k)\,\forall
k\in\mathbb{Z}\quad\textrm{and}\quad\nu\in(0,\bar{R})\,.$ (13)

######  Remark 2.10.

Assume that $U\in Y_{r}$ is a weak solution of (6), then

$\begin{gathered}|U(z)|\leq\sum_{k\in\mathbb{Z}}|U_{k}|\leq\sqrt{\sum_{k\in\mathbb{Z}}|U_{k}|^{2}(r+k)^{2}}\sqrt{\sum_{k\in\mathbb{Z}}(r+k)^{-2}}\\\
=\sqrt{\frac{\pi}{2}}{\rm cosec}\,\pi r\,\|U\|_{Y_{r}}.\end{gathered}$

Let $\widetilde{R}:=\max_{x\in\mathbb{R}_{+}}x\Phi(x)$. Then

$\begin{gathered}|\nu|\|U^{\prime}\|_{L^{2}}=|\nu|\|U\|_{Y_{r}}\leq\|{\mathcal{L}}_{r}U\|_{L^{2}}+\|f(|U|^{2}U)\|_{L^{2}}\\\
\leq\widetilde{R}\|U\|_{L^{2}}+c_{1}\left\||U|^{2s+1}+|U|\right\|_{L^{2}}\\\
\leq\left(\widetilde{R}r_{1}^{2}+c_{1}r_{1}^{2}+c_{1}\frac{\pi^{s+1}}{2^{s}}{\rm
cosec}^{2s+1}\,\pi r\,\|U\|_{Y_{r}}^{2s}\right)\|U\|_{Y_{r}}.\end{gathered}$

So if $U\neq 0$ then we obtain

$|\nu|\leq\widetilde{R}r_{1}^{2}+c_{1}r_{1}^{2}+c_{1}\frac{\pi^{s+1}}{2^{s}}{\rm
cosec}^{2s+1}\,\pi r\,\|U\|_{Y_{r}}^{2s},$

i.e.

$\|U\|_{Y_{r}}\geq\sqrt{2}\sqrt[2s]{\frac{|\nu|-\widetilde{R}r_{1}^{2}-c_{1}r_{1}^{2}}{c_{1}\pi^{s+1}{\rm
cosec}^{2s+1}\,\pi r}}$

for

$|\nu|\geq\widetilde{R}r_{1}^{2}+c_{1}r_{1}^{2}.$

Hence $\|U\|_{Y_{r}}\to\infty$ as $|\nu|\to\infty$ for a possible nonzero
solution $U\in Y_{r}$ of (6).

###  2.4 Examples

We first note

$\Phi(x)=\frac{2}{x}\sum_{j\in\mathbb{N}}a_{j}(1-\cos
xj)=\frac{2}{x}\left[\sum_{j\in\mathbb{N}}a_{j}-\Re\sum_{j\in\mathbb{N}}a_{j}{\,\textrm{\rm
e}}^{xj\imath}\right].$ (14)

Now we turn the the following concrete examples.

######  Example 2.11.

First we suppose that $a_{j}$ is decaying rapidly to $0$. Let
$a_{j}=\frac{1}{j!}$. Then

$\begin{gathered}\sum_{j\in\mathbb{N}}\frac{1}{j!}{\,\textrm{\rm
e}}^{xj\imath}={\,\textrm{\rm e}}^{{\,\textrm{\rm
e}}^{x\imath}}-1={\,\textrm{\rm e}}^{\cos x+\imath\sin x}-1\\\ ={\,\textrm{\rm
e}}^{\cos x}\left[\cos\sin x+\imath\sin\sin x\right]-1.\end{gathered}$

So by (14) we derive

$\Phi(x)=\frac{2}{x}\left[\sum_{j\in\mathbb{N}}\frac{1}{j!}-{\,\textrm{\rm
e}}^{\cos x}\cos\sin x+1\right]=\frac{2}{x}\left[{\,\textrm{\rm
e}}-{\,\textrm{\rm e}}^{\cos x}\cos\sin x\right].$

By Remark 1.1, $\Phi\in C^{1}(\mathbb{R},\mathbb{R})$ with the graph on
$[-4\pi,4\pi]$:

A numerical solution shows that $\Phi$ has a maximum
$\bar{R}=\Phi(x_{0})\doteq 3.15177$ at $x_{0}\doteq 1.03665$.

######  Example 2.12.

Now we suppose that $a_{j}$ is decaying exponentially to $0$. Let
$a_{j}={\,\textrm{\rm e}}^{-j}$, hence we have the discrete Kac-Baker
interaction kernel [13, 14]. Then

$\begin{gathered}\sum_{j\in\mathbb{N}}{\,\textrm{\rm e}}^{-j}{\,\textrm{\rm
e}}^{xj\imath}=\sum_{j\in\mathbb{N}}{\,\textrm{\rm
e}}^{(x\imath-1)j}=\frac{{\,\textrm{\rm e}}^{x\imath-1}}{1-{\,\textrm{\rm
e}}^{x\imath-1}}\\\ =\frac{\cos x+\imath\sin x}{{\,\textrm{\rm e}}-\cos
x-\imath\sin x}=\frac{{\,\textrm{\rm e}}\cos x-1+{\,\textrm{\rm e}}\imath\sin
x}{{\,\textrm{\rm e}}^{2}+1-2{\,\textrm{\rm e}}\cos x}.\end{gathered}$

So by (14) we derive

$\Phi(x)=\frac{2}{x}\left[\sum_{j\in\mathbb{N}}{\,\textrm{\rm
e}}^{-j}-\frac{{\,\textrm{\rm e}}\cos x-1}{{\,\textrm{\rm
e}}^{2}+1-2{\,\textrm{\rm e}}\cos x}\right]=\frac{2{\,\textrm{\rm
e}}({\,\textrm{\rm e}}+1)(1-\cos x)}{({\,\textrm{\rm e}}-1)x({\,\textrm{\rm
e}}^{2}+1-2{\,\textrm{\rm e}}\cos x)}.$

By Remark 1.1, $\Phi\in C^{1}(\mathbb{R},\mathbb{R})$ with the graph on
$[-4\pi,4\pi]$:

A numerical solution shows that $\Phi$ has a maximum
$\bar{R}=\Phi(x_{0})\doteq 0.992045$ at $x_{0}\doteq 0.991541$.

######  Example 2.13.

In this example, we suppose that $a_{j}$ is decaying polynomially to $0$ (cf.
[30]), by considering several cases:

1\. Let $a_{j}=\frac{1}{j^{4}}$. Then

$\Phi(x)=\frac{2}{x}\sum_{j\in\mathbb{N}}\left(\frac{1}{j^{4}}-\frac{1}{j^{4}}\cos
xj\right)=\frac{\left(|x|-2\pi\left[\frac{|x|}{2\pi}\right]\right)^{2}}{24x}\left(2\pi-|x|+2\pi\left[\frac{|x|}{2\pi}\right]\right)^{2}.$

Here $[\cdot]$ is the integer part function. By Remark 1.1, $\Phi\in
C^{1}(\mathbb{R},\mathbb{R})$ with the graph on $[-4\pi,4\pi]$:

$\Phi$ has a maximum $\bar{R}=\Phi(x_{0})=\frac{4\pi^{3}}{81}\doteq 1.53117$
at $x_{0}=2\pi/3\doteq 2.0944$. Similar results hold for $a_{j}=j^{-\beta}$
with $\beta>3$ by Remark 1.1.

2\. Let $a_{j}=\frac{1}{j^{3}}$. So we consider the dipole-dipole interaction
(cf. [5, 13, 25, 30]). By Remark 1.1, $\Phi\in C(\mathbb{R},\mathbb{R})$ with
the graph on $[-4\pi,4\pi]$:

$\Phi$ has a maximum $\bar{R}=\Phi(x_{0})\doteq 1.68311$ at $x_{0}\doteq
1.76076$. Next we know that [41]

$\sum_{j\in\mathbb{N}}\frac{1}{j}\cos
xj=-\ln\left|2\sin\frac{x}{2}\right|,\quad 0<x<2\pi.$

Then

$\sum_{j\in\mathbb{N}}\frac{1}{j^{2}}\sin
xj=-\int\limits_{0}^{x}\ln\left|2\sin\frac{s}{2}\right|ds.$

Using $x/2\leq\sin x\leq x$ for $x\geq 0$ small, we derive

$x-x\ln x=-\int\limits_{0}^{x}\ln
s\,ds\leq\sum_{j\in\mathbb{N}}\frac{1}{j^{2}}\sin
xj\leq-\int\limits_{0}^{x}\ln\frac{s}{2}ds=x-x\ln\frac{x}{2}.$

By L’Hopital’s rule, we obtain

$\lim_{x\to 0_{+}}\frac{\Phi(x)}{x}=\lim_{x\to
0_{+}}\frac{4\sum_{j\in\mathbb{N}}\frac{1}{j^{3}}\sin^{2}xj}{x^{2}}=\lim_{x\to
0_{+}}\frac{2\sum_{j\in\mathbb{N}}\frac{1}{j^{2}}\sin 2xj}{x}=+\infty.$

Hence $\Phi$ has no derivative at $x_{0}=0$.

