# The discrete nonlinear Schrödinger equation with linear gain and nonlinear
loss: the infinite lattice with nonzero boundary conditions and its finite
dimensional approximations

G. Fotopoulos<EMAIL_ADDRESS>Academic Support Department,
Abu Dhabi Polytechnic, P.O. Box 111499, Abu Dhabi, UAE  Department of
Mathematics, Xiamen University Malaysia, 43900, Selangor, Malaysia  N. I.
Karachalios<EMAIL_ADDRESS>Department of Mathematics, University of Thessaly,
Lamia 35100, Greece  V. Koukouloyannis<EMAIL_ADDRESS>Department of
Mathematics, University of the Aegean, Karlovasi, 83200 Samos, Greece  P.
Kyriazopoulos<EMAIL_ADDRESS>Department of Mathematics,
University of Thessaly, Lamia 35100, Greece  K. Vetas<EMAIL_ADDRESS>Department of Mathematics, University of Thessaly, Lamia 35100, Greece

###### Abstract

The study of nonlinear Schrödinger-type equations with nonzero boundary
conditions define challenging problems both for the continuous (partial
differential equation) or the discrete (lattice) counterparts. They are
associated with fascinating dynamics emerging by the ubiquitous phenomenon of
modulation instability. In this work, we consider the discrete nonlinear
Schrödinger equation with linear gain and nonlinear loss. For the infinite
lattice supplemented with nonzero boundary conditions which describe solutions
decaying on the top of a finite background, we give a rigorous proof that for
the corresponding initial-boundary value problem, solutions exist for any
initial condition, if and only if, the amplitude of the background has a
precise value $A_{*}$ defined by the gain-loss parameters. We argue that this
essential property of this infinite lattice can’t be captured by finite
lattice approximations of the problem. Commonly, such approximations are
defined by lattices with periodic boundary conditions or as it is shown
herein, by a modified problem closed with Dirichlet boundary conditions. For
the finite dimensional dynamical system defined by the periodic lattice, the
dynamics for all initial conditions are captured by a global attractor.
Analytical arguments corroborated by numerical simulations show that the
global attractor is trivial, defined by a plane wave of amplitude $A_{*}$.
Thus, any instability effects or localized phenomena simulated by the finite
system can be only transient prior the convergence to this trivial attractor.
Aiming to simulate the dynamics of the infinite lattice as accurately as
possible, we study the dynamics of localized initial conditions on the
constant background and investigate the potential impact of the global
asymptotic stability of the background with amplitude $A_{*}$ in the long-time
evolution of the system.

Dissipative discrete nonlinear Schrödinger systems, Ablowitz–Ladik lattice,
non-zero boundary conditions, modulation instability

##  I Introduction

In the present paper, we continue our studies initiated in [1] on the
dynamical behavior of the solutions of the following discrete nonlinear
Schrödinger [2] (DNLS)-type equation

$\mathrm{i}\dot{u}_{n}+k(u_{n+1}-2u_{n}+u_{n-1})+|u_{n}|^{2}u_{n}=\mathrm{i}\gamma
u_{n}+\mathrm{i}\delta|u_{n}|^{2}u_{n}.$ (1.1)

Eq. (1.1) is a fundamental nonlinear lattice model incorporating linear and
nonlinear gain/loss effects. The parameter $\gamma$ describes linear loss
($\gamma<0$) [or gain ($\gamma>0$)], while $\delta$ describes nonlinear loss
($\delta<0$) [or gain ($\delta>0$)]. The presence of these effects is of
crucial importance, particularly in the context of nonlinear optics, where the
model (1.1) may describe the evolution of localized modes in optical
waveguides, see [3, 4, 5, 6] (and the references therein). In this context,
$\gamma$ describes a linear absorption ($\gamma<0$) [or linear amplification
($\gamma>0$)], while $\delta$ stands for nonlinear amplification ($\delta>0$)
[or gain saturation ($\delta<0$)]. The importance of these effects in various
discrete and continuous set-ups has been also highlighted in [7, 8, 9, 10, 11,
12] as they may have a prominent role in the potential stability/instability
of the localized modes.

In [1] we studied the dynamics of the model (1.1) in the gain/loss regimes for
$\gamma$ and $\delta$, where collapse in finite-time (or blow-up) is
manifested. We identified therein, distinct types of collapse and estimates
for the blow-up time which in various cases proved to be sharp. We remark that
the potential destabilization by finite-time collapse has been proved to be a
major characteristic of the dynamical behavior of DNLS models incorporating
gain/loss effects. [5, 6].

Herein, we will focus on the study of the dynamics in the case of linear loss
$\gamma>0$ and nonlinear gain $\delta<0$. While the loss/loss regime
$\gamma<0$, $\delta<0$ is yet of physical significance, it is characterized by
the decay of solutions for all initial data. Therefore, our primary concern
will be the former regime which is characterized by non-trivial long-time
asymptotic behavior.

We will consider equation (1.1) both in infinite and finite and lattices. The
first case concerns the system supplemented with nonzero boundary conditions
at infinity

$\lim_{|n|\rightarrow\infty}u_{n}(t)=A\exp(\mathrm{i}A^{2}t),\;\;A>0,$ (1.2)

where $A$ is the amplitude of the constant background. This problem is
drastically different from the case of zero boundary conditions and is
associated with the emergence of fascinating dynamics and a wide class of
localized phenomena which may occur on the top of the finite background $A$,
due to the ubiquitous phenomenon of modulation instability (MI).

The second case concerns the system supplemented with periodic boundary
conditions. For instance, we will consider an arbitrary number of $N+1$
oscillators which are placed equidistantly at the interval $\Omega=[-L,L]$ of
length $2L.$ We denote $k=1/h^{2}$ the discretization parameter, where
$h=2L/N$ is the lattice spacing. The spatial coordinate of the oscillators is
given then by $x_{n}=-L+nh$, $n=0,1,2,\ldots,N$. Then we supplement Eq. (1.1)
with periodic boundary conditions

$u_{n}=u_{n+N}$ (1.3)

and initial conditions $u_{n}(0)=u_{n,0}\in{\ell}^{p}_{\mathrm{per}}$, the
space of periodic sequences (see [1] and below). We remark that apart of the
physical significance of the periodic boundary conditions (1.3), the periodic
problem serves also as a finite lattice approximation of the infinite lattice
problem in the case of the nonzero boundary conditions (1.2) (but also for the
vanishing boundary conditions $A=0$) when $L$ is sufficiently large.

However, regarding even its local in time solvability, the problem (1.1)-(1.2)
is essentially different than its periodic counterpart. We prove that the
problem (1.1)-(1.2) admits a unique, at least a local in time solution if and
only if $A=A_{*}:=\sqrt{-\gamma/\delta}$. In other words, spatially localized
solutions on the top of the finite background $A>0$ exist if and only if $A$
is fixed to the “critical value” $A_{*}$. Apart of its independent interest,
this result raises a serious warning concerning the approximations of the
infinite lattice problem by the finite lattice, as the periodic one: when $L$
is sufficiently large one may simulate the dynamics of the problem with non-
zero boundary conditions (1.2) for arbitrary $A>0$, by triggering initial
conditions of the form

$\displaystyle\lim_{|n|\rightarrow\infty}u_{n}(0)=A,$ (1.4)

with a sufficiently fast rate of decay of $u_{n}(0)$ on $A$, in order to make
the error induced by the periodic boundary conditions negligible since the
initial data and the corresponding solutions will satisfy them only
asymptotically. It is evident that the numerical solutions which can be
produced for an arbitrary $A$, do not capture the crucial restriction for the
solvability of the infinite lattice problem stated above.

In the light of the above result for the infinite lattice, the periodic
problem (1.1)-(1.3) deserves much attention. It is shown that the relevant
finite dimensional dynamical system posseses a global attractor and its
spatially averaged $l^{2}$-norm is uniformly bounded by $A_{*}$. In this
paper, to the best of our knowledge, we provide a novel argument that
characterizes the global attractor. Specifically, we establish that the global
attractor is a time-periodic orbit defined by the unique non-trivial plane
wave solution with a constant amplitude denoted as $A_{*}$, a frequency of
$\tilde{\omega}$, and a wave number of $q$. This solution satisfies the
dispersion relation given by:

$\displaystyle\tilde{\omega}=4k\sin^{2}\Big{(}\frac{hq}{2}\Big{)}-A_{*}^{2}.$

To implement this argument, we first demonstrate that the attractor represents
the $\boldsymbol{\omega}$-limit set of all initial conditions stemming from
solutions in the form of:

$u_{w}(x_{n},t)=A(t)\exp\left[\mathrm{i}(qx_{n}-\Omega(t)\right],\quad
t\in\mathbb{R}.$

What sets our argument apart is that we prove convergence not only in terms of
amplitudes, where $A(t)\rightarrow A_{*}$ as $t\rightarrow\infty$ but also
through a detailed analysis of the exact frequency function $\Omega(t)$. We
determine its behavior in relation to its slant asymptotes, ultimately
establishing that $\Omega(t)\rightarrow\Omega_{\infty}(t):=\tilde{\omega}t$.
We extend this result to cover the remaining initial conditions through a
finite-dimensional representation using their discrete Fourier series.
Secondly, we conduct a modulational instability (MI) analysis for the plane
wave attractor, demonstrating its potential for modulational instability. By
combining the convergence results with the MI analysis, we establish that the
MI is transient.

To illustrate the convergence and transient MI effects, we perform numerical
simulations for two representative scenarios: plane-wave initial conditions
and localized ones superimposed on a finite background. These numerical
findings, which align closely with the theoretical results, also showcase the
evolution of the initial conditions’ spectrum. Of particular interest is the
case of unstable wave numbers, where the system ultimately selects a node from
within the stability band upon reaching convergence to the attractor.

The analysis presented above, particularly in the context of numerical
simulations for the periodic lattice, underscores that any localized phenomena
arising due to modulational instability (MI) are inherently transient.
However, within a finite time frame before reaching ultimate equilibrium,
these localized phenomena can be notably intriguing.

For localized initial conditions superimposed on top of $A=A_{*}$, we observe
the emergence of waveforms similar to Peregrine solitons as initial events.
Subsequently, at later time intervals, spatiotemporal patterns develop due to
the modulation instability of the underlying background. These patterns bear a
remarkable resemblance to those observed in the dynamics of localized initial
data in the integrable Ablowitz-Ladik lattice [13]. They exhibit the universal
structure described in [14] during the nonlinear stage of modulational
instability. This structure consists of two outer, quiescent sectors
corresponding to the supporting background, separated by a wedge-shaped
central region characterized by modulated periodic oscillations. The proximity
of these dynamics is noteworthy, considering that DNLS (1.1) represents a
system that has moved “two steps forward” in breaking the integrability
barrier defined by the AL system. DNLS (1.1) is a dissipative perturbation of
the non-integrable Hamiltonian DNLS. The stability of the quiescent sectors
can be fully justified by the analytical results. Notably, $A=A_{*}$
represents the amplitude of the supporting background, which, as highlighted
earlier, is the only amplitude supporting localized solutions in the infinite
lattice. Moreover, for the periodic lattice used in the numerical simulations,
$A=A_{*}$ corresponds to the amplitude of the globally asymptotically stable
background. In the case where $A\neq A_{*}$ where solutions for the infinite
lattice do not exist, the numerical simulations for the periodic lattice show
that the MI pattern persists on the top, this time of the evolving background.
This continues until the amplitude reaches its limiting value $A_{*}$, at
which point the entire MI structure disappears and is replaced by the
attracting plane wave.