Next, let $a_{j}=j^{-\beta}$ for $2<\beta<3$. By Remark 1.1, $\Phi$ is still
continuous. Since $\Phi(0)=0$ and

$\lim_{x\to 0_{+}}\frac{\Phi(x)}{x}\geq\lim_{x\to
0_{+}}\frac{4\sum_{j\in\mathbb{N}}\frac{1}{j^{3}}\sin^{2}xj}{x^{2}}=+\infty,$

$\Phi(x)$ is continuous but not $C^{1}$-smooth on $\mathbb{R}$.

3\. Let $a_{j}=\frac{1}{j^{2}}$. Then

$\Phi(x)=\frac{2}{x}\sum_{j\in\mathbb{N}}\left(\frac{1}{j^{2}}-\frac{1}{j^{2}}\cos
xj\right)=\frac{\left(|x|-2\pi\left[\frac{|x|}{2\pi}\right]\right)}{2x}\left(2\pi-|x|+2\pi\left[\frac{|x|}{2\pi}\right]\right).$

By Remark 1.1, $\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$ with the
graph on $[-4\pi,4\pi]$:

$\Phi$ is discontinuous at $x_{0}=0$ where it has a supremum $\bar{R}=\pi$.

4\. Let $a_{j}=j^{-\beta}$ for $1<\beta<2$. For $\beta=7/4$, $\Phi$ has the
graph on $[-4\pi,4\pi]$:

Hence $\Phi$ is discontinuous at $x_{0}=0$ with $\lim_{x\to
0_{+}}\Phi(x)=+\infty$. We show that this holds for any $1<\beta<2$. First
suppose $3/2<\beta<2$. Then the series

$\Upsilon(x):=\sum_{j\in\mathbb{N}}\frac{1}{j^{\beta-1}}\sin jx$

converges uniformly on any $[\varepsilon,2\pi-\varepsilon]$ for
$0<\varepsilon<\pi$. But since
$\sum_{j\in\mathbb{N}}\frac{1}{j^{2(\beta-1)}}<\infty$, so $\Upsilon\in
L^{2}\subset L^{1}$. On the other hand, we know [41] that

$\Upsilon(x):=\Gamma(2-\beta)\cos\frac{\pi(\beta-1)}{2}\cdot x^{\beta-2}+O(1)$

on $(0,\pi]$. Hence

$\sum_{j\in\mathbb{N}}\frac{1-\cos
jx}{j^{\beta}}=\int_{0}^{x}\Upsilon(s)ds=\frac{\Gamma(2-\beta)}{\beta-1}\cos\frac{\pi(\beta-1)}{2}\cdot
x^{\beta-1}+O(x)$

on $[0,\pi]$. Consequently, we obtain

$\Phi(x)=\frac{2\Gamma(2-\beta)}{\beta-1}\cos\frac{\pi(\beta-1)}{2}\cdot
x^{\beta-2}+O(1)$

on $(0,\pi]$, which implies $\lim_{x\to 0_{+}}\Phi(x)=+\infty$ for any
$3/2<\beta<2$. Finally, if $1<\beta\leq 3/2$, then

$\Phi(x)\geq\frac{2}{x}\sum_{j\in\mathbb{N}}\frac{1-\cos
jx}{j^{7/4}}=\frac{8}{3}\Gamma\left(\frac{1}{4}\right)\cos\frac{3\pi}{8}\cdot\frac{1}{\sqrt[4]{x}}+O(1)\to+\infty$

as $x\to 0_{+}$. Hence, $\lim_{x\to 0_{+}}\Phi(x)=+\infty$ for any
$1<\beta<2$.

Summarizing, we have the following result.

######  Lemma 2.14.

Let $a_{j}=j^{-\beta}$ for $1<\beta$. Then

  * (i)

$\Phi\in C^{1}(\mathbb{R},\mathbb{R})$ for $\beta>3$, and
${\mathcal{R}}\Phi=[-\bar{R},\bar{R}]$ for some $\bar{R}<\infty$.

  * (ii)

$\Phi\in C(\mathbb{R},\mathbb{R})$ and $\Phi\notin
C^{1}(\mathbb{R},\mathbb{R})$ for $2<\beta\leq 3$, and
${\mathcal{R}}\Phi=[-\bar{R},\bar{R}]$ for some $\bar{R}<\infty$.

  * (iii)

$\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$ and $\Phi\notin
C(\mathbb{R},\mathbb{R})$ for $\beta=2$, and ${\mathcal{R}}\Phi=(-\pi,\pi)$.

  * (iv)

$\Phi\in C(\mathbb{R}\setminus\\{0\\},\mathbb{R})$ and $\Phi\notin
C(\mathbb{R},\mathbb{R})$ for $1<\beta<2$, and
${\mathcal{R}}\Phi=(-\infty,+\infty)$.

######  Remark 2.15.

We see that if the interaction is strong, so the case (iv) of Lemma 2.14
holds, then there are continuum many quasi periodic traveling wave solutions
$U(z)$ of Theorem 1.2 for any $\nu\neq 0$, $T>0$ and $r\in(0,1)$ such that
$r\notin\left\\{z-[z]\mid z\in\frac{T}{2\pi}\Phi^{-1}(-\nu)\right\\}$, with
$\|U\|_{Y_{r}}\to\infty$ as $|\nu|\to\infty$ by Remark 2.10. On the other
hand, if the interaction is weak, then we can show in addition quasi periodic
traveling waves with speeds in intervals $(-\infty,-\bar{R})$ and
$(\bar{R},\infty)$ for any $T>0$ and $r\in(0,1)$.

######  Remark 2.16.

For the reader convenience, we present the above graphs of function $\Phi$ to
visualize their quantitative and qualitative changes according to different
choices of values of sequences $\\{a_{j}\\}_{j\in\mathbb{Z}}$ in (3), and
hence with different consequences from Theorems 1.2 and 1.3 for the existence
and bifurcations of quasi periodic traveling wave solutions of (3). Moreover,
these graphs can be compared with similar ones for traveling waves for higher
dimensional DNLS in Section 4 and for traveling waves with frequencies in
Section 5. Finally these examples are motivated by applications mentioned in
the corresponding references.

##  3 Bifurcation of Traveling Wave Solutions

In this section we proceed with the study of (6) when nonresonance of Theorem
1.2 fails, i.e. $r\in\\{\bar{r}_{1},\bar{r}_{2},\cdots,\bar{r}_{m}\\}$. We
scale in (6) the velocity by $\nu\leftrightarrow\nu/(1+\lambda)$ to get
equation

$-\nu\imath
U^{\prime}(z)=(1+\lambda)\left(\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+f(|U(z)|^{2})U(z)\right)\,,$
(15)

where $\lambda$ is a small parameter, i.e.
$u_{n}(t)=U\left(n-\frac{\nu}{1+\lambda}t\right)$ is a solution of (3). We are
interested in the existence of quasi periodic solutions $U(z)$ of (15) stated
in Theorem 1.3.

###  3.1 Preliminaries

In this subsection we recall some results from critical point theory of [29].
Let $H$ be a Hilbert space with a scalar product $(\cdot,\cdot)$ and the
corresponding norm $\|\cdot\|$. Let $\Theta:S^{1}\to L(H)$ be an isometric
representation of the unit circle $S^{1}$ over $H$, i.e. the following
properties are satisfied

  * (R)

$\Theta(0)=\bm{I}$ \- the identity,
$\Theta(\theta_{1}+\theta_{2})=\Theta(\theta_{1})\Theta(\theta_{2})$ for any
$\theta_{1},\theta_{2}\in S^{1}$, $(\theta,h)\to\Theta(\theta)h$ is
continuous, and $\|\Theta(\theta)h\|=\|h\|$ for any $\theta\in S^{1}$ and
$h\in H$.

We set

$\textrm{Fix}(S^{1}):=\left\\{h\in
H\mid\Theta(\theta)h=h\,\forall\theta\in\Theta\right\\}.$

We consider $J_{1},J_{2}\in C^{2}(H,\mathbb{R})$ such that

  * (H1)

$J_{2}(0)=0$ and $\nabla J_{1}(0)=\nabla J_{2}(0)=0$.

  * (H2)

$\textrm{Hess}\,J_{1}(0)$ is a Fredholm operator, i.e.
$\dim\textrm{Hess}\,J_{1}(0)<\infty$, ${\mathcal{R}}\textrm{Hess}\,J_{1}(0)$
is closed and $\textrm{codim}\,{\mathcal{R}}\textrm{Hess}\,J_{1}(0)<\infty$.