Motivated by the above results and our current investigations on the closeness
between integrable and non-integrable lattice systems [15, 16, 17], we examine
the potential proximity between the solutions of the Ablowitz-Ladik lattice
and the DNLS system (1.1) when both systems are supplemented with periodic
boundary conditions. While for bounded solutions the distance is always
uniformly bounded, in the same fashion as in [15, 16, 17], we prove that for
finite times this distance grows at most linearly. Since the first events
reminiscent to rogue waves occur at small time intervals, this growth is
proved to be effective in measuring the distance between the rogue wave events
emerging in the dynamics of (1.1) and those defined by the analytical discrete
Peregrine soliton solution of the Ablowitz-Ladik lattice.

The paper is structured as follows. In Section II, we provide the proof of the
existence of solutions for the infinite lattice with nonzero boundary
conditions. For the definition and properties of the sequence spaces that
define the functional analytic set-up of the problem, we refer to [1] and the
references therein. In Section III, we present the results concerning the
dynamics of the periodic lattice. Sections IV and V are devoted to the
numerical results. In particular, in Section IV, we present the results of the
numerical studies concerning the transient modulation instability of the plane
wave attractor. In Section V, we present the numerical results for localized
initial conditions, distinguishing between the case $A=A_{*}$, the only case
permitted for the infinite lattice, and the case $A\neq A_{*}$, which can be
relevant in the periodic lattice. In this section, we also provide the proof
of the estimates for the distance between the solutions of the DNLS (1.1) and
the Ablowitz-Ladik periodic lattices, along with the corresponding numerical
investigations. We conclude with Section VI, which summarizes our findings.

##  II The case of the infinite lattice with nonzero boundary conditions

Solutions of the initial value problem for the infinite lattice (1.1) with the
nonzero boundary conditions and $\gamma>0$, $\delta<0$ exist if and only if
$A=A_{*}$. This is the main result of this section stated in the following
theorem.

######  Theorem II.1.

Consider the DNLS system (1.1) with $\gamma>0,\delta<0$ in the infinite
lattice supplemented with the nonzero boundary conditions (1.2). We also
assume that the initial condition satisfies (1.4). Then the system has a
solution if and only if $A=A_{*}$. The solution is unique.

###### Proof.

First, we apply to the DNLS (1.1) the gauge transformation

$u_{n}(t)=\psi_{n}(t)\exp(\mathrm{i}A^{2}t),$ (2.5)

yielding the system in the form

$\mathrm{i}\dot{\psi}_{n}+k(\psi_{n+1}-2\psi_{n}+\psi_{n-1})-A^{2}\psi_{n}+|\psi_{n}|^{2}\psi_{n}=\mathrm{i}\gamma\psi_{n}+\mathrm{i}\delta|\psi_{n}|^{2}\psi_{n},$
(2.6)

which satisfies the boundary conditions

$\lim_{|n|\rightarrow\infty}\psi_{n}(t)=A.$ (2.7)

Next, we apply to the system (2.6), the change of variables

$U_{n}={\psi}_{n}-A,\,\,\,n\in{\mathbb{Z}},$ (2.8)

The system for the new variable $U_{n}$ reads as

$\displaystyle\mathrm{i}\dot{U}_{n}+k(U_{n+1}-2U_{n}+U_{n-1})-A^{2}(U_{n}+A)$
$\displaystyle+|U_{n}+A|^{2}(U_{n}+A)$
$\displaystyle=\mathrm{i}\gamma(U_{n}+A)+\mathrm{i}\delta|U_{n}+A|^{2}(U_{n}+A),\;\;$
(2.9)

where $U_{n}$, due to the non-zero boundary conditions (2.7), should satisfy
the vanishing boundary conditions at infinity

$\lim_{|n|\rightarrow\infty}U_{n}(t)=0.$ (2.10)

Thus, the system should be examined for its solvability in the sequence spaces
$\ell^{p}$, for any $p\geq 1$. Let us recall the crucial embedding

$\ell^{q}\subset\ell^{p},\quad\|u\|_{\ell^{p}}\leq\|u\|_{\ell^{q}},\quad 1\leq
q\leq p\leq\infty.$ (2.11)

For the local existence of solutions, we may apply the generalized Picard-
Lindelöf Theorem [18, Theorem 3.A, pg. 78]. The discrete Laplacian
$\Delta_{d}U_{n}=k(U_{n+1}-2U_{n}+U_{n-1})$ is a bounded linear operator
$\Delta_{d}:\ell^{p}\rightarrow\ell^{p}$, for any $p\geq 1$, that is, there
exists a constant $C>0$, such that

$\|\Delta_{d}U\|_{\ell^{p}}\leq C\|U\|_{\ell^{p}},\quad\mbox{for all}\quad
U\in\ell^{p}.$ (2.12)

Next, we consider the nonlinear operators

$\displaystyle F_{1}(U_{n})$
$\displaystyle=-A^{2}(U_{n}+A)+|U_{n}+A|^{2}(U_{n}+A),$ $\displaystyle
F_{2}(U_{n})$
$\displaystyle=\mathrm{i}\gamma(U_{n}+A)+\mathrm{i}\delta|U_{n}+A|^{2}(U_{n}+A).$

For the operator $F_{1}$, we observe that

$\displaystyle F_{1}(U_{n})$
$\displaystyle=-A^{2}U_{n}-A^{3}+(|U_{n}|^{2}+AU_{n}+A\overline{U}_{n}+A^{2})(U_{n}+A)$
$\displaystyle=|U_{n}|^{2}U_{n}+A(U_{n}^{2}+2|U_{n}|^{2})+A^{2}(U_{n}+\overline{U}_{n}).$

Using the inequality

$\left||u_{n}|^{2}u_{n}-|v_{n}|^{2}v_{n}\right|\leq|v_{n}|^{2}|u_{n}-v_{n}|+|u_{n}|\left(|v_{n}|+|u_{n}|\right)|u_{n}-v_{n}|,$

and the embedding (2.11), we may show that
$F_{1}:\ell^{p}\rightarrow\ell^{p}$, for any $p\geq 1$, is well defined and is
Lipschitz continuous on bounded sets of $\ell^{p}$. That is, for any $U,V\in
B_{R}$, with $B_{R}$ a closed ball of $\ell^{p}$ of center $0$ and radius $R$,
we derive the existence of constants $K_{1}(A,R)$ and $K_{1,1}(A,R)$, such
that

$\displaystyle\|F_{1}(U)\|_{\ell^{p}}$ $\displaystyle\leq K_{1}(A,R),$ (2.13)
$\displaystyle\|F_{1}(U)-F_{1}(V)\|_{\ell^{p}}$ $\displaystyle\leq
K_{1,1}(A,R)\|U-V\|_{\ell^{p}}.$ (2.14)

Similarly, for the operator $F_{2}$, we have that

$F_{2}(U_{n})=\mathrm{i}\gamma U_{n}+\mathrm{i}\gamma A+\mathrm{i}\delta
A^{3}+\mathrm{i}\delta\left\\{|U_{n}|^{2}U_{n}+A(U_{n}^{2}+2|U_{n}|^{2})+A^{2}(2U_{n}+\overline{U}_{n}\right\\}.$
(2.15)

Assume that $A=A_{*}$. Then, $\mathrm{i}\gamma A+\mathrm{i}\delta A^{3}=0$ and
we can show that the operator $F_{2}$ has the properties (2.13)-(2.14). We may
proceed by recasting the system (II) in the form of the abstract evolution
equation

$\displaystyle\dot{U}=\mathcal{L}(U),\quad\mathcal{L}(U)=\mathrm{i}[-\Delta_{d}(U)-F_{1}(U)+F_{2}(U)],$
(2.16)

and we will implement the aforementioned Picard-Lindelöf Theorem on its
equivalent integral formula

$U(t)=U(0)+\int_{0}^{t}\mathcal{L}[U(s)]ds,$ (2.17)

to establish the existence of a unique solution $U\in l^{p}$, for any $p\geq
1$, when the initial condition $U(0)\in\ell^{p}$. This is the case, since the
operator $\mathcal{L}:\ell^{p}\rightarrow\ell^{p}$ has the properties
(2.13)-(2.14) and the conditions of the Picard-Lindelöf Theorem are satisfied.
The solution has the following properties: There exists some $T^{*}(U(0))>0$
such that the corresponding initial value problem for (II)-(2.10), has a
unique solution which is continuously differentiable with respect to time,
i.e., $U\in C^{1}([0,T],\ell^{p})$ for all $0<T<T^{*}(U(0))$. In addition, the
following alternatives hold: Either $T^{*}(U(0))=\infty$ (global existence) or
$T^{*}(U(0))<\infty$ and $\lim_{t\uparrow
T^{*}(U(0))}\|U(t)\|_{\ell^{p}}=\infty$ (collapse). Furthermore the solution
$U$ depends continuously on the initial condition $U(0)\in\ell^{p}$, with
respect to the norm of $C([0,T],\ell^{p})$. Using the change of variables
(2.5) and (2.8), we establish the existence of a unique solution of the
original problem (1.1)-(1.2),

$u_{n}(t)=(U_{n}(t)+A_{*})\exp(\mathrm{i}A_{*}^{2}t).$ (2.18)

For the reverse direction of the proof, we have to show that if a solution of
the initial boundary value (1.1)-(1.2)-(1.4) problem exists, then $A=A_{*}$.
We argue by contradicion: Assume that a solution of the problem
(1.1)-(1.2)-(1.4) exists for all $t\in[0,T]$, for some $T>0$ and that $A\neq
A_{*}$. Then, through the same transformations (2.5)-(2.8), a solution $U$ of
the problem (II)-(2.7) should exist for all $t\in[0,T]$, and $U\in
C^{1}([0,T],\ell^{p})$, which means that $\|\dot{U}(t)\|_{\ell^{p}}<\infty$,
for all $t\in[0,T]$. Hence, due to (2.16),
$\|\mathcal{L}(U)\|_{\ell^{p}}<\infty$ for all $t\in[0,T]$. This can’t be
valid because if $A\neq A_{*}$, the operator $F_{2}$ is not bounded on bounded
sets of $\ell^{p}$ as it follows from (2.15). Indeed, for $U$ the solution of
the problem which satisfies $\|U(t)\|_{\ell^{p}}\leq R$, for all $t\in[0,T]$,
the boundedness of $F_{2}$ in bounded sets of $\ell^{p}$ would imply the
counterpart of (2.13),

$\|F_{2}(U(t))\|_{\ell^{p}}\leq K_{2}(A,R),\quad\mbox{for all}\quad
t\in[0,T].$ (2.19)

Since from (2.15),

$\mathrm{i}\gamma A+\mathrm{i}\delta A^{3}=-F_{2}(U_{n})+\mathrm{i}\gamma
U_{n}+\mathrm{i}\delta\left\\{|U_{n}|^{2}U_{n}+A(U_{n}^{2}+2|U_{n}|^{2})+A^{2}(2U_{n}+\overline{U}_{n}\right\\},$
(2.20)