  * (H3)

$\dim\ker\textrm{Hess}\,J_{1}(0)\geq 2$ and $\textrm{Hess}\,J_{2}(0)$ is
positive definite on $\ker\textrm{Hess}\,J_{1}(0)$.

  * (H4)

$J_{1}$ and $J_{2}$ are $S^{1}$-invariant, i.e.
$J_{1,2}(\Theta(\theta)h)=\Theta(\theta)J_{1,2}(h)$ for any $\theta\in\Theta$
and $h\in H$.

  * (H5)

$\ker\textrm{Hess}\,J_{1}(0)\cap\textrm{Fix}(S^{1})=\\{0\\}$.

Now we can state the following [29, Theorem 6.7].

######  Theorem 3.1.

Under the above assumptions (H1)-(H5), for each sufficiently small
$\varepsilon>0$, equation

$\nabla J_{1}(h)+\lambda\nabla J_{2}(h)=0$ (16)

has at leat $\frac{1}{2}\dim\ker\textrm{\rm Hess}\,J_{1}(0)$ of $S^{1}$-orbit
solutions

$\left\\{(\lambda_{k}(\varepsilon),\Theta(\theta))h_{k}(\varepsilon)\mid\theta\in
S^{1}\right\\},\quad k=1,2,\cdots,\frac{1}{2}\dim\ker\textrm{\rm
Hess}\,J_{1}(0)$

such that $J_{2}(h_{k}(\varepsilon))=\varepsilon$ and $h_{k}(\varepsilon)\to
0$, $\lambda_{k}(\varepsilon)\to 0$ as $\varepsilon\to 0$. Clearly
$h_{k}(\varepsilon)\neq 0$.

######  Remark 3.2.

When $\textrm{Hess}\,J_{2}(0)$ is negative definite on
$\ker\textrm{Hess}\,J_{1}(0)$, then Theorem 3.1 holds for $\varepsilon<0$
small.

######  Remark 3.3.

By (H4), $\ker\textrm{Hess}\,J_{1}(0)$ is invariant with respect to $\Theta$.
Using (H5), $\dim\ker\textrm{Hess}\,J_{1}(0)$ is even.

Now assume $H=H_{+}\oplus H_{-}$ be an orthogonal and $\Theta$-invariant
decomposition with the corresponding orthogonal projections $P_{\pm}:H\to
H_{\pm}$. Then $\Theta(\theta)P_{\pm}=P_{\pm}\Theta(\theta)$ for any
$\theta\in\Theta$. Let us consider an equation

$\zeta(\bm{I}_{+}-\bm{I}_{-})h+(1+\lambda)({\mathcal{K}}h+\nabla{\mathcal{F}}(h))=0,$
(17)

where $\zeta\neq 0$ is a constant, $\lambda$ is a small parameter,
$\bm{I}_{\pm}:H_{\pm}\to H_{\pm}$ are the identities. We suppose

  * (A)

${\mathcal{K}}:H\to H$ is compact self-adjoint and ${\mathcal{F}}\in
C^{2}(H,\mathbb{R})$ with ${\mathcal{F}}(0)=0$, $\nabla{\mathcal{F}}(0)=0$,
$\textrm{\rm Hess}\,{\mathcal{F}}(0)=0$, and ${\mathcal{K}}$, ${\mathcal{F}}$
are $S^{1}$-invariant. Moreover, ${\mathcal{K}}H_{\pm}\subset H_{\pm}$.

Then

$\begin{gathered}J_{1}(h)=\frac{\zeta}{2}(\|P_{+}h\|^{2}-\|P_{-}h\|^{2})+\frac{1}{2}({\mathcal{K}}h,h)+{\mathcal{F}}(h),\\\
J_{2}(h)=\frac{1}{2}({\mathcal{K}}h,h)+{\mathcal{F}}(h).\end{gathered}$

Hence

$\begin{gathered}J_{1}(0)=J_{2}(0)=0,\quad\nabla J_{1}(0)=\nabla
J_{2}(0)=0,\\\
\textrm{Hess}\,J_{1}(0)=\zeta(\bm{I}_{+}-\bm{I}_{-})+{\mathcal{K}},\quad\textrm{Hess}\,J_{2}(0)={\mathcal{K}}.\end{gathered}$

So assumptions (H1), (H2) and (H4) are satisfied. Since
$P_{\pm}{\mathcal{K}}={\mathcal{K}}P_{\pm}$, equation

$\textrm{Hess}\,J_{1}(0)h=\zeta(\bm{I}_{+}-\bm{I}_{-})h+{\mathcal{K}}h=0$

splits into

${\mathcal{K}}h_{+}=-\zeta h_{+},\quad{\mathcal{K}}h_{-}=\zeta h_{-},\quad
h_{\pm}=P_{\pm}h.$

Consequently, supposing either

  * (B+)

$\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}=\\{0\\}$,
$\dim\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}\geq 2$ and
$\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}\cap\textrm{Fix}(S^{1})=\\{0\\}$

or

  * (B-)

$\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}=\\{0\\}$,
$\dim\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}\geq 2$ and
$\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}\cap\textrm{Fix}(S^{1})=\\{0\\}$

we get either

$\ker\textrm{Hess}\,J_{1}(0)=\ker(\zeta\bm{I}-{\mathcal{K}})\cap H_{-}$

or

$\ker\textrm{Hess}\,J_{1}(0)=\ker(\zeta\bm{I}+{\mathcal{K}})\cap H_{+}$

and so (H5) holds as well. Finally, we derive

$\textrm{Hess}\,J_{2}(0)|\ker\textrm{Hess}\,J_{1}(0)=\pm\zeta\bm{I}$

and thus (H3) is also verified (cf. Remark 3.2). Summarizing, Theorem 3.1 and
Remark 3.2 is applicable to (17):

######  Corollary 3.4.

Under assumptions (A) and (B±), for each sufficiently small $\varepsilon\neq
0$, $\pm\varepsilon\zeta>0$, equation (17) has at leat
$\frac{1}{2}\dim\ker(\zeta\bm{I}\mp{\mathcal{K}})\cap H_{\mp}$ of
$S^{1}$-orbit solutions

$\left\\{(\lambda_{k}(\varepsilon),\Theta(\theta))h_{k}(\varepsilon)\mid\theta\in
S^{1}\right\\},\quad
k=1,2,\cdots,\frac{1}{2}\dim\ker(\zeta\bm{I}\mp{\mathcal{K}})\cap H_{\mp}$

such that
$\frac{1}{2}({\mathcal{K}}h_{k}(\varepsilon),h_{k}(\varepsilon))+{\mathcal{F}}(h_{k}(\varepsilon))=\varepsilon$
and $h_{k}(\varepsilon)\to 0$, $\lambda_{k}(\varepsilon)\to 0$ as
$\varepsilon\to 0$. Clearly $h_{k}(\varepsilon)\neq 0$.

######  Remark 3.5.

If $\textrm{Fix}(S^{1})=\\{0\\}$ then (B+) holds if

  * (i)

$-\zeta\notin\sigma({\mathcal{K}}/H_{+})$,
$\zeta\in\sigma({\mathcal{K}}/H_{-})$ and $\zeta$ has a multiplicity at least
$2$,

while (B-) holds if

  * (ii)

$-\zeta\in\sigma({\mathcal{K}}/H_{+})$,
$\zeta\notin\sigma({\mathcal{K}}/H_{-})$ and $-\zeta$ has a multiplicity at
least $2$,

respectively.

###  3.2 Proof of Theorem 1.3

We again assume for simplicity $T=2\pi$. So let $r=\bar{r}_{1}\in(0,1)$ and
the equation

$-\nu=\Phi\left(\bar{r}_{1}+k\right)$

has solutions $k_{1},k_{2},\cdots,k_{m_{1}}\in\mathbb{Z}$ which are either all
nonnegative, or all negative. Next (15) has the form (cf. (9))

$2(\nu\bm{I}_{+}-\nu\bm{I}_{-})-(1+\lambda)\left(\widetilde{K}_{r}{\mathcal{L}}_{r}U+\Psi_{r}(U)\right)=0$
(18)

and

$\begin{gathered}H=X_{r},\quad\zeta=2\nu,\quad H_{\pm}=X_{\pm},\\\
{\mathcal{K}}=-\widetilde{K}_{r}{\mathcal{L}}_{r},\quad{\mathcal{F}}(u)=-\int_{0}^{2\pi}F(|U(z)|^{2})dz.\end{gathered}$

Isometric representation $\Theta$ is naturally given as

$\Theta(\theta)U(z):=U(z+\theta),$

i.e.