(2.19) and the fact that the rest of the terms of the right-hand side of
(2.20) are bounded in $\ell^{p}$ for all $t\in[0,T]$, would imply that the
left hand side of (2.20) is summable, and this is impossible. Therefore, the
initial-boundary value problem (1.1)-(1.2)-(1.4) has a solution if and only if
$A=A_{*}$. ∎

###  II.1 Remarks on Theorem II.1 for finite lattice approximations

The finite lattice approximations are defined either in the spaces of periodic
sequences with period $N$, denoted by

${\ell}^{p}_{per}:=\bigg{\\{}U=(U_{n})_{n\in\mathbb{Z}}\in\mathbb{R}:\quad
U_{n}=U_{n+N},\quad\|U\|_{\ell^{p}_{\mathrm{per}}}:=\bigg{(}h\sum_{n=0}^{N-1}|U_{n}|^{p}\bigg{)}^{\frac{1}{p}}<\infty\bigg{\\}},\quad
1\leq p\leq\infty,$

or in the case of the Dirichlet boundary conditions, in the finite dimensional
subspaces of $\ell^{2}$,

${\ell}^{p}_{0}:=\bigg{\\{}U=(U_{n})_{n\in\mathbb{Z}}\in\mathbb{R}:\quad
U_{0}=U_{N}=0,\quad\|U\|_{\ell^{p}_{0}}:=\bigg{(}h\sum_{n=1}^{N-1}|U_{n}|^{p}\bigg{)}^{\frac{1}{p}}<\infty\bigg{\\}},\quad
1\leq p\leq\infty.$

If $\mathcal{X}^{p}$ is either one of the above spaces, their norms are
equivalent, according to the inequality

$\|U\|_{\mathcal{X}^{q}}\leq\|U\|_{\mathcal{X}^{p}}\leq
N^{\frac{(q-p)}{qp}}\|U\|_{\mathcal{X}^{q}},\quad 1\leq p\leq q<\infty.$
(2.21)

In this finite dimensional set-up, it is evident from the definition of the
operator $F_{2}$ in (2.15) or (2.20) that Theorem (II.1) is valid for any
$A>0$ since the operator $F_{2}:\mathcal{X}^{p}\rightarrow\mathcal{X}^{p}$ is
well defined and locally Lipschitz continuous for any $A>0$. Therefore, finite
lattice approximations do not capture the essential necessary and sufficient
condition on $A$ for the existence of solutions of the infinite lattice which
is proved in Theorem II.1.

##  III The case of the lattice with periodic boundary conditions

In this section we consider the case of the finite lattice (1.1) when
supplemented with the periodic boundary conditions (1.3). This finite
dimensional system, apart of its physical significance may serve as one of the
main finite lattice approximations of the problem of the infinite lattice
supplemented with the nonzero boundary conditions discussed in the previous
section. In this connection, the value of the background amplitude $A_{*}$,
which is the only one which may support localized solutions on top of the
background in the infinite lattice, chiefly governs the dynamics. The phase
space of the system is the Hilbert space $\ell^{2}_{per}$ endowed with the
scalar product

$(\phi,\psi)_{\ell^{2}_{\mathrm{per}}}=h\,\mathrm{Re}\sum_{n=0}^{N-1}\phi_{n}\overline{\psi}_{n},\quad\phi,\,\psi\in\ell^{2}_{\mathrm{per}}.$

######  Theorem III.1.

  1. 1.

(The global attractor in $\ell^{2}_{per}$). Let
$u_{n}(0)=u_{n,0}\in\ell^{2}_{per},$ and $\gamma>0$, $\delta<0.$ Then, the
solution of the initial value problem for (1.1)–(1.3) $u_{n}\in
C^{1}([0,\infty),\ell^{2}_{per})$. Furthermore, for the dynamical system

$\varphi(t,u_{n,0}):\ell^{2}_{per}\rightarrow\ell^{2}_{per},\leavevmode\nobreak\
\leavevmode\nobreak\ \varphi(t,u_{n,0})=u_{n}(t),$ (3.1)

there exists a global attractor and the averaged power $\displaystyle
P_{a}[u(t)]:=\frac{1}{N}\sum_{n=0}^{N-1}|u_{n}(t)|^{2}$ of the solutions
satisfy the estimate

$\lim\sup_{t\to\infty}P_{a}[u(t)]\leq A_{*}^{2}.$ (3.2)

  2. 2.

The system (1.1)-(1.3) admits solutions of the form

$w_{n}(t):=w(x_{n},t)=A(t)\exp\left[\mathrm{i}(qx_{n}-\Omega(t)\right],\quad
t\in\mathbb{R},$ (3.3)

where $q=K\pi/L$, $K\in\mathbb{N}$ is the constant wavenumber and the time
variable frequency function $\Omega(t)$ and amplitude $A(t)$ satisfy the
system of equations

$\displaystyle\Omega(t)$
$\displaystyle=4k\sin^{2}\Big{(}\frac{hq}{2}\Big{)}t-\Theta(t),\quad\dot{\Theta}(t)=A^{2}(t)$
(3.4) $\displaystyle\dot{A}(t)$ $\displaystyle=\gamma A(t)+\delta A^{3}(t).$
(3.5)

for all $t\in\mathbb{R}$. The solutions (3.3) have the following property
under the flow (3.1): The limit set

$\boldsymbol{\omega}(w_{n}(0))=\gamma_{*},$ (3.6)

where $\gamma_{*}$ is the time-periodic orbit defined by the only non-trivial
plane wave solution of constant amplitude $A_{*}$ with frequency
$\tilde{\omega}$ and wave number $q$ satisfying (3.4), when we set in (3.4)
$A(t)\equiv A_{*}$, that is,

$\tilde{\omega}=4k\sin^{2}\Big{(}\frac{hq}{2}\Big{)}-A_{*}^{2}.$ (3.7)

For the solutions (3.3), the limiting bound (3.2) is exact, in the sense that

$\lim_{t\to\infty}P_{a}[w_{n}(t)]=A_{*}^{2}.$ (3.8)

###### Proof.

1\. We multiply equation (1.1) by $-\mathrm{i}u_{n}$ in the
$\ell^{2}_{per}$-inner product. Using that
$\displaystyle(\Delta_{d}u,\mathrm{i}u)_{\ell^{2}_{\mathrm{per}}}=-\mathrm{i}\sum_{n=0}^{N-1}|u_{n+1}-u_{n}|^{2}$,
by keeping the real parts of the resulting equation, we derive the following
power balance law

$\frac{d}{dt}\bigg{(}h\sum_{n=0}^{N-1}|u_{n}|^{2}\bigg{)}=2\gamma\,h\sum_{n=0}^{N-1}|u_{n}|^{2}+2\delta\,h\sum_{n=0}^{N-1}|u_{n}|^{4}.$
(3.9)

Note that for $\gamma=\delta=0$ we have the conservation of the power, which
is one of the conserved quantities of the conservative DNLS .

Let $\gamma>0$ and $\delta:=-\tilde{\delta}<0$. The second term of the right-
hand side of Eq. (3.9) can be estimated as

$-2\tilde{\delta}\,h\sum_{n=0}^{N-1}|u_{n}|^{4}\leq-\frac{2\tilde{\delta}}{Nh}\bigg{(}h\sum_{n=0}^{N-1}|u_{n}|^{2}\bigg{)}^{2},$

where we used the Cauchy–Schwarz inequality

$\sum_{n=0}^{N-1}|u_{n}|^{2}\leq\sqrt{N}\bigg{(}\sum_{n=0}^{N-1}|u_{n}|^{4}\bigg{)}^{1/2}.$

Consequently, Eq. (3.9) turns to a Bernoulli’s differential inequality

$\frac{d}{dt}\bigg{(}h\sum_{n=0}^{N-1}|u_{n}|^{2}\bigg{)}\leq
2\gamma\,h\sum_{n=0}^{N-1}|u_{n}|^{2}-\frac{2\tilde{\delta}}{Nh}\bigg{(}h\sum_{n=0}^{N-1}|u_{n}|^{2}\bigg{)}^{2},$
(3.10)

and equivalently, when diving by $Nh$, the equation for $P_{a}[u(t)]$

$\frac{d}{dt}P_{a}[u]\leq 2\gamma P_{a}[u]-2\tilde{\delta}P^{2}_{a}[u].$
(3.11)

Setting $\phi=1/P_{a}[u]$ we see that $\phi$ satisfies

$\frac{d}{dt}\phi+2\gamma\phi\geq 2\tilde{\delta},$

which (with the aid of the integrating factor $\exp(2\gamma t)$) gives

$\phi(t)\geq\phi(0)\exp(-2\gamma
t)+\frac{\tilde{\delta}}{\gamma}\left[1-\exp\left(-2\gamma t\right)\right]$

and finally

$P_{a}[u(t)]\leq\frac{1}{P_{a}[u(0)]^{-1}\exp(-2\gamma
t)+\frac{\tilde{\delta}}{\gamma}\left[1-\exp\left(-2\gamma
t\right)\right]},\quad\forall t\geq 0.$ (3.12)

The latter means that $P_{a}[u(t)]$ is uniformly bounded. Letting
$t\to\infty,$ we conclude that $P_{a}[u(t)]$ satisfies the estimate (3.2).

From (3.2), we deduce that the dynamical system (3.1) has bounded orbits
$\forall t\in[0,\infty)$. Let
$\mathcal{B}:=\\{u_{n,0}\in\ell^{2}_{per}:P_{a}[u]\leq R^{2}\\}$ be an
arbitrary closed ball in $\ell^{2}_{per}$. The ball
$\mathcal{B}_{a}(0,\rho):=\\{u\in\ell^{2}_{per}:P_{a}[u]\leq\rho^{2},\;\;\;\rho^{2}>A_{*}^{2}\\}$
is an absorbing set for the dynamical system $\varphi(t,u_{n,0})$: there
exists $T^{*}(\mathcal{B},\mathcal{B}_{a})>0$ such that for any $t\geq
T^{*}(\mathcal{B},\mathcal{B}_{a})$ it holds that $P_{a}[u]\leq\rho^{2}$ and
$\varphi(t,\mathcal{B})\subset\mathcal{B}_{a}.$ Hence, we may define the
$\boldsymbol{\omega}$-limit set in $\ell^{2}_{per}$, for any bounded set
$\mathcal{B}$,

$\boldsymbol{\omega}(\mathcal{B})=\bigcap_{s\geq 0}\overline{\bigcup_{t\geq
s}\varphi(t,\mathcal{B})}.$

Since the dynamical system is finite-dimensional the above limit set defines
its compact global attractor.

2\. We start by seeking solutions of the form

$u_{n}(t)=W(t)\exp\left[\mathrm{i}(qx_{n}-\tilde{\omega}t)\right],\quad\tilde{\omega}\in\mathbb{R}.$
(3.13)

By substituting (3.13) to (1.1) we derive the equation for $W(t)$

$\mathrm{i}\dot{W}+\lambda W+|W|^{2}W=\mathrm{i}\gamma
W+\mathrm{i}\delta|W|^{2}W,\quad\lambda=\tilde{\omega}-4k\sin^{2}\Big{(}\frac{hq}{2}\Big{)}.$
(3.14)

The linear term $\lambda W$ will be removed by the gauge transformation

$W(t)=\Phi(t)\exp(\mathrm{i}\lambda t),$ (3.15)

applied to (3.14). Then, the equation for $\Phi(t)$ is

$\mathrm{i}\dot{\Phi}+|\Phi|\Phi=\mathrm{i}\gamma\Phi+\mathrm{i}\delta|\Phi|^{2}\Phi.$
(3.16)

The last transformation we will apply is

$\Phi(t)=A(t)\exp[\mathrm{i}\Theta(t)],\quad A,\Theta\in C^{1}(\mathbb{R}),$
(3.17)

to (3.16). We derive the equation for $A(t)$,

$\mathrm{i}\dot{A}-\dot{\Theta}A+A^{3}=\mathrm{i}\gamma A+\mathrm{i}\delta
A^{3}.$ (3.18)

The choice

$\dot{\Theta}=A^{2},$ (3.19)

eliminates the second and the third term of the left-hand side of (3.18), in
order to obtain (3.5). Then returning to the starting form of solutions
(3.13), by applying the transformations (3.15) and (3.17) one after another,
we build the solution (3.3) with the system (3.4)-(3.5). Note that in this
process, the term $\tilde{\omega}t$ was eliminated resulting in equation (3.4)
for $\Omega(t)$. We could start with the ansatz
$u_{n}(t)=W(t)\exp(\mathrm{i}qx_{n})$ instead of (3.13) to build exactly the
same solution, however we started with (3.13) for notational purposes relevant
to the solution of constant amplitude $A(t)=A_{*}$ which will be discussed
below. Indeed, we will discuss the limit of the orbit (3.3) as
$t\rightarrow\infty$, by using (3.4)-(3.5).

Let $A(0)$ be the initial amplitude of the solution (3.3). Equation (3.5) has
the unique solution

$A^{2}(t)=\frac{\gamma A^{2}(0)}{(\gamma+\delta A^{2}(0))\exp(-2\gamma
t)-\delta A^{2}(0)},$ (3.20)

for which, it holds that

$\lim_{t\to\infty}A^{2}(t)=A_{*}^{2}.$ (3.21)

Next, by integrating (3.19) in the interval $[0,t]$, we get

$\Theta(t)-\Theta(0)=\int_{0}^{t}A^{2}(s)ds.$ (3.22)

The function $\Theta(t)-\Theta(0)$ has the following properties:

$\displaystyle\lim_{t\rightarrow\infty}\big{(}\Theta(t)-\Theta(0)\big{)}$
$\displaystyle=\infty,$ (3.23)
$\displaystyle\lim_{t\rightarrow+\infty}\frac{\Theta(t)-\Theta(0)}{t}$
$\displaystyle=\lim_{t\rightarrow\infty}\dot{\Theta}(t)=A_{*}^{2},$ (3.24)
$\displaystyle\lim_{t\rightarrow\infty}\big{[}\big{(}\Theta(t)-\Theta(0)\big{)}-A^{2}_{*}t\big{]}$
$\displaystyle=-\frac{1}{2\delta}\ln\Big{(}\frac{A^{2}(0)}{A_{*}^{2}}\Big{)}=:b>0.$
(3.25)

Note that when $A^{2}(0)=A_{*}^{2}$, we have $A^{2}(t)=A_{*}^{2}$ for all
$t\geq 0$, and in this case, we have that $b=0$. From (3.24) and (3.25), we
have that as $t\rightarrow\infty$, the function $\Theta(t)$ has the slant
asymptote

$\Theta_{\infty}(t)=A_{*}^{2}t+\Theta(0)+b.$ (3.26)

Without loss of generality we can select $\Theta(0)=-b$ (since we can always
adjust by the relevant phase factor). Then, in the limit $t\rightarrow\infty$,
we replace the function $\Omega(t)$ by

$\displaystyle\Omega_{\infty}(t)$
$\displaystyle=4k\sin^{2}\Big{(}\frac{hq}{2}\Big{)}t-\Theta_{\infty}(t)=4k\sin^{2}\Big{(}\frac{hq}{2}\Big{)}t-A_{*}^{2}t=\tilde{\omega}t,$
(3.27) $\displaystyle\tilde{\omega}$
$\displaystyle=4k\sin^{2}\Big{(}\frac{hq}{2}\Big{)}-A_{*}^{2}.$ (3.28)

Now, we may use (3.21) in the solution (3.3) and (3.27)-(3.28) (or
alternatively (3.18) and (3.24)), to see that as $t\rightarrow\infty$,
$\gamma_{*}$ is the orbit defined by the solution

$w_{\infty,n}(t)=A_{*}\exp\left[\mathrm{i}(qx_{n}-\Omega_{\infty}(t)\right]=A_{*}\exp\left[\mathrm{i}(qx_{n}-\tilde{\omega}t\right],$
(3.29)

where $\tilde{\omega}$ and $q$ satisfy (3.28). This is exactly the claimed
dispersion relation (3.7) for the unique plane wave of constant amplitude
$A_{*}$ (3.29), to exist. Usually we derive the dispersion relation (3.28) by
substitution of the solution (3.29) to the DNLS (1.1). Here, it is interesting
to find, that the solution (3.29) can be derived when using (3.20) for
$A(0)^{2}=A_{*}^{2}$ for its amplitude, and importantly, by the interpretation
of the function $\Theta_{\infty}(t)$ for $\beta=0$ and $\Theta(0)=0$ in
(3.26). The assertion (3.6) is proved. The proof of (3.8) follows by the
calculation of

$P_{a}[w_{n}(t)]=\frac{1}{N}\sum_{n=0}^{N-1}|w_{n}(t)|^{2}=A^{2}(t),$ (3.30)

and the limit (3.21).

∎

Remark: It is important to stress the following: The system (1.1) do not
admits solutions of the form (3.3) with constant amplitude $A$, unless this
amplitude is $A=A_{*}$. In other words, there is no other constant $A>0$ than
$A_{*}$, which satisfies both equations (3.4) and (3.5) simultaneously. On the
other hand, we may always trigger initial conditions of the form

$u_{n}(0)=A\exp(\mathrm{i}qx_{n}),\quad A\neq A_{*},$ (3.31)

and arbitrary $q$, which can be considered generically as perturbations of the
unique plane wave solution of constant amplitude. The fate of such initial
conditions will be determined in the next paragraph.

###  III.1 Consequences of Theorem III.1 with respect to a linear instability
analysis

The main consequence of Theorem II.1 is that the dynamical system defined by
the periodic lattice (1.1)-(1.3) is globally asymptotically stable for any
initial condition in $\ell^{2}_{per}$: for arbitrary initial conditions all
the orbits are converging to a state whose averaged norm satisfies the bound
(3.2). In the special case where the initial condition is defined by a
solution of the form (3.3), the attractor is trivial, defined by the only
plane wave solution which can have constant amplitude, which is $A^{*}$. This
global asymptotic stability do not excludes the emergence of instability
effects. Nevertheless, these effects can be only transient, prior the
convergence of the system to its ultimate state. We investigate the potential
of these transient instability effects in the framework of the linear
modulation instability analysis of plane waves. The motivation is that the
fate of such initial conditions is predicted by the Theorem II.1 as stated
above.

The modulational instability (MI) of plane-waves is well studied in the case
of the Hamiltonian DNLS [19]. We will perform here this analysis for the
system (1.1) with the presence of the linear gain and nonlinear loss effects.
The significance of these findings lies on the fact that they prove that the
simplest initial data of the form (3.31) have interesting dynamics prior the
convergence to the attractor, and that even this attractor can be transiently,
modulationally unstable.

To justify the above statement, we consider the perturbation of the only plane
wave solution (3.3) with constant amplitude $A(t)\equiv A_{*}$,

$u_{n}(t)=(A_{*}+b_{n}(t))\exp\left[\mathrm{i}(\theta_{n}(t)+\psi_{n}(t))\right],$
(3.32)

where $\theta_{n}(t)=qx_{n}-\tilde{\omega}t$, $x_{n}=-L+nh,n=0,1,\ldots,N$ and
$b_{n},\psi_{n}$ are the amplitude and the angle perturbations, respectively.
Substitution of (3.32) to (1.1), yields the system

$\left\\{\begin{aligned}
-A_{*}\dot{\psi}_{n}&-kA_{*}(\psi_{n+1}-\psi_{n-1})\sin(hq)+k(\Delta_{d}b_{n})\cos(hq)+2A_{*}^{2}b_{n}=0,\\\
\dot{b}_{n}&+kA_{*}(\Delta_{d}\psi_{n})\cos(hq)+k(b_{n+1}-b_{n-1})\sin(hq)-2\delta
A_{*}^{2}b_{n}=0.\end{aligned}\right.$ (3.33)

For the derivation of (3.33), we used the dispersion relation (3.28) and that
$\gamma+\delta A_{*}^{2}=0$.

To simplify the system (3.33), we consider amplitude perturbations of the form
$b_{n}(t)=b_{0}\exp(\mathrm{i}(Qx_{n}-\Omega_{p}t))$ and phase perturbations
of the form $\psi_{n}(t)=\psi_{0}\exp(\mathrm{i}(Qx_{n}-\Omega_{p}t))$. Then
the system (3.33) becomes

$\left\\{\begin{aligned}
&\left(\Omega-2k\sin(hQ)\sin(hq)\right)A_{*}\psi_{0}\mathrm{i}+(-4k\sin^{2}(hQ/2)\cos(hq)+2A_{*}^{2})b_{0}=0,\\\
&-4kA_{*}(\sin^{2}(hQ/2)\cos(hq)\psi_{0}\mathrm{i}+\left(\Omega-2k\sin(hQ)\sin(hq)-2\delta
A_{*}^{2}\mathrm{i}\right)b_{0}=0.\end{aligned}\right.$ (3.34)

The linear homogeneous system (3.34) for $(b_{0},\psi_{0})$ has a nontivial
solution if and only if its determinant is zero. This request provides the
modulational instability formula:

$\Lambda^{2}-2\mathrm{i}\delta
A^{2}\Lambda-\Gamma\left(\Gamma-2A^{2}\right)=0,$ (3.35)

with $\Lambda=\Omega-2k\sin(hQ)\sin(hq)$ and $\Gamma=4k\sin^{2}(Q/2)\cos(hq)$.
We elaborate further on the formula (3.35), to derive the conditions for the
modulation instability of the examined plane wave solution. Since (3.35) is a
quadratic equation in $\Lambda$ with complex coefficients, it has only complex
roots. The discriminant of (3.35) is

$\Delta_{\Lambda}=4(\Gamma(\Gamma-2A_{*}^{2})-\delta^{2}A_{*}^{4}),$

and accordingly, the roots are

$\Lambda_{\pm}=\delta
A_{*}^{2}\mathrm{i}\pm\frac{\sqrt{\Delta_{\Lambda}}}{2}=\delta
A_{*}^{2}\mathrm{i}\pm\sqrt{\Gamma(\Gamma-2A_{*}^{2})-\delta^{2}A_{*}^{4}}.$
(3.36)

We are interested in the case $\text{Im}(\Omega)=\text{Im}(\Lambda_{\pm})>0$
for the exponential growth of the perturbations, and thus the emergence of
instability. If the quantity under the square root is positive, we have that

$\text{Im}(\Lambda_{\pm})=\delta<0.$

If the quantity under the square root is negative, we have that

$\text{Im}(\Lambda_{-})=\delta-\sqrt{\delta^{2}A_{*}^{4}-\Gamma(\Gamma-2A_{*}^{2})}<0.$

Thus, we are only interested for the case

$\displaystyle\text{Im}(\Lambda_{+})>0\iff$ $\displaystyle\delta
A_{*}^{2}+\sqrt{\delta^{2}A_{*}^{4}-\Gamma(\Gamma-2A_{*}^{2})}>0$
$\displaystyle\iff$ $\displaystyle\Gamma(\Gamma-2A_{*}^{2})<0.$ (3.37)
$\displaystyle\iff$ $\displaystyle
4k\sin^{2}(hQ/2)\cos(hq)[4k\sin^{2}(hQ/2)\cos(hq)-2A_{*}^{2}]<0.$

If $\cos(hq)<0$, then (3.37) is violated and the plane wave solution is
stable. Therefore, instability occurs when

$\cos(hq)>0.$ (3.38)

Indeed, in order to verify (3.38), it suffices to show that there exists at
least one $Q$ such that (3.37) holds. This $Q=Q^{*}$ satisfies the equation

$\sin(hQ^{*}/2)=\frac{h^{2}A_{*}^{2}}{2\cos(hq)},$

and implies the requested condition for instability (3.37), if (3.38) holds.