$\Theta(\theta)\left(\sum_{k\in\mathbb{Z}}U_{k}{\,\textrm{\rm
e}}^{(\bar{r}_{1}+k)z\imath}\right)=\sum_{k\in\mathbb{Z}}U_{k}{\,\textrm{\rm
e}}^{\theta k\imath}{\,\textrm{\rm e}}^{(\bar{r}_{1}+k)z\imath}.$

Note $\textrm{Fix}(S^{1})=\\{0\\}$. It is easy to verify (R) for $\Theta$. By
results of Section 2, we get both ${\mathcal{K}}H_{\pm}\subset H_{\pm}$ and
assumption (A) holds, and moreover

$\sigma\left({\mathcal{K}}/H_{\pm}\right)=\left\\{\pm 2\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{\pm}\right\\}.$

Note $\mathbb{Z}_{+}=\\{0\\}\cup\mathbb{N}$ and $\mathbb{Z}_{-}=-\mathbb{N}$.
Hence (i) of Remark 3.5 is satisfied if

$-\nu\notin\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{+}\right\\},\quad-\nu\in\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{-}\right\\},$

while (ii) if

$-\nu\in\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{+}\right\\},\quad-\nu\notin\left\\{\Phi(\bar{r}_{1}+k)\mid
k\in\mathbb{Z}_{-}\right\\}.$

But these are precisely assumptions of Theorem 1.3. So its proof is complete
by Corollary 3.4 and Remark 3.5.

##  4 Traveling Waves for Higher Dimensional DNLS

In this section, we first show how to extend previous results for
2-dimensional DNLS (2D DNLS) [10, 11, 22] of forms

$\begin{gathered}\imath\dot{u}_{n,m}=\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\Delta_{i,j}u_{n,m}+f(|u_{n,m}|^{2})u_{n,m},\quad(n,m)\in\mathbb{Z}^{2}\\\
=2\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\left(u_{n+i,m+j}-u_{n,m}\right)+f(|u_{n,m}|^{2})u_{n,m},\end{gathered}$
(19)

where $u_{n,m}\in\mathbb{C}$,
$\mathbb{Z}_{0}^{2}:=\mathbb{Z}^{2}\setminus\\{(0,0)\\}$,
$\Delta_{i,j}u_{n,m}:=u_{n+i,m+j}+u_{n-i,m-j}-2u_{n,m}$ are $2$-dimensional
discrete Laplacians, $f$ satisfies (H1) and $a_{i,j}=a_{-i,-j}$ along with
$\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}|a_{i,j}|<\infty$ and all $a_{i,j}$
are not zero.

Again, (19) conserves two dynamical invariants

$\begin{gathered}\sum\limits_{(n,m)\in\mathbb{Z}^{2}}|u_{n,m}|^{2}\quad-\textrm{the
norm},\\\
\sum\limits_{(n,m)\in\mathbb{Z}^{2}}\left[-\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\left|u_{n+i,m+j}-u_{n,m}\right|^{2}+F(|u_{n,m}|^{2})\right]\quad-\textrm{the
energy}.\end{gathered}$

We look for traveling wave solutions of (19) of the form

$u_{n,m}(t)=U(n\cos\theta+m\sin\theta-\nu t)$ (20)

with a direction $(\cos\theta,\sin\theta)$ [21]. Hence we are interested in
the equation

$-\nu\imath
U^{\prime}(z)=\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\partial_{i,j}U(z)+f(|U(z)|^{2})U(z)\,,$
(21)

where $z=n\cos\theta+m\sin\theta-\nu t$, $\nu\neq 0$ and

$\partial_{i,j}U(z):=U(z+i\cos\theta+j\sin\theta)+U(z-i\cos\theta-j\sin\theta)-2U(z).$

We see that (21) has a very similar form like (6). So we can directly repeat
the above arguments, where now instead of $\Phi(x)$ we get

$\Phi_{\theta}(x):=\frac{4}{x}\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\sin^{2}\frac{x(i\cos\theta+j\sin\theta)}{2}.$

Set $\bar{R}_{\theta}:=\sup_{\mathbb{R}}\Phi_{\theta}$. Summarizing, Theorems
1.2 and 1.3 have the following analogies:

######  Theorem 4.1.

Let (H1) hold and $T>0$, $\theta\in[0,2\pi)$. Then for almost each
$\nu\in\mathbb{R}\setminus\\{0\\}$ and any rational $r\in\mathbb{Q}\cap(0,1)$,
there is a nonzero periodic traveling wave solution (20) of (19) with $U\in
C^{1}(\mathbb{R},\mathbb{C})$ satisfying (5). Moreover, for any
$\nu\in\mathbb{R}\setminus\\{0\\}$, there is at most a finite number of
$\bar{r}_{1,\theta},\bar{r}_{2,\theta},\cdots,\bar{r}_{m_{\theta},\theta}\in(0,1)$
such that equation

$-\nu=\Phi_{\theta}\left(\frac{2\pi}{T}(\bar{r}_{j,\theta}+k)\right)$

has a solution $k\in\mathbb{Z}$. Then for any
$r\in(0,1)\setminus\\{\bar{r}_{1,\theta},\bar{r}_{2,\theta},\cdots,\bar{r}_{m_{\theta},\theta}\\}$
there is a nonzero quasi periodic traveling wave solution (20) of (19) with
the above properties. In particular, for any $|\nu|>\bar{R}_{\theta}$ and
$r\in(0,1)$, there is such a nonzero quasi periodic traveling wave solution.

######  Theorem 4.2.

Suppose $f\in C^{2}(\mathbb{R}_{+},\mathbb{R})$ with $f(0)=0$. If there are
$\bar{r}_{1,\theta}\in(0,1)$, $T>0$, $\theta\in[0,2\pi)$ and
$\nu\in{\mathcal{R}}\Phi_{\theta}\setminus\\{0\\}$ such that all integer
number solutions $k_{1},k_{2},\cdots,k_{m_{1,\theta}}$ of equation

$-\nu=\Phi_{\theta}\left(\frac{2\pi}{T}(\bar{r}_{1,\theta}+k)\right)$

are either nonnegative or negative, and $m_{1,\theta}>0$. Then for any
$\varepsilon>0$ small there are $m_{1,\theta}$ branches of nonzero quasi
periodic traveling wave solutions (20) of (19) with $U_{j,\varepsilon}\in
C^{1}(\mathbb{R},\mathbb{C})$, $j=1,2,\cdots,m_{1,\theta}$, and nonzero
velocity $\nu_{\varepsilon}$ satisfying $U_{j,\varepsilon}(z+T)={\,\textrm{\rm
e}}^{2\pi\bar{r}_{1}\imath}U(z)_{j,\varepsilon}$, $\forall z\in\mathbb{R}$
along with $\nu_{\varepsilon}\to\nu$ and $U_{j,\varepsilon}\rightrightarrows
0$ uniformly on $\mathbb{R}$ as $\varepsilon\to 0$.

######  Example 4.3.

We consider the discrete 2D Kac-Baker interaction kernel
$a_{i,j}={\,\textrm{\rm e}}^{-|i|-|j|}$ for $(i,j)\in\mathbb{Z}^{2}_{0}$. Then
$\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}{\,\textrm{\rm
e}}^{-|i|-|j|}=\frac{4{\,\textrm{\rm e}}}{(e-1)^{2}}$ and

$\Phi_{\theta}(x)=\left[\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm
e}}-1)^{2}}-\frac{({\,\textrm{\rm e}}^{2}-1)^{2}}{(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm e}}\cos(x\cos\theta))(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm e}}\cos(x\sin\theta))}\right]\frac{4}{x}.$

A numerical evaluation shows that function $(x,\theta)\to\Phi_{\theta}(x)$ has
a maximum $\bar{R}\doteq 9.75047$ at $x_{0}\doteq 1.08205$ and
$\theta_{0}\doteq 0.785398$. To justify this theoretically, we take
$a=x\cos\theta$ and $b=x\sin\theta$ to transform $\Phi_{\theta}(x)$ into

$\Phi(a,b)=\left[\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm
e}}-1)^{2}}-\frac{({\,\textrm{\rm e}}^{2}-1)^{2}}{(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm e}}\cos a)(1+{\,\textrm{\rm e}}^{2}-2{\,\textrm{\rm
e}}\cos b)}\right]\frac{4}{\sqrt{a^{2}+b^{2}}}\,.$

Note $\Phi(a,b)=\Phi(\pm a,\pm b)=\Phi(b,a)$. A numerical evaluation shows
that function $\Phi_{\theta}(a,b)$ has a maximum $\bar{R}\doteq 9.75047$ at
$a_{0}=b_{0}\doteq 0.765123$ which correspond to $x_{0}$ and $\theta_{0}$. On
the other hand, if $a^{2}+b^{2}\geq 4$ then $\Phi(a,b)\leq
2\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm e}}-1)^{2}}\doteq
9.36539<9.75047$, so $\Phi(a,b)$ achieves its maximum in the disc
$D_{2}:=\left\\{a^{2}+b^{2}\leq 4\right\\}$. Next, solving the system
$\frac{\partial}{\partial a}\Phi(a,b)=\frac{\partial}{\partial b}\Phi(a,b)=0$
we derive $\frac{\sin a_{0}}{a_{0}}=\frac{\sin b_{0}}{b_{0}}$ at the maximum
point $(a_{0},b_{0})\in D_{2}$, $a_{0}>0$, $b_{0}>0$. But the function
$\frac{\sin w}{w}$ is decreasing on $[0,2]$, so $a_{0}=b_{0}$, and thus
$\theta_{0}=\pi/4$. An elementary but awkward calculus shows for function

$\Phi_{\pi/4}(x)=\left[\frac{({\,\textrm{\rm e}}+1)^{2}}{({\,\textrm{\rm
e}}-1)^{2}}-\frac{({\,\textrm{\rm e}}^{2}-1)^{2}}{\left(1+{\,\textrm{\rm
e}}^{2}-2{\,\textrm{\rm
e}}\cos\left(x\frac{\sqrt{2}}{2}\right)\right)^{2}}\right]\frac{4}{x}$

with the graph on $[-20,20]$:

that $x_{0}\in(0,2)$ is the only root of $\Phi_{\pi/4}^{\prime}(x_{0})=0$ on
$(0,2)$, and then $\bar{R}=\Phi_{\pi/4}(x_{0})$. So $\bar{R}$ is computed also
analytically in this case.