We will illustrate the above transient modulation instability of the plane
wave attractor with numerical simulations, in the next section.

##  IV Numerical study I: Transient modulation instability effects and
convergence of plane wave initial conditions to the plane wave attractor

This section is devoted to the first part of the numerical investigations of
the paper. We illustrate the transient modulation instability of the plane
wave attractor, which is verified by the modulation instability analysis of
the section III.1. For this purpose, we study the dynamics of the simplest
plane wave initial conditions, defining perturbations of the plane wave
attractor,

$u_{n}(0)=(A_{*}+A_{p})\exp\Big{(}\frac{\mathrm{i}K\pi x_{n}}{L}\Big{)},$
(4.1)

where $A_{p}$ is the perturbation of the amplitude $A_{*}$ and $K$ is the wave
number of the plane wave initial condition (4.1). The spatial parameters of
the lattice are fixed to $L=50$, $h=1$, thus $N=100$. The linear gain and
nonlinear loss strengths are $\gamma=-\delta=1.5$. These values of $\gamma$
and $\delta$ are chosen to achieve a faster convergence to the attractor.

Due to the periodic boundary conditions (1.3) and discreteness, the physically
meaningful $K\in[0,N/2]$, $K\in\mathbb{N}$. In terms of the modulation
instability analysis of section III.1, $q=\pi K/L$, and the modulation
instability condition (3.38) is satisfied for $0<K<25$, defining the unstable
modes. The rest of the modes $K$ are modulationally stable.

(a) | (b) |  (c)
---|---|---
|  |
Figure 1: Convergence to the plane wave attractor with amplitude $A_{*}=1$ for
the stable mode $K=45$. Rest of parameters: $L=50$, $h=1$,
$\gamma=-\delta=1.5$. Panel (a): Fourier spectrum of the initial condition
(4.1) for $A_{p}=2$ and $A_{p}=-0.999$. Panel (b): Fourier spectrum of the
solutions at $t=5$, for $A_{p}=2$ and $A_{p}=-0.999$. Panel (c): Convergence
of the orbits of the central node of the lattice $u_{0}(t)$ in the phase plane
$\big{(}\textrm{Re}(u_{0}(t)),\,\textrm{Im}(u_{0}(t))\big{)}$ to the limit
cycle of radius $A_{*}=1$; internal (red) orbit for $A_{p}=-0.999$ and
external (blue) orbit for $A_{p}=2$. Details are given in the text.

The first example concerns the dynamics of the stable modes. We choose the
stable mode $K=45$. According to the results of Section III, for arbitrary
perturbations $A_{p}$, the orbit starting from the initial condition (4.1)
will converge to the plane wave attractor with amplitude $A_{*}=1$ and the
initially chosen mode $K=45$. The dynamics for the amplitude perturbations
$A_{p}=2$ and $A_{p}=-0.999$ is depicted in Fig. 1. Panel (a) depicts the
Fourier spectrum of the initial condition (4.1) for $A_{p}=2$ and
$A_{p}=-0.999$, where $K=45$ is the chosen stable mode. Panel (b) depicts the
Fourier spectrum of the solutions at $t=5$ where convergence to the attractor
is attained, for both cases of the perturbations $A_{p}$. It shows that the
attractor preserves the initial wave number $K=45$, as predicted. No other
modes are excited transiently (intermediate snapshots are omitted as they are
identical to (b)), since the initial mode is stable. Panel (c) illustrates the
convergence to the attractor in the plane
$\big{(}\mathrm{Re}(u_{0}(t)),\mathrm{Im}(u_{0}(t))\big{)}$ which is the phase
plane for the central node $u_{0}(t)$ of the lattice located at $n=0$. The
dynamics is portrayed as a convergence to the asymptotically stable limit
cycle of radius $A_{*}$; the orbit in the interior of the limit cycle is the
one for $A_{p}=-0.999$ and the orbit in the exterior of the limit cycle is the
one for $A_{p}=2$.

(a) |  (b)
---|---
|
(c) |  (d)
|
Figure 2:  Snapshots of the evolution of the Fourier spectrum of the solution
with initial condition (4.1) for $A_{p}=2$ and $K=8$. Rest of parameters:
$L=50$, $h=1$, $\gamma=-\delta=1.5$. Details are given in the text.

The second example concerns the case of the unstable modes. We choose the
unstable mode $K=8$. According to the results of Section III.1 the
corresponding orbit should converge to the plane wave attractor with amplitude
$A_{*}=1$. Concerning the wave number of the attracting state, the analytical
arguments provide the prediction for its choice: For the linear gain strength
$\gamma=1.5$ the convergence to the amplitude $A_{*}$ should be fast. On the
other hand, the initial unstable wave number should introduce transient
modulation instability effects. While the convergence to the steady-state
amplitude $A_{*}$ has a fast exponential rate, the emergence of instability
effects will occur later, perturbing the phase of the attracting state,
transiently. Nevertheless, the ultimate state should choose a stable wave
number, since it is the globally asymptotically stable state for the system.
This prediction is fully illustrated in Figures 2 and 3 depicting the dynamics
for $A_{p}=2$. Figure 2 shows the evolution of the Fourier spectrum of the
solution. The initial wave number $K=8$, of panel (a) is preserved as shown in
panel (b) for $t=5$; at this time the system has almost converged to the plane
wave with amplitude $A_{*}=1$ at a fast exponential rate as shown in the panel
(a) of Figure 3, and MI effects are not yet visible. These are developed later
as depicted in panel (c) of Figure 2, demonstrating the excitation of the
whole Fourier spectrum at $t=125$. Panel (b) of Figure 3 portrays the deformed
orbit for $t\in[0,125]$ which is a time period within the stage of modulation
instability. The instability can last for large times, however, eventually,
the global attractor is attained and its wave number should be selected by the
stability band. This is exactly the case, depicted in panel (d) of Figure 2,
showing the snapshot of the Fourier spectrum at $t\in[3600,3700]$. At this
time the only active wavenumber is the stable mode $K=44$ which is the one
choosed by the plane wave attractor of amplitude $A_{*}$.

(a) |  (b) |  (c)
---|---|---
|  |
Figure 3: Snapshots of the orbits of the central node of the lattice
$u_{0}(t)$ in the phase plane
$\big{(}\textrm{Re}(u_{0}(t)),\,\textrm{Im}(u_{0}(t))\big{)}$ for the solution
of Figure 2. Details are given in the text.

##  V Numerical Study II: Simulating the infinite lattice by its finite
lattice approximations

This section is devoted to the second part of the numerical investigations of
the paper. We study the dynamics of localized initial conditions on the top of
a background $A>0$ of the form

$u_{n}(0)=A+f(x),$ (5.2)

where $f(x)$ is a decaying function $\lim_{|x|\rightarrow\infty}f(x)=0$. The
case of localized data is relevant to the simulations of the infinite lattice
(1.1)-(1.2) and thus, they will be explored in detail regarding their global
and transient dynamics.

###  V.1 Localized initial data (5.2) and the global attractor

The analytical results of sections III and III.1 corroborated with the
numerical findings of section IV, provide a strong indication that the global
attractor for all the initial data under the dynamical system (3.1) is the
plane wave attractor (3.6). These indications are enhanced by the fact that
any solution of the DNLS (1.1) in $\ell^{2}_{per}$ can be represented by the
finite sum of plane waves which is its discrete Fourier series,

$\displaystyle u_{n}(t)$
$\displaystyle=\frac{1}{N}\sum_{K=0}^{N-1}A_{K}(t)\exp\Big{(}\frac{\mathrm{i}K2\pi
n}{N}\Big{)},$ $\displaystyle A_{K}(t)$
$\displaystyle=h\sum_{n=0}^{K-1}u_{n}(t)\exp\Big{(}-\frac{\mathrm{i}K2\pi
n}{N}\Big{)}$

and the fact that any plane wave initial condition is attracted by the orbit
$\gamma_{*}$.

Figure 4 depicts the dynamics of the Fourier spectrum of the solution for the
initial condition (5.2) with

$f(x)=\frac{\lambda_{1}}{\lambda_{2}+\lambda_{3}x^{2}},\quad\lambda_{i}>0,\quad
i=1,2,3,$ (5.3)

the standard example of an initial condition decaying on $A$ at a quadratic
rate. For the initial condition (5.2)-(5.3), we choose $A=0.5$,
$\lambda_{1}=\lambda_{2}=1,\lambda_{3}=4$. The gain and loss parameters are
$\gamma=-\delta=0.1$; note that $A\neq A_{*}$. The rest of the lattice
parameters are $L=50$, $h=1$, $N=100$.

(a) | (b) | (c)
---|---|---
|  |
(d) | (e) | (f)
|  |
Figure 4: Snapshots of the Fourier spectrum of the solution for the initial
condition (5.2)-(5.3) with $A=0.5$. Lattice parameters:, $L=50$, $h=1$,
$\gamma=-\delta=0.1$. Details are given in the text.

The transient dynamics prior to the convergence to $\gamma_{*}$ is
illustrated. This involves the excitation of the full spectrum at the early
stages of the evolution as it is shown in panel (b) for $t=40$, while
progressively, the active Fourier modes are reduced, as it is shown in the
snapshots (c) for $t=80$, (d) for $t=200$ and (e) for $t=600$. When the orbit
$\gamma_{*}$ is reached, the only active mode is the stable mode $K=50$, see
panel (f). The same dynamics was observed for other choices of the lattice
parameters and gain and loss strengths.

###  V.2 Transient dynamics of localized initial data (5.2): MI dynamics and
excitation of extreme wave events in the dynamics of the DNLS (1.1).

For the periodic lattice (1.1)-(1.3) the numerical simulations illustrate that
the dynamics prior the convergence to the global attractor are associated with
the emergence of characteristic patterns of MI dynamics and the emergence of
extreme wave events. This is a consequence of the transient modulation
instability discussed above. Our numerical investigations consider two
examples of initial conditions of the form (5.2), with different rates of
localization. The first concerns (5.2) with the quadratically decaying
function $f(x)$ (5.3), and the second with the exponentially decaying function

$f(x)=\sigma\mathrm{sech}(\rho x),\quad\sigma,\;\rho>0.$ (5.4)

In the light of Theorem II.1 we distinguish between two cases for $A$ in
(5.2). The case where $A=A_{*}$ and the case where $A\neq A_{*}$.