Summarizing, Theorems 4.1 and 4.2 can be applied in this case for any suitable
nonzero $\nu$, and resonant traveling waves with maximum velocities which are
achieved in the diagonal directions $\pm\theta_{0}=\pm\pi/4$.

Finally, it is now clear how to proceed to 3D DNLS or even to higher
dimensional DNLS, so we omit further details.

##  5 Traveling Waves with Frequencies

We could consider more general traveling wave solutions than above of forms

$\begin{gathered}u_{n}(t)=U(n-\nu t){\,\textrm{\rm e}}^{\imath\omega t},\\\
u_{n,m}(t)=U(n\cos\theta+m\sin\theta-\nu t){\,\textrm{\rm e}}^{\imath\omega
t}\end{gathered}$ (22)

with velocity $\nu\neq 0$ and frequency $\omega\neq 0$ (see [32]). Then, there
is a dispersion relation between the velocity $\nu$ and frequency $\omega$ as
follows. Inserting (22) into (3) and (19), respectively, we are interested in
equations

$\begin{gathered}-\nu\imath
U^{\prime}(z)=\sum\limits_{j\in\mathbb{N}}a_{j}\partial_{j}U(z)+\omega
U(z)+f(|U(z)|^{2})U(z),\\\ -\nu\imath
U^{\prime}(z)=\sum\limits_{(i,j)\in\mathbb{Z}^{2}_{0}}a_{i,j}\partial_{i,j}U(z)+\omega
U(z)+f(|U(z)|^{2})U(z),\end{gathered}$ (23)

respectively. We see that (6), (21) and (23) are very similar, so we can
repeat the above arguments to (23) when instead of $\Phi(x)$ and
$\Phi_{\theta}(x)$ now we have

$\Phi(x,\omega):=\Phi(x)-\frac{\omega}{x}\,,\quad\Phi_{\theta}(x,\omega):=\Phi_{\theta}(x)-\frac{\omega}{x}\,,$
(24)

respectively. Consequently, we have analogies of Theorems 1.2, 1.3, 4.1 and
4.2 to (23) but we do not state them since they are obvious.

######  Example 5.1.

We consider the discrete Kac-Baker interaction kernel from Example 2.12. Then

$\Phi(x,\omega)=\frac{2{\,\textrm{\rm e}}({\,\textrm{\rm e}}+1)(1-\cos
x)}{({\,\textrm{\rm e}}-1)x({\,\textrm{\rm e}}^{2}+1-2{\,\textrm{\rm e}}\cos
x)}-\frac{\omega}{x}\,.$

To be more concrete, we first take $\omega=1$, and then $\Phi(x,1)$ has the
graph on $[-4\pi,4\pi]$:

with $\lim_{x\to 0_{\pm}}\Phi(x,1)=\mp\infty$. A numerical evaluation shows
that function $\Phi(x,1)$ has a maximum $\bar{R}\doteq 0.282071$ on
$(0,\infty)$ at $x_{0}\doteq 1.9905$. Consequently, the analogy of Theorem 1.2
can be applied now to any $\nu\neq 0$ while the analogy of Theorem 1.3 can be
applied for almost any $\nu\in\mathbb{R}\setminus[-0.282071,0.282071]$, while
for nonzero $\nu\in[-0.282071,0.282071]$ could be problematic in general.

On the other hand for $\omega=-1$, $\Phi(x,-1)$ has the graph on
$[-4\pi,4\pi]$:

with $\lim_{x\to 0_{\pm}}\Phi(x,-1)=\pm\infty$. Consequently, the analogy of
Theorem 1.2 can again be applied now to any $\nu\neq 0$ while the analogy of
Theorem 1.3 can now be applied for almost any $\nu\neq 0$. Of course now we
have totally different situations than in Example 2.12 for traveling waves
without frequencies by comparing the above graphs with that one in Example
2.12.

###  5.1 Acknowledgements

Michal Fečkan is partially supported by the Grants VEGA-MS 1/0098/08 and VEGA-
SAV 2/7140/27. Vassilis Rothos is partially supported by Research Grant-
International Relations of AUTH.

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mycases $\displaystyle{##}$$\displaystyle{##}$ {.

#  Stochastic Maximum Principle for optimal advertising models with delay and
non-convex control space.

Giuseppina Guatteri
Dipartimento di Matematica, Politecnico di Milano
via Bonardi 9, 20133
Milano, Italia
<EMAIL_ADDRESS>

Federica Masiero
Dipartimento di Matematica e Applicazioni
Università di Milano-Bicocca
via Cozzi 55, 20125 Milano, Italia
<EMAIL_ADDRESS>
###### Abstract

In this paper we study optimal advertising problems that models the
introduction of a new product into the market in the presence of carryover
effects of the advertisement and with memory effects in the level of goodwill.
In particular, we let the dynamics of the product goodwill to depend on the
past, and also on past advertising efforts. We treat the problem by means of
the stochastic Pontryagin maximum principle, that here is considered for a
class of problems where in the state equation either the state or the control
depend on the past. Moreover the control acts on the martingale term, as in
[6], but now the space of controls $U$ can be chosen to be non-convex. The
maximum principle is thus formulated using a first-order adjoint Backward
Stochastic Differential Equations (BSDEs), which can be explicitly computed
due to the specific characteristics of the model, and a second-order adjoint
relation.

Key words: Stochastic optimal Control; Delay Equations; Advertisement Models;
Stochastic Maximum Principle

AMS classification: 93E20, 49K45, 93C23

##  1 Introduction

We consider a stochastic model for problems of optimal advertising under
uncertainty, so we have to study a stochastic version of advertising models.
We start from the stochastic model introduced in [1] (see also [2] and [3]):
we consider carryover effects of the advertising, which in the model reads as
delay in the control, and with memory effects of the level of goodwill, which
in the model reads as delay in the state. We refer also to [4] for optimal
advertising problems with memory effects in the level of goodwill.

In our model the delay in the effect of advertising affects the martingale
term of the state equation,
Namely we consider, following [1], the controlled stochastic differential
equation in $\mathbb{R}$ with pointwise delay in the state and in the control

$\left\\{\begin{array}[c]{l}dx(t)=\left[b_{0}u(t)-a_{0}x(t)-{a_{d}}x(t-\mathtt{d})\right]\,dt+\sigma_{1}x(t)dW^{1}_{t}+\sigma_{2}u(t-\mathtt{d})dW^{2}_{t},\\\
\\\ x(\tau)=x_{0}(\tau),\quad\tau\in[-\mathtt{d},0].\\\ \end{array}\right.$
(1.1)

Following the usual notations, see e.g. [1], $x$ is the level of goodwill,
$a_{0}$ and $a_{d}$ are factors related to the image deterioration in case of
no advertising, $b_{0}$ is a constant representing an advertising
effectiveness factor. The diffusion term $\sigma_{1}x(t)dW^{1}_{t}$ represents
for the word of mouth communication, with $\sigma_{1}\geq 0$ the so-called
”advertising volatility”; while the second diffusion term keeps track of the
delayed effect of the advertising effort $u$, and the constant $\sigma_{2}\geq
0$ is the effectiveness volatility of the communication. Besides, $x_{0}(0)$
is the initial goodwill level, while $x_{0}(\tau),\,\tau\in[-\mathtt{d},0)$ is
the history of the goodwill level before the advertising campaign is started.

The functional to maximize, over all controls in $\mathcal{U}$, is the
following profit, defined on finite horizon:

$\bar{J}(t,x,u)=\mathbb{E}\int_{t}^{T}\left(-c\left(u(s)\right){+}l\left(x(s))\right)\right)\;ds+\mathbb{E}\,r(x(T)),$
(1.2)

where $c$ is the cost of advertising, $l$ is the current reward, and $r$ is
the reward from the final goodwill. Our purpose is to derive a maximum
principle for such a problem with non convex control space $U$, extending the
results already present in the literature for the convex case, see e.g. [5],
where the stochastic maximum principle for control problems with pointwise
delay in the state and in the control is studied, see also [6] and [7] where
more general models are treated but the convexity of the control space $U$ is
still required. We underline the fact that stochastic control problems with
diffusion depending on the control are difficult to treat; concerning problems
related to advertising we mention the recent paper [8], where the author
solves the problem by means of the dynamic programming approach, differently
from here the case of pointwise delay cannot be treated, moreover it is taken
into account a diffusion depending linearly on the control, but not on the
delayed control. We stress that one novelty of this paper is to handle the new
difficulty coming form the non convexity of $U$: also in the non delayed case
the non convexity of $U$ makes the approach based on stochastic maximum
principle more complicated. First we need to utilize the spike variation of
the control and introduce the second variation to handle the control acting on
the martingale term. We will therefore follow [9] to introduce this additional
step. Thanks to the specific characteristics of the state equation, we can
derive a quasi-explicit form for both the first adjoint, which results in an
anticipating backward equation, see [10], and the second adjoint, which is
written using the optimal cost and a simple auxiliary process. Note that the
specific case we address is not considered in [11] and does not completely
fall within the hypotheses of [9], as we also consider the delay the control
appearing in the martingale term.

The structure of the paper is the following: in section 2 we present the
notation, the assumptions and the control problem. Section 3 is devoted to
collect the results on the first variation of the optimal state and also on
the second variation, which we have to study since the space of controls is
not convex. Section 4 concerns the first and second adjoint relations
necessary to formulate the stochastic maximum principle, which is stated and
proved section 4.3.

##  2 Stochastic control problem for delay equations appearing in advertising
models

###  2.1 Assumptions and preliminaries

Let $W(t)$ be a $2$ dimensional brownian motion defined on a complete
probability space $(\Omega,{\cal F},\mathbb{P})$ and $(\mathcal{F}_{t})_{t\geq
0}$ the natural filtration generated by $W$, augmented in the usual way. We
fix a finite time horizon $T>0$ and we set, for every $q\geq 1$ and for every
$0\leq u\leq v\leq T$, the following spaces:

$L^{0}_{\mathcal{F}}(\Omega\times[u,v];\mathbb{R}^{k})=:\\{X:\Omega\times[u,v]\rightarrow\mathbb{R}^{k},\text{
$(\mathcal{F}_{t})$-progressive measurable\\} }\\\ $ (2.1)
$L^{q}_{\mathcal{F}}(\Omega\times[u,v];\mathbb{R}^{k})=:\\\
\\{X:\Omega\times[u,v]\rightarrow\mathbb{R}^{k},\text{
$(\mathcal{F}_{t})$-progr.meas.,
}\left(\mathbb{E}\int_{u}^{v}|X(t)|^{q}dt\right)^{1/q}<\infty\\}$ (2.2)
$L^{q}_{\mathcal{F}}(\Omega;C([u,v];\mathbb{R}^{k}))=:\\\
\\{X:\Omega\times[u,v]\rightarrow\mathbb{R}^{k},\text{
$(\mathcal{F}_{t})$-progr.meas.,
}\Big{(}\mathbb{E}\sup_{t\in[u,v]}|X(t)|^{q}\Big{)}^{1/q}<\infty\\}$ (2.3)

###  2.2 Formulation of the problem

We recall the state equation we are interested in:

$\left\\{\begin{array}[c]{l}dx(t)=\left[b_{0}u(t)-a_{0}x(t)-{a_{d}}x(t-\mathtt{d})\right]\,dt+\sigma_{1}x(t)dW^{1}_{t}+\sigma_{2}u(t-\mathtt{d})dW^{2}_{t},\\\
\\\ x(\tau)=x_{0}(\tau),\quad\tau\in[-\mathtt{d},0),\;x(0)=x_{0}(0).\\\
\end{array}\right.$ (2.4)

We assume the following on the coefficients:

######  Hypothesis 2.1.

  * (i)

$a_{0},b_{0},\sigma_{1},\,\sigma_{2}\in\mathbb{R}$;

  * (ii)

the control strategy $u$, the advertisement in this case, belongs to the space

$\mathcal{U}:=\left\\{z\in
L^{0}_{\mathcal{F}}(\Omega\times[0,T],\mathbb{R}):z(t)\in
U,\;\mathbb{P}-a.s.\right\\}$

where $U$ is a bounded subset of $\mathbb{R}$ possibly non convex, in
particular $U$ can be a bounded subset of $\mathbb{N}$ and this represents the
realistic situation in which the advertisement is multiple of a given
quantity;

  * (iii)

$\mathtt{d}>0$ is the delay with which the past of the state affects the
system;

We recall that the purpose is to maximize, over all controls in $\mathcal{U}$,
the following profit, on finite horizon:

$\bar{J}(t,x,u)=\mathbb{E}\int_{t}^{T}\left(-c\left(u(s)\right){+}l\left(x(s))\right)\right)\;ds+\mathbb{E}\,r(x(T)),$
(2.5)

######  Hypothesis 2.2.

We assume the following:

  * (i)

according to the literature, see e.g. [12], $c:U\rightarrow\mathbb{R}$ is non
linear, convex and locally lipschitz;

  * (ii)

$l:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ represents the current reward,
it is twice differentiable with at most linear growth;

  * (iii)

$r:\mathbb{R}\rightarrow\mathbb{R}$ represents the foreseen reward from the
final goodwill and it is twice differentiable and strictly increasing, with
bounded second derivatives.

From now on, since we recall and apply the results in [9] where a minimization
problem is considered, we focus on the problem (equivalent to the original one
of maximizing the profit $\bar{J}$ given in (2.5) since $J=-\bar{J}$) of
minimizing, over all admissible control in ${\cal U}$, the cost functional

$J(t,x,u)=\mathbb{E}\int_{t}^{T}\left(c(u(s))-l(x(s))\right)ds-\mathbb{E}\,r(x(T)),$
(2.6)

We are going to formulate necessary conditions for optimality.

##  3 First and second order variations of the optimal state equation

Since $U$ is not necessarily convex we use the spike variation method, see
[13], see also [9] for the case of delay equation.
Let $(u,x)$ be an optimal couple: that is ${u}$ is an optimal control and
${x}$ its corresponding optimal trajectory, that is solution to equation
(2.4). The spike variation works as follows: for $\varepsilon>0$ we consider
an interval $V_{\varepsilon}\subset[0,T]$, with
$m(V_{\varepsilon})=\varepsilon$, where $m$ is the Lebesgue measure. Let $v\in
U$ we set

$u^{\varepsilon}(t)=\left\\{\begin{array}[]{ll}{u}(t)&t\in[0,T]\setminus
V_{\varepsilon}\\\ v&t\in V_{\varepsilon}\end{array}\right.$ (3.1)

We are going to derive a maximum principle for the control problem with state
equation (2.4) and cost functional (2.6) in the case of non convex space of
controls. We will denote by $x^{\varepsilon}$ the solution of (2.4)
corresponding to $u^{\varepsilon}$ and $\delta u$ the spike variation of the
control, i.e. $\delta u(t):=(u(t)-v))I_{V_{\varepsilon}}(t)$.
We can write the equation for the first order variation of the state:

$\left\\{\begin{array}[]{ll}dy^{\varepsilon}(t)=\left[-a_{0}y^{\varepsilon}(t)-a_{d}y^{\varepsilon}(t-\mathtt{d})\right]\,dt+\sigma_{1}y^{\varepsilon}(t)dW^{1}_{t}+\sigma_{2}\delta
u(t-\mathtt{d})dW^{2}_{t}\quad t\in[0,T],\\\ \\\ y^{\varepsilon}(0)=0,\quad
y^{\varepsilon}(\tau)=0,\quad-\mathtt{d}\leq\tau<0.\end{array}\right.$ (3.2)

where $\delta
u^{\varepsilon}(t-\mathtt{d})=(u(t-\mathtt{d})-v)I_{V_{\varepsilon}}(t-\mathtt{d})$
and the second variation is, for $t\in[0,T]$,

$\left\\{\begin{array}[]{ll}dz^{\varepsilon}(t)=\left[-a_{0}z^{\varepsilon}(t)-a_{d}z^{\varepsilon}(t-\mathtt{d})\right]\,dt+b_{0}\delta
u(t)dt+\sigma_{1}z^{\varepsilon}(t)dW^{1}_{t}\\\ \\\
z^{\varepsilon}(0)=0,\quad
z^{\varepsilon}(\tau)=0,\quad-\mathtt{d}\leq\tau<0.\end{array}\right.$ (3.3)

It is well known that such equations are well posed in
$L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}))$ (see e.g. [14] for a general
theory on stochastic delay equations).