#### The case $A=A_{*}$.

We discuss the dynamics for the lattice parameters $h=1$, $N=400$, $L=200$ and
gain and loss strengths $\gamma=0.0025$, $\delta=-0.01$. According to Theorem
II.1, the infinite lattice supplemented with the non-zero boundary conditions
(1.2) support localized solutions if and only if
$A=A_{*}=\sqrt{-\gamma/\delta}=0.5$. This is the suitable choice of the
background of the initial condition (5.2), in order to simulate the dynamics
of the infinite lattice.

(a) |  (b)
---|---
|
(c) |  (d)
|
Figure 5:  Top row: Spatiotemporal evolution of the density $|u_{n}(t)|^{2}$ for the DNLS equation (1.1), with initial conditions (5.2) for $A=A_{*}=0.5$ when $\gamma=0.0025$ and $\delta=-0.01$. Panel (a) for $f(x)$ (5.3) with $\lambda_{1}=\lambda_{2}=1$ and $\lambda_{3}=4$. Panel (b) for $f(x)$ (5.4) with $\sigma=0.6$, $\rho=1$. Bottom row: The spatiotemporal evolution of the density $|\phi_{n}(t)|^{2}$ of the solutions of the AL-lattice (5.6). Panel (c) for $f(x)$ (5.3) and panel (d) for $f(x)$ (5.4) with parameters as above. Rest of the lattice parameters for both systems: $L=200$, $h=1$. More details in the text.  (a) |  (b)
---|---
|
(c) |  (d)
|
Figure 6:  Parameters of the lattice and initial conditions as in Figure 5. Top row: initial condition (5.2) with $f(x)$ (5.3). The continuous (blue) curve in panel (a) traces the temporal evolution of the density $|u_{0}(t)|^{2}$ of the central node of the DNLS (1.1). The dashed (red) curve shows the same evolution for the density of the central node $|\phi_{AL}(x_{0},t,2.40,A_{*})|^{2}$ of the analytical dPS solution (5.5). Panel (b) depicts a comparison of the profile of the first extreme wave event for the DNLS (1.1) (see panel (a) of Figure 5), against the profile of the analytical dPS solution (5.5) with $q=A_{*}$, $t_{0}=2.40$. Bottom row: Same comparisons as above but for the initial condition (5.2) with $f(x)$ (5.4). In this case, the compared dPS solution is $\phi_{AL}(x_{0},t,3.30,A_{*})$. More details in the text.  (a) |  (b)
---|---
|
Figure 7: Spatiotemporal evolution of the density $|u_{n}(t)|^{2}$ for the
DNLS equation (1.1) with initial conditions (5.2) with $A=0.5$ when
$\gamma=0.01$, $\delta=-0.01$, i.e., $A\neq A_{*}$ Panel (a) for $f(x)$ (5.3)
with $\lambda_{1}=\lambda_{2}=1$ and $\lambda_{3}=4$. Panel (b) for $f(x)$
(5.4) with $\sigma=0.6$, $\rho=1$. Rest of the lattice parameters for both
systems: $L=200$, $h=1$. More details in the text.

The top row of Figure 5 depicts the contour plots of the spatiotemporal
evolution of the density $|u_{n}(t)|^{2}$ of the DNLS (1.1), for the initial
conditions (5.2), with $A=A_{*}$. Panel (a) depicts the dynamics for the
quadratically decaying $f(x)$ (5.3) with $\lambda_{1}=\lambda_{2}=1$ and
$\lambda_{3}=4$. The choice of $\lambda_{3}=4$ is made so that the initial
condition has the same spatial decaying rate as the discrete Peregrine soliton
(dPS) [20]

$\phi_{AL}(x_{n},t,t_{0},q)=q\Big{(}1-\frac{4(1+q^{2})(1+4\mathrm{i}q^{2}(t-t_{0}))}{1+4x_{n}^{2}q^{2}+16q^{4}(1+q^{2})(t-t_{0})^{2}}\Big{)}\exp\big{(}2\mathrm{i}q^{2}(t-t_{0})\big{)},$
(5.5)

the rational analytical solution of the integrable Ablowitz–Ladik (AL) lattice

$\mathrm{i}\dot{\phi}_{n}+k(\phi_{n+1}-\phi_{n}+\phi_{n-1})+|\phi_{n}|^{2}(\phi_{n-1}+\phi_{n+1})=0,$
(5.6)

in the case where $k=1$. We remark that this choice for $\lambda_{i}$ is not
restrictive, since similar dynamics exhibited for various cases of
$\lambda_{i}$. Actually, regarding the general structure of the pattern, the
localization rate of the initial condition is proved to be irrelevant as it is
shown in panel (b), where apart of the differences of the centered
oscillations, almost the same pattern is structured for the exponentially
decaying initial condition $f(x)$ (5.4), with $\sigma=0.6$ and $\rho=1$.

The patterns of the DNLS (1.1) share major characteristics of the ones
emerging in the dynamics the AL- lattice when the same initial conditions are
used, as it is shown in the bottom row of Figure 5, where panel (c)
corresponds to the initial condition with $f(x)$ (5.3) and panel (d) to the
initial condition with $f(x)$ (5.4). In particular, the patterns of the AL
system are specific examples, here in the case of the integrable AL-lattice,
of the universal behavior of modulationally unstable media [13]. Let us recall
that in [13], the long-standing open question about the nonlinear stage of
modulation instability on the infinite line for the integrable NLS equation

$\mathrm{i}u_{t}+u_{xx}+|u|^{2}u=0,$ (5.7)

with non-zero boundary conditions,

$\lim_{|x|\rightarrow\infty}u(x,t)=A\exp(\mathrm{i}A^{2}t),\quad A>0$ (5.8)

was fully resolved. It was proved in [14], that for any given initial
condition $u(x,0)$ representing a sufficiently localized initial perturbation
of the constant background $A$, the solution $u(x,t)$ of (5.7) is
$u(x,t)=u_{asymp}(x,t)+\mathrm{O}(1/\sqrt{t})$ as $t\rightarrow\infty$.
Cruciallly, $u_{asymp}(x,t)$ has different forms in different sectors of the
$xt$-plane, namely: (i) in the two “plane-wave” regions, $x<-4\sqrt{2}At$ and
$x>4\sqrt{2}At$, we have that $|u_{asymp}(x,t)|=A$, that is, the solution has
the same amplitude as the undisturbed background $A$ (ii) in the “modulated
elliptic wave” region $-4\sqrt{2}At<x<4\sqrt{2}At$ the solution is expressed
by a slow modulation of the elliptic solutions of (5.7). In [13], analytical
arguments corroborated with numerical simulations provided evidence that this
behavior is universal, as it is shared by a wide class of NLS-type equations
and systems, including the AL-lattice (5.6).

In this context, of the universality of the MI behavior, we claim that the
findings depicted for the DNLS (1.1) in Figure 5 are important. First we
should stress that the DNLS (1.1) is a system at least “two steps forward”
regarding the breaking of the integrability barrier defined by the AL system,
since it is a dissipative perturbation of the non-integrable Hamiltonian DNLS.
Second, the theoretical results of Theorems II.1 and III.1 accompanied with
the MI analysis of the plane-wave of amplitude $A_{*}$ are fully explaining
the dynamics observed in panels (a) and (b). The illustrated dynamics show
that the breaking of integrability induced by the DNLS (1.1) is not dramatic.
It shares the major characteristics of the dynamics of the integrable AL-
lattice, that is, the structure comprised of the two outer, quiescent sectors
separated by the wedge-shaped central region characterized by the oscillatory
behavior. In the outer sectors, the amplitude of $u_{n}(t)$ is $A_{*}$, of the
undisturbed background, the only one which may sustain localized solutions in
the case of the infinite lattice. On the other hand, since the numerical
solutions are generated by solving numerically the periodic lattice, this
background will never lose stability in terms of the amplitude in the outer
regions since the solutions are initiated on the stable attractor of the
finite system regarding its amplitude. Yet for the finite dimensional system,
the MI dynamics in the wedge-shaped central region can be only transient as it
is proved in Theorem III.1 and analyzed in the discussion of section V.1.
Importantly, in the case of the finite lattice the wedge will eventually
disappear as the system will select a stable wave number defining the global
attractor, and the asymptotic state of the solution will be the plane wave of
amplitude $A_{*}$ with the stable wavenumber.

Focusing in shorter time intervals, where the formation of the first events
occur, it is interesting to observe that these are reminiscent of the dPS.
This similarity is illustrated in Figure 6. In this figure, a comparison of
the dynamics of the DNLS (1.1) against the analytical dPS solution (5.5) of
the AL-lattice (5.6) is made, for short times around the emergence of the
first extreme events spotted in panels (a) and (b) of Figure 5. Panels (a) and
(b) correspond to the case of the initial condition (5.2) with the
quadratically decaying $f(x)$ (5.3), and panels (c) and (d) to the case of the
sech-type $f(x)$ (5.4). Panels (a) and (c) depict the temporal evolution of
the density of the central node $|u_{0}(t)|^{2}$ of DNLS (1.1), traced by the
continuous (blue) curve. The dashed (red) curves depict the same evolution for
the density of the central node $|\phi_{AL}(x_{0},t,t_{0},A_{*})|^{2}$ of the
analytical dPS solution (5.5). For the algebraic $f(x)$ (5.3), the time of
occurrence of the rogue wave is $t_{0}=2.40$ and for the sech-type $f(x)$
(5.4), the time is $t_{0}=3.30$. These instants $t_{0}$ are used in the
formula (5.5) in order to define the dPS solutions which are used for the
comparison. We observe the remarkable similarity of the time growth and decay
rates of the events, close to that of the analytical dPS solutions. Panels (b)
and (d) depict a comparison of the profiles of the first extreme wave events
for the DNLS (1.1), against the profiles of the analytical dPS solution (5.5)
with $q=A_{*}$, and the corresponding times $t_{0}$ mentioned above. Again,
the similarity of the profiles is remarkable, and importantly, the coincidence
of the supporting backgrounds of the solutions $A=A_{*}$. It is interesting to
observe that the profile emerging from the $\mathrm{sech}$-profiled initial
condition is more proximal to the dPS analytical profile.

(a) |  (b)
---|---
|
(c) |  (d)
|
Figure 8:  The counterpart of the study of Figure 6 in the case $A\neq A_{*}$.
Parameters of the DNLS lattice (1.1) and initial conditions as in Figure 7.
Details in the text.

#### The case $A\neq A_{*}$.

The parameters and initial conditions are as in the previous case, with the
major difference that $\gamma=-\delta=0.01$ which define the value for
$A_{*}=1$. Therefore, the initial conditions are on top of the background with
amplitude $A=0.5\neq A_{*}$. It is crucial to remark that the results are not
relevant for the infinite lattice with nonzero boundary conditions since
according to Theorem III.1, solutions do not exist when $A\neq A_{*}$.
However, we see in Figure 7, that for finite times, the characteristic MI
pattern is present. The important feature is that since the initial condition
is on the top of $A\neq A_{*}$, the wedge-shaped region is not formed on an
undisturbed plane wave region. In full agreement with the theoretical
analysis, since $A<A^{*}$, the wedge is developed on a background of
increasing amplitude according to the ode solution (3.20); this is evident by
the change of ”shading” in the outer regions of the wedge. The background
eventually converges to that of the global attractor $A^{*}$, accompanied by
the disappearance of the wedge, as predicted theoretically by Theorem II.1.

However, since we have chosen $A=0.5<A_{*}=1$, for short times prior those for
which the effect of the increasing amplitude will become significant, we
expect a similarity with the dynamics illustrated in Figure 6 for the case
$A=0.5=A_{*}$. This fact is shown in Figure 8 which illustrates the same
comparison of the first emerging extreme wave events spotted in Figure 7
against the analytical dPS solutions. The dynamics are found to be almost
indistinguishable to the one presented in Figure 6. Nevertheless, when
examining the dynamics for larger times, we can clearly see a distinction
between the scenario where $A=0.5=A_{*}$ and the scenario where $A=0.5\neq
A_{*}=1$. To illustrate this, we focus on the sech-type initial conditions
(5.4) and refer to Figures 5b and 7b. Panel (a) of Figure 9 shows a snapshot
of the solution at $t=38.5$ when $A=0.5=A_{*}$. It is evident that the
amplitude of the background $A_{*}$ is preserved in full accordance with
Theorem II.1. In contrast, in panel (b) of Figure 9 which corresponds to the
case $A=0.5\neq A_{*}=1$ and shows a snapshot of the solution at $t=32.8$, the
background of the solution is elevated, in accordance with Theorem III.1.

Thus, we may expect that two systems with initial conditions (5.2) with
$A_{1}=A_{2}$ may exhibit similar dynamics for short time intervals when
$A_{*}=A_{1}$ and $A_{*}\neq A_{2}$ which defer for large times, particularly
regarding their long-time asymptotics. Accordingly, we may expect a similarity
of the dynamics of both of the above systems to the dynamics of the AL-lattice
for short-time intervals. We will discuss this issue theoretically, in the
next section.

(a) |  (b)
---|---
|
Figure 9:  The profile of the density $|u_{n}(t)|^{2}$ (blue curve) of the
third extreme event of the DNLS (1.1) with initial conditions (5.2) and $f(x)$
(5.4) with $\sigma=0.6$, $\rho=1$. The dashed (red) curve shows the background
of the initial condition for $A=0.5$. Panel (a): $A_{*}=A$ and $t=38.5$. Panel
(b): $A_{*}=1$ and $t=32.8$.

###  V.3 Return to the analytical considerations: measuring the distance
between the solution the DNLS (1.1) and of the Ablowitz-Ladik lattice

The following results provide a theoretical consideration of the numerical
observations on the distance between the analytical rogue wave solution of the
AL-lattice and the rogue wave alike structure emerged as a first event in the
dynamics of the DNLS (1.1). First, we prove the following property of the
solutions of the AL-lattice (5.6) when supplemented with the periodic boundary
conditions (1.3).

######  Lemma V.1.

For the initial condition $\phi_{n}(0)\in\ell^{2}_{per}$ of the AL-lattice
(5.6), we consider the quantity

$\mathcal{N}(0)=h\sum_{n=0}^{N-1}\ln(1+|\phi_{n}(0)|^{2})<\infty.$ (5.9)

Then, the corresponding solution of the AL-lattice satisfies the estimate

$\|\phi(t)\|_{\ell^{2}_{per}}^{2}\leq
h\exp(h^{-1}\mathcal{N}(0))-h,\qquad\forall t\geq 0.$ (5.10)

###### Proof.

The fact that $\mathcal{N}(0)$ is finite follows by the elementary inequality
$\ln(1+x)\leq x$ for all $x>0$ which implies that

$\mathcal{N}(0)\leq
h\sum_{n=0}^{N-1}|\phi_{n}(0)|^{2}=\|\phi(0)\|^{2}_{\ell^{2}_{per}}$ (5.11)

Setting $\lambda_{n}^{2}(t)=\ln(1+|\phi_{n}(t)|^{2})$, we write
$\mathcal{N}(t)$ in the form

$\mathcal{N}(t)=h\sum_{n=0}^{N-1}\ln(1+|\phi_{n}(t)|^{2})=h\sum_{n=0}^{N-1}\lambda^{2}_{n}(t).$
(5.12)

We recall that $\mathcal{N}(t)$ is one of the conserved quantities of the AL-
lattice and has this property still in the case of the periodic boundary
conditions (1.3), [21]. From (5.12), and by using the equivalence of norms
(2.21) for $q=2j$ and $p=2$, which implies that

$h\sum_{n=0}^{N-1}|\lambda_{n}|^{2j}\leq
h^{1-j}\bigg{(}h\sum_{n=0}^{N-1}|\lambda_{n}|^{2}\bigg{)}^{j},$

we get the estimate:

$\displaystyle h\sum_{n=0}^{N-1}|\phi_{n}|^{2}$
$\displaystyle=h\sum_{n=0}^{N-1}\big{(}\exp(|\lambda_{n}|^{2})-1\big{)}=\sum_{n=0}^{N-1}\,\bigg{(}\sum_{j=0}^{\infty}\frac{|\lambda_{n}|^{2j}}{j!}-1\bigg{)}$
$\displaystyle=h\sum_{n=0}^{N-1}\,\sum_{j=1}^{\infty}\frac{|\lambda_{n}|^{2j}}{j!}=\sum_{j=1}^{\infty}\frac{h}{j!}\,\sum_{n=0}^{N-1}|\lambda_{n}|^{2j}$
$\displaystyle\leq\sum_{j=1}^{\infty}\frac{h^{1-j}}{j!}\,\bigg{(}\sum_{n=0}^{N-1}h|\lambda_{n}|^{2}\bigg{)}^{j}=h\sum_{j=1}^{\infty}\frac{\left(h^{-1}\mathcal{N}\right)^{j}}{j!}$
$\displaystyle=h\bigg{(}\sum_{j=0}^{\infty}\frac{\left(h^{-1}\mathcal{N}\right)^{j}}{j!}-1\bigg{)}=h\exp(h^{-1}\mathcal{N})-h.$
(5.13)

From the conservation of $\mathcal{N}$, that is
$\mathcal{N}(t)=\mathcal{N}(0)$ for all $t\geq 0$, it follows that

$h\sum_{n=0}^{N-1}|\phi_{n}(t)|^{2}\leq
h\exp(h^{-1}\mathcal{N}(0))-h,\qquad\forall t\geq 0,$

which is the claimed estimate (5.10). ∎

######  Theorem V.1.

Let $u_{n}(0),\;\phi_{n}(0)\in\ell^{2}_{per}$ be the initial conditions of the
DNLS (1.1) and of the AL-lattice (5.6), respectively. For the initial
condition of the DNLS (1.1) we assume additionally that

$P_{a}[u(0)]:=\frac{1}{N}\sum_{n=0}^{N-1}|u_{n}(0)|^{2}<A_{*}^{2}.$ (5.14)

Then, the distance
$\|Y(t)\|_{\ell^{2}_{per}}=\|u(t)-\phi(t)\|_{\ell^{2}_{per}}$ satisfies for
all $t\geq 0$, the following estimates:

  * •

estimate I:

$\|u(t)-\phi(t)\|_{\ell^{2}_{per}}\leq\|u(0)-\phi(0)\|_{\ell^{2}_{per}}+\gamma
F_{1}(t)+\big{(}\sqrt{\delta^{2}+1}\big{)}F_{2}(t)+2\left[\exp(\mathcal{N}(0))-1\right]^{3/2}t:=\mathcal{F}(t),\quad$
(5.15)

where the functions $F_{1}(t),F_{2}(t)$ are defined explicitly as

$\displaystyle F_{1}(t)$
$\displaystyle=\int_{0}^{t}\sqrt{B(s)}ds=\frac{1}{\sqrt{\beta\gamma}}\ln\Big{(}\frac{\sqrt{\gamma\nu}-\sqrt{\beta}}{\sqrt{\beta}e^{-\gamma
t}+\sqrt{(e^{2\gamma t}-1)\beta+\gamma\nu}}\Big{)},$ $\displaystyle F_{2}(t)$
$\displaystyle=\int_{0}^{t}B(s)^{3/2}ds$
$\displaystyle=\frac{1}{\beta^{3/2}}\bigg{[}\sqrt{\gamma}\mathrm{ArcSinh}\Big{(}\frac{\sqrt{\beta}e^{\gamma
t}}{\sqrt{\nu\gamma-\beta}}\Big{)}-\frac{\sqrt{\beta\gamma}e^{\gamma
t}}{\sqrt{\beta(e^{2\gamma
t}-1)+\gamma\nu}}-\sqrt{\gamma}\mathrm{ArcSinh}\Big{(}\frac{\sqrt{\beta}}{\sqrt{\nu\gamma-\beta}}\Big{)}+\frac{\sqrt{\beta}}{\sqrt{\nu}}\bigg{]},$
$\displaystyle\nu$
$\displaystyle=\|u(0)\|_{\ell^{2}_{per}}^{-2},\quad\beta=\frac{\tilde{\delta}}{Nh}=-\frac{\delta}{Nh}>0.$

  * •

estimate II:

$\displaystyle\|u(t)-\phi(t)\|_{\ell^{2}_{per}}$
$\displaystyle\leq\|u(0)-\phi(0)\|_{\ell^{2}_{per}}+\alpha
t:=\mathcal{F}_{b}(t),$ (5.16) $\displaystyle\alpha$ $\displaystyle=\gamma
A_{*}\sqrt{Nh}+\big{(}\sqrt{\delta^{2}+1}\big{)}\sqrt{h}A_{*}^{3}N^{3/2}+2\sqrt{h}\left[\exp(h^{-1}\mathcal{N}(0))-1\right]^{3/2}.$
(5.17)

###### Proof.

We derive the equation for $Y_{n}(t)=u_{n}(t)-\phi_{n}(t)$, by subtracting the
equations (1.1)-(5.6). Then, $Y_{n}(t)$ satisfies

$\mathrm{i}\dot{Y}_{n}+\Delta_{d}Y_{n}=\mathrm{i}\gamma
u_{n}+(\mathrm{i}\delta-1)|u_{n}|^{2}u_{n}+|\phi_{n}|^{2}(\phi_{n-1}+\phi_{n+1}).$
(5.18)

We multiply the equation (5.18) by $\overline{Y}_{n}$ in the
$\ell^{2}_{per}$-inner product. By keeping the imaginary parts of the
resulting equation, we get the balance law for $Y$,

$\frac{1}{2}\frac{d}{dt}\|Y\|_{\ell^{2}_{per}}^{2}=\gamma
h\sum_{n=0}^{N-1}u_{n}\overline{Y}_{n}+h\mathrm{Im}\Big{\\{}(\mathrm{i}\delta-1)\sum_{n=0}^{N-1}|u_{n}|^{2}u_{n}\overline{Y}_{n}\Big{\\}}+h\,\mathrm{Im}\Big{\\{}\sum_{n=0}^{N-1}|\phi_{n}|^{2}(\phi_{n-1}+\phi_{n+1})\overline{Y}_{n}\Big{\\}}.$
(5.19)

Using the Cauchy–Schwarz inequality, the first term at the right-hand side of
(5.19) can be estimated as

$h\sum_{n=0}^{N-1}u_{n}\overline{Y}_{n}=\sum_{n=0}^{N-1}\sqrt{h}u_{n}\sqrt{h}\overline{Y}_{n}\leq\|u\|_{\ell^{2}_{per}}\|Y\|_{\ell^{2}_{per}}.$
(5.20)