The following asymptotic behaviors hold, see [9, Theorem 3.3] for the delayed
system and the classic result in [15, Theorem 4.4],

$\mathbb{E}\sup_{t\in[0,T]}|y^{\varepsilon}(t)|^{2}=O(\varepsilon),\qquad\mathbb{E}\sup_{t\in[0,T]}|z^{\varepsilon}(t)|^{2}=O(\varepsilon),$
(3.4)
$\mathbb{E}\sup_{t\in[0,T]}|x^{\varepsilon}(t)-x(t)-y^{\varepsilon}(t)|^{2}=O(\varepsilon^{2}),$
(3.5)
$\mathbb{E}\sup_{t\in[0,T]}|x^{\varepsilon}(t)-x(t)-y^{\varepsilon}(t)-z^{\varepsilon}(t)|^{2}=o(\varepsilon^{2}),$
(3.6)

Moreover, $\forall\,p\geq 1$,

$\displaystyle\mathbb{E}\sup_{t\in[-\mathtt{d},T]}|y^{\varepsilon}(t)|^{p}<+\infty,\qquad\mathbb{E}\sup_{t\in[-\mathtt{d},T]}|z^{\varepsilon}(t)|^{p}<+\infty,$
(3.7)

##  4  Maximum Principle for the stochastic delay equation

In this Section, first we formulate the first order adjoint equation in 4.1,
then we pass to the second order adjoint in 4.2, and finally we formulate the
stochastic maximum principle in 4.3.

###  4.1 First order duality relation

Following [9] we define the first order adjoint equation of (3.2), that is the
anticipated backward equation

$\left\\{\begin{array}[]{l}p(t)=r_{x}(x(T))+\displaystyle\int_{t}^{T}l_{x}\left(x(s)\right)\,ds-
a_{0}\displaystyle\int_{t}^{T}p(s)\,ds-
a_{d}\displaystyle\int_{t}^{T}\mathbb{E}^{{\cal F}_{s}}p(s+\mathtt{d})\,ds\\\
\\\ \qquad+\sigma_{1}\displaystyle\int_{t}^{T}\mathbb{E}^{{\cal
F}_{s}}q^{1}(s+\mathtt{d})\,ds+\displaystyle\int_{t}^{T}q^{1}(s)dW^{1}_{s}+\displaystyle\int_{t}^{T}q^{2}(s)dW^{2}_{s}\\\
\\\ p(T-\tau)=0,\;\text{for all }\tau\in[-\mathtt{d},0),\\\ \\\
q^{1}(T-\tau)=q^{2}(T-\tau)=0\quad\text{a.e.
}\tau\in[-\mathtt{d},0).\end{array}\right.$ (4.1)

that admits a unique solution $(p,q)\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}))\times
L^{2}_{\mathcal{F}}(\Omega\times[0,T];\mathbb{R}^{2})$, see e.g. [10] where we
write $q=\left(q^{1},q^{2}\right)$.

Since equation (4.1) is linear, in view of the future times conditions we have
an explicit, recursive, formulation for the solution $(p,q)$:

######  Proposition 4.1.

Assume Hypotheses 2.1 and 2.2. Define for every $\
k=0,\dots,\Big{[}\frac{T}{\mathtt{d}}\Big{]}$,

$p^{k}(t):=p_{|_{[(T-(k+1)\mathtt{d})\vee 0,(T-k\mathtt{d})]}(t)}$

and

$q^{k,1}(t):=q^{1}_{|_{[(T-(k+1)\mathtt{d})\vee
0,(T-k\mathtt{d})]}}(t),q^{k,2}(t):=q^{2}_{|_{[(T-(k+1)\mathtt{d})\vee
0,(T-k\mathtt{d})]}}(t).$

Then $(p^{k},q^{k})$ solve:

$\left\\{\begin{array}[]{ll}p^{0}(t)=e^{-a_{0}(T-t)}\mathbb{E}^{\mathcal{F}_{t}}(r_{x}(x(T)))+\mathbb{E}^{\mathcal{F}_{t}}\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}l_{x}\left(x(s)\right)\,ds,\quad
t\in[T-\mathtt{d},T],\\\ \\\
q^{0,1}(t)=e^{-a_{0}(T-t)}L_{1}(t)-\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}K_{1}(s,t)\,ds,\qquad\quad\text{
for a.e. }t\in[T-\mathtt{d},T],\\\
q^{0,2}(t)=e^{-a_{0}(T-t)}L_{2}(t)-\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}K_{2}(s,t)\,ds,\qquad\quad\text{
for a.e. }t\in[T-\mathtt{d},T].\\\
p^{k}(t)=e^{-a_{0}(T-k\mathtt{d}-t)}p^{k-1}(T-k\mathtt{d})-a_{d}\displaystyle\int_{t}^{T-k\mathtt{d}}e^{-a_{0}(s-t)}\mathbb{E}^{{\cal
F}_{s}}p^{k-1}(s+\mathtt{d})\,ds\\\ \\\
+\sigma_{1}\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}\mathbb{E}^{{\cal
F}_{s}}q^{k-1,2}(s+\mathtt{d})\,ds+\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}q^{k,1}(s)dW^{1}_{s}\\\
\\\ +\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}q^{k,2}(s)dW^{2}_{s},\quad
t\in[(T-(k+1)\mathtt{d})\vee
0,(T-k\mathtt{d})],\,k:1,\dots,\Big{[}\frac{T}{\mathtt{d}}\Big{]}\\\
\end{array}\right.$ (4.2)

Where $L_{1}$ and $L_{2}$ are given by:

$\mathbb{E}^{{\cal
F}_{t}}(r_{x}(X(T)))=\mathbb{E}(r_{x}(X(T)))+\int_{t}^{T}L_{1}(s)\,dW^{1}(s)+\int_{t}^{T}L_{2}(s)\,dW^{1}(s).$
(4.3)

and $K_{1}$ and $K_{2}$ are given by:

$\mathbb{E}^{{\cal
F}_{t}}(l_{x}(s))=\mathbb{E}(l_{x}(s))+\int_{T-\mathtt{d}}^{t}K_{1}(s,\tau)\,dW^{1}(\tau)+\int_{T-\mathtt{d}}^{t}K_{2}(s,\tau)\,dW^{2}(\tau).$
(4.4)

Proof. Notice that, to avoid trivialities, we are taking $\mathtt{d}<T$.
Let us consider the first order adjoint equation (4.1), (, , for
$t\in(T-\mathtt{d},T]$: in view of the fact that $p(t)=q^{i}(t)=0,\,i=1,2$ for
a.e. $t\in[T,T+\mathtt{d}]$, for $t\in[T-\mathtt{d},T]$ it can be rewritten as
a linear BSDE

$\left\\{\begin{array}[]{l}p(t)=r_{x}(x(T))+\displaystyle\int_{t}^{T}l_{x}\left(x(s)\right)\,ds-
a_{0}\displaystyle\int_{t}^{T}p(s)\,ds+\displaystyle\int_{t}^{T}q^{1}(s)dW^{1}_{s}\\\
\\\ \qquad+\displaystyle\int_{t}^{T}q^{2}(s)dW^{2}_{s}\\\ \\\
p(T-\tau)=0,\;\text{for all }\tau\in[-\mathtt{d},0),\\\ \\\
q^{1}(T-\tau)=q^{2}(T-\tau)=0\quad\text{a.e.
}\tau\in[-\mathtt{d},0),\end{array}\right.$ (4.5)

and its solution for $t\in[T-\mathtt{d},T],$ is given by

$\displaystyle p^{0}(t)$
$\displaystyle=e^{-a_{0}(T-t)}r_{x}(x(T))+\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}l_{x}(x(s))\,ds+\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}q^{0,1}(s)dW^{1}_{s}$
$\displaystyle+\displaystyle\int_{t}^{T}e^{-a_{0}(s-t)}{q^{0,2}}(s)dW^{2}_{s}$
(4.6)

Formula (4.2) is just the rearrangement of the variation of constant formula,
while (4.3) and (4.4) is an application of the Martingale Representation
Theorem, see also [16].

Using (3.2), (3.2) and (4.1), we deduce the first duality relation:

######  Proposition 4.2.

Assume 2.1 and 2.2 then:

$\displaystyle\mathbb{E}p(T)y^{\varepsilon}(T))=\mathbb{E}r_{x}(x(T))y^{\varepsilon}(T)=$
$\displaystyle-\mathbb{E}\displaystyle\int_{0}^{T}[l_{x}(x(s))y^{\varepsilon}(s)+q^{2}(s)\delta
u(s-\mathtt{d})]\,ds$ (4.7)
$\displaystyle\mathbb{E}p(T)z^{\varepsilon}(T))=\mathbb{E}r_{x}(x(T))z^{\varepsilon}(T)=-\mathbb{E}\displaystyle\int_{0}^{T}l_{x}(x(s))z^{\varepsilon}(s)\,ds+\mathbb{E}\displaystyle\int_{0}^{T}p(s)\delta
u(s)\,ds$ (4.8)

Moreover the cost can be written as follows:

$\displaystyle J(u)-J(u^{\varepsilon})=\mathbb{E}\int_{0}^{T}\left(-\delta
l(t)+b_{0}p(t)\delta u(t)-\sigma_{2}q^{2}(t)\delta u(t-\mathtt{d})\right)\,dt$
(4.9)
$\displaystyle+\frac{1}{2}\mathbb{E}\,r_{xx}(x(T))(y^{\varepsilon}(T))^{2}+\frac{1}{2}\mathbb{E}\int_{0}^{T}l_{xx}(t,x(t))(y^{\varepsilon}(t))^{2}\,dt$

Proof. See [9, Propositions 4.1 and 4.2].

###  4.2 Second order duality relation

Following [9, Theorem 4.9] we introduce the process $P$, called $P_{0,0}$ in
[9], that in this simplified case can be explicitely defined by means of a
auxiliary process $\widetilde{y}^{s,1}$ (here $s$ indicates the initial time
and $1$ is the initial condition) that solves the following equation

$\left\\{\begin{array}[]{ll}d{\widetilde{y}}^{s,1}(t)=(\displaystyle-
a_{0}{\widetilde{y}}^{s,1}(t)-a_{d}\widetilde{y}^{s,1}(t-\mathtt{d}))\,dt+\displaystyle\sigma_{1}\widetilde{y}^{s,1}(t)dW(t),\quad
t\in[s,T],\\\ \\\ {\widetilde{y}}^{s,1}(s)=1,\quad
y(\tau)=0,\quad-\mathtt{d}\leq\tau<s.\end{array}\right.$ (4.10)

By means of the solution process $\widetilde{y}^{s,1}\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}))$, the process $P$ can be now
defined by

$\displaystyle
P(s)=\mathbb{E}^{\mathcal{F}_{s}}r_{xx}(x(T))\widetilde{y}^{s,1}(T)^{2}+\mathbb{E}^{\mathcal{F}_{s}}\displaystyle\int_{s}^{T}l_{xx}(t)\widetilde{y}^{s,1}(t)^{2}\,dt.$
(4.11)

Also in this case, the solution $\widetilde{y}^{1,s}$ of equation (4.10) can
be explicitly defined by recursion:

######  Proposition 4.3.

Assume Hypotheses 2.1 and 2.2. Define

$\widetilde{y}^{k}(t):=\widetilde{y}^{s,1}_{|_{[(s+k\mathtt{d}),(s+(k+1)\mathtt{d})\wedge
T]}}(t),\text{ for every }k:0,\dots,\Bigg{[}\frac{T}{\mathtt{d}}\Bigg{]}$

Then $\widetilde{y}^{k}$ solves

$\left\\{\begin{array}[]{ll}\widetilde{y}^{0}(t)=e^{(a_{o}-\frac{\sigma_{1}^{2}}{2}(t-s)+\sigma_{1}(W^{1}(t)-W^{1}(s))}\quad
t\in[s,(s+\mathtt{d})\wedge T]\\\ \\\
\widetilde{y}^{k}(t)=e^{-a_{0}(t-(s+k\mathtt{d}))}\widetilde{y}^{k-1}(s+k\mathtt{d})-a_{d}\displaystyle\int^{t}_{(s+k\mathtt{d})}e^{-a_{0}(t-r)}\widetilde{y}^{k-1}(r-\mathtt{d})\,ds\\\
+\sigma_{1}\displaystyle\int^{t}_{(s+k\mathtt{d})}e^{a_{0}(t-r)}\widetilde{y}^{k}(s)dW^{1}_{s}\quad
t\in[(s+k\mathtt{d}),(s+(k+1)\mathtt{d})\wedge
T],\,k=1,\dots,\Big{[}\frac{T}{\mathtt{d}}\Big{]}\end{array}\right.$ (4.12)

Proof. Thanks to the initial condition in (4.10), we have that for every
$t\in[s,s+\mathtt{d}\wedge T]$:

$\left\\{\begin{array}[]{l}d\widetilde{y}^{s,1}(t)=-a_{0}\widetilde{y}^{s,1}(t)\,dt+\sigma_{1}\widetilde{y}^{s,1}(t)dW^{1}_{t}\\\
\widetilde{y}^{s,1}(s)=1,\end{array}\right.$ (4.13)

Therefore
$\widetilde{y}^{0}(t)=\widetilde{y}^{s,1}_{|_{[s,(s+\mathtt{d})\wedge
T]}}(t)=e^{(a_{o}-\frac{\sigma_{1}^{2}}{2}(t-s)+\sigma_{1}(W^{1}(t)-W^{1}(s))}$.
Then the general case for $t>s+\mathtt{d}$ follows by the application of the
constant variation formula on any interval
$[(s+k\mathtt{d}),(s+(k+1)\mathtt{d})\wedge T]$.

The expansion of the cost then becomes:

######  Proposition 4.4.

Assume 2.1 and 2.2. Then following expansion holds true:

$\displaystyle J(u)-J(u^{\varepsilon})=\mathbb{E}\int_{0}^{T}\left(-\delta
c(s)+b_{0}p(s)\delta u(s)-\sigma_{2}q^{2}(s)\delta
u(s-\mathtt{d})\right)(t)\,ds$ (4.14)
$\displaystyle+\frac{1}{2}\mathbb{E}\int_{0}^{T}\sigma_{2}^{2}(\delta
u(s-\mathtt{d}))^{2}P(s)\,ds+o(\varepsilon)$

Proof. The proof is in [9] [Proposition 4.9, Theorem 4.10, Theorem 4.11] and
is a consequence of an approximation procedure based on the regularization of
the dirac measure $\delta_{\mathtt{d}}$ and the well known reformulation of
the delayed problem in infinite dimensions.

###  4.3 The Stochastic Maximum principle

We are now in the position to state our main result.

######  Theorem 4.5.

Under assumptions 2.1 and 2.2, any optimal couple $(x,u)$ satisfies the
following variational inequality :

$b_{0}(u(t)-v)p(t)-\sigma_{2}(u(t)-v)\mathbb{E}^{{\cal
F}_{t}}q^{2}(t+\mathtt{d})-(c(u(t))-c(v))\\\ \\\
+\frac{1}{2}\sigma_{2}(u(t)-v)^{2}\mathbb{E}^{{\cal F}_{t}}P(t+\mathtt{d})\leq
0,{\qquad\forall\,v\in U,\ a.e.\,\mathbb{P}\text{-a.s.}}$ (4.15)

Where $(p,q)\in L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}))\times
L^{2}_{\mathcal{F}}(\Omega\times[0,T];\mathbb{R}^{2})$, is the solution of the
first order adjoint equation (4.1), and $P$ is the second adjoint process
given by (4.11).

Proof. Let $t\in[0,T]$, and $V_{\varepsilon}=[t,t+\varepsilon]$, then

$\displaystyle J(u)-J(u^{\varepsilon})=\mathbb{E}\int_{0}^{T}\left(-\delta
c(s)+b_{0}p(s)\delta u(s)-\sigma_{2}q^{2}(s)\delta
u(s-\mathtt{d})\right)(t)\,ds$ (4.16)
$\displaystyle+\frac{1}{2}\mathbb{E}\int_{0}^{T}\sigma_{2}^{2}(\delta
u(s-\mathtt{d}))^{2}P(s)\,ds=$
$\displaystyle\mathbb{E}\int_{0}^{T}\left(-[(c(u(s))-v)+b_{0}p(s)(u(s)-v)]I_{V_{\varepsilon}}(s)-\sigma_{2}q^{2}(s)(u(s-\mathtt{d})-v)I_{V_{\varepsilon}}(s-\mathtt{d})\right)\,ds$
$\displaystyle+\frac{1}{2}\mathbb{E}\int_{0}^{T}\sigma_{2}^{2}(u(s-\mathtt{d})-v)^{2}P(s)I_{V_{\varepsilon}}(s-\mathtt{d})\,ds=$
$\displaystyle-\mathbb{E}\int_{t}^{t+\varepsilon}\left((c(u(s))-v)I_{V_{\varepsilon}}(s)-b_{0}p(s)(u(s)-v)\right)\,ds$
$\displaystyle-\mathbb{E}\int_{t}^{t+\varepsilon}\mathbb{E}^{{\cal
F}_{t}}\sigma_{2}q^{2}(s^{\prime}+\mathtt{d})(u(s^{\prime})-v)\,ds+\frac{1}{2}\mathbb{E}\int_{t}^{t+\varepsilon}\mathbb{E}^{{\cal
F}_{t}}\sigma_{2}^{2}(u(s^{\prime})-v)^{2}P(s^{\prime}+\mathtt{d})]\,ds^{\prime}.$

Letting $\varepsilon$ tends to $0$, by standard arguments we deduce (4.15).

Acknowledgements: The authors are grateful to the GNAMPA (Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni) for the financial
support.

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