For the second term of the right-hand side of (5.19) we have the estimate:

$\displaystyle
h\mathrm{Im}\Big{\\{}(\mathrm{i}\delta-1)\sum_{n=0}^{N-1}|u_{n}|^{2}|u_{n}||\overline{Y}_{n}|\Big{\\}}$
$\displaystyle\leq\big{(}\sqrt{\delta^{2}+1}\big{)}\big{(}h^{-1}h\sup_{0\leq
n\leq N-1}|u_{n}|^{2}\big{)}h\sum_{n=0}^{N-1}|u_{n}||Y_{n}|$
$\displaystyle\leq\big{(}\sqrt{\delta^{2}+1}\big{)}h^{-1}\|u\|^{2}_{\ell^{2}_{per}}h\sum_{n=0}^{N-1}|u_{n}||Y_{n}|$
(5.21)
$\displaystyle\leq\big{(}\sqrt{\delta^{2}+1}\big{)}h^{-1}\|u\|_{\ell^{2}_{per}}^{3}\|Y\|_{\ell^{2}_{per}}.$

For the third term, let us note that due to the periodic boundary conditions
(1.3), we have that

$\sum_{n=0}^{N-1}|\phi_{n-1}|^{2}=\sum_{n=0}^{N-1}|\phi_{n+1}|^{2}=\sum_{n=0}^{N-1}|\phi_{n}|^{2}.$
(5.22)

Then, by using (5.22), the third term of the right-hand side of (5.19) can be
estimated as follows:

$\displaystyle
h\mathrm{Im}\Big{\\{}\sum_{n=0}^{N-1}|\phi_{n}|^{2}|\phi_{n-1}+\phi_{n+1}||\overline{Y}_{n}|\Big{\\}}\leq\Big{(}h^{-1}h\sup_{0\leq
n\leq
N-1}|\phi_{n}|^{2}\Big{)}\Big{\\{}h\sum_{n=0}^{N-1}|\phi_{n-1}|\,|Y_{n}|+h\sum_{n=0}^{N-1}|\phi_{n+1}|\,|Y_{n}|\Big{\\}}$
$\displaystyle\leq
h^{-1}\|\phi\|^{2}_{\ell^{2}_{per}}\bigg{\\{}\Big{(}h\sum_{n=0}^{N-1}|\phi_{n-1}|^{2}\Big{)}^{1/2}\Big{(}h\sum_{n=0}^{N-1}|Y_{n}|^{2}\Big{)}^{1/2}+\Big{(}h\sum_{n=0}^{N-1}|\phi_{n+1}|^{2}\Big{)}^{1/2}\Big{(}h\sum_{n=0}^{N-1}|Y_{n}|^{2}\Big{)}^{1/2}\bigg{\\}}$
(5.23) $\displaystyle\leq
2h^{-1}\|\phi\|_{\ell^{2}_{per}}^{3}\|Y\|_{\ell^{2}_{per}}.$

Now, using the estimates (5.20), (V.3) and (V.3) together with the equation

$\frac{1}{2}\frac{d}{dt}\|Y\|_{\ell^{2}_{per}}^{2}=\|Y\|_{\ell^{2}_{per}}\frac{d}{dt}\|Y\|_{\ell^{2}_{per}},$

we derive from (5.19), the differential inequality

$\displaystyle\frac{d}{dt}\|Y\|_{\ell^{2}_{per}}\leq\gamma\|u\|_{\ell^{2}_{per}}+\big{(}\sqrt{\delta^{2}+1}\big{)}h^{-1}\|u\|_{\ell^{2}_{per}}^{3}+2h^{-1}\|\phi\|_{\ell^{2}_{per}}^{3}.$
(5.24)

Proof of estimate I: We bound from above the right hand side of (5.24) by
using the estimate

$\|u(t)\|^{2}_{\ell^{2}_{per}}\leq\frac{\gamma}{\gamma\exp(-2\gamma
t)\|u(0)\|^{-2}_{\ell^{2}_{per}}+\frac{\tilde{\delta}}{Nh}[1-\exp(-2\gamma
t)]}:=B(t),\quad\forall t\geq 0,$ (5.25)

(which can be easily deduced for $\|u(t)\|_{\ell^{2}_{per}}$ from the bound
(3.12)) and the estimate (5.10) for $\|\phi(t)\|_{\ell^{2}_{per}}$. We remark
that under the assumption (5.14), the function $B(t)$ satisfies

$0<B(t)<A_{*}^{2}Nh,\quad\mbox{for all $t>0$},$ (5.26)

and recall from Theorem III.1, that

$\lim_{t\rightarrow\infty}B(t)=A^{2}_{*}Nh.$ (5.27)

Then we integrate in the arbitrary interval $[0,t]$ to get the claimed
estimate (5.15). We also note that the assumption (5.14) can be written
alternatively as

$\nu\gamma>\beta,$ (5.28)

and due to (5.28) the functions $F_{1}(t)$ and $F_{2}(t)$ are well defined for
all $t>0$.
Proof of the estimate II: Since the bound (5.26) holds for all $t>0$, the
right-hand side of (5.24) can be estimated alternatively, giving the
differential inequality

$\displaystyle\frac{d}{dt}\|Y\|_{\ell^{2}_{per}}\leq\gamma
A_{*}\sqrt{Nh}+\big{(}\sqrt{\delta^{2}+1}\big{)}\sqrt{h}A_{*}^{3}(N)^{3/2}+2\sqrt{h}\big{[}\exp(h^{-1}\mathcal{N}(0))-1\big{]}^{3/2}.$
(5.29)

Integration of (5.29) in the arbitrary interval $[0,t]$ proves the alternative
estimate (5.16) with the at most linear growth rate (5.17). ∎

###  V.4 Remarks on the numerical observations of section V.2 under Theorem
V.1

The numerical observations of section (V.2) can be explained under the light
of Theorem V.1. It is more tractable for simplicity, to consider the estimate
II given in (5.16)-(5.17). The explicit linear growth rate $\alpha$ (5.17) and
its dependence on $\gamma>0$, $\delta<0$, $N$, suggests a moderate linear
growth for short time intervals when $|\delta|$ and $|\gamma|$ are small, and
$N$ is physically relevantly large. In particular, we observe that the
spatially averaged distance (which is equivalent and close to the
$\ell^{\infty}$ metric in the case of the finite lattices as it is justified
from (2.21))

$D_{a}(t):=\frac{1}{\sqrt{Nh}}\|Y(t)\|_{\ell^{2}_{per}},$ (5.30)

has a linear growth rate, at most of $\mathcal{O}(1)$ for short time
intervals, if

$\gamma^{3}<\min\Big{\\{}-\frac{\delta}{Nh},-\frac{\delta^{3}}{(\delta^{2}+1)hN^{3}}\Big{\\}}\approx\mathcal{O}(1).$
(5.31)

The smallness requirement (5.31) is relevant in physical setups where $\gamma$
and $-\delta$ are small, as chosen for our numerical experiments. It is
highlighted in the top row of Figure 10, depicting plots of the function
(5.30) when $t\in[0,10]$ for the initial conditions (5.2) (panel (a)) and
(5.4) (panel (b)). The rest of parameters are selected as in Figures 5 and 6.
In accordance with the theoretical estimates, for these set of lattice
parameters and time interval, $D_{a}(t)$ is smaller than $\mathcal{O}(1)$; we
remark the almost linear growth after the minimum (attained close to the rogue
wave event), which is still less than of $\mathcal{O}(1)$, for this time
interval.

Moreover, the estimates (5.16)-(5.17) suggest that the distance between the
structures for the above set of parameters, should be smaller when focused to
the core of the MI pattern around $x_{0}=0$ for small time intervals. This
prediction is illustrated in the bottom row of Figure 10, depicting the time
evolution of the averaged distance

$D_{a,r}:=\frac{1}{\sqrt{hN_{r}}}\Big{(}h\sum_{-10}^{10}|u_{n}(t)-\phi_{n}(t)|^{2}\Big{)}^{\frac{1}{2}},$
(5.32)

where $N_{r}$ denotes the number of nodes in $x_{n}\in[-10,10]$.

(a) | (b)
---|---
|
(c) | (d)
|
Figure 10: Parameters as in Figures 5 and 6. Panels (a) and (b): Plot of the
spatially averaged distance $D_{a}(t)$ (5.30) when $t\in[0,10]$, for the
initial conditions (5.2) and (5.4), respectively. Panels (c) and (d): Plot of
$D_{a,r}(t)$ (5.32) when $t\in[0,10]$ and $x_{n}\in[-10,10]$, for the same
initial conditions.

##  VI Conclusions

In this work, we studied the dynamics of the Discrete Nonlinear Schrödinger
Equation incorporating linear gain and nonlinear loss effects, a significant
model in various physical contexts. In the case of the infinite lattice we
considered the problem with nonzero boundary conditions proving that solutions
exist if and only if the supporting background has a prescribed critical value
$A_{*}$ defined by the gain and loss strengths. We argued that this property
cannot be captured by finite lattice approximations, as those defined by the
system supplemented with periodic boundary conditions, which are essential for
the numerical simulations of the problem. For the periodic lattice, which is
of physical and mathematical significance itself, we proved that the dynamics
are dominated by the convergence to a global attractor which is the unique
plane wave of constant amplitude, namely $A_{*}$, with a novel argument which
studies not only the limits of the amplitudes of the initial conditions but
also their frequencies and spectrum. The convergence arguments accompanied by
a modulation instability analysis justify that modulation instability, the
fundamental mechanism for the emergence of localized structures can be only
transient in the case of the periodic lattice. The dynamics of localized
initial data are investigated numerically: in the case where $A=A_{*}$ which
is the only relevant in order to approximate the infinite lattice, we identify
the persistence of the major characteristics of the modulation instability
patterns manifested in the dynamics of integrable NLS systems, as the
Ablowitz-Ladik lattice: the oscillatory wedge shaped sector and the two outer
plane wave regions of the undisturbed background of amplitude $A_{*}$, which
is globally asymptotically stable in the case of the finite lattice. In the
case $A\neq A_{*}$ which is only relevant for the finite lattice, the
modulation instability pattern still persists, this time on a varying
background transiently, prior the convergence of the dynamics to the plane
wave attractor. At the early stages of the dynamics we identified the
emergence of rogue wave events which are remarkably proximal to the analytical
discrete Peregrine rogue wave solution of the Ablowitz-Ladik lattice. This
proximity is justified analytically by suitable estimates for the distance
between the solutions of the DNLS system and the Ablowitz-Ladik lattice.

As a summary, we argue that the breaking of integrability by the considered
dissipative DNLS system is not dramatic since the major features of the
integrable dynamics are preserved. The above results showcase that the
behavior due to modulation instability can be universal and observed not only
in Hamiltonian NLS systems but also in even more realistic set-ups described
by non integrable systems where dissipation and forcing effects are present.

###### Acknowledgements.

The authors G.F. and N.I.K. acknowledge that this work was partially supported
by the Xiamen University Malaysia Research Fund (Grant No:
XMUMRF/2022-C9/IMAT/0020).

Authors Declarations
The authors have no conflicts to disclose.

Authors Contributions Statement
All authors contributed equally to the study conception, design and writing of
the manuscript. Material preparation, data collection and analysis were
performed equally by all authors. All authors read and approved the final
manuscript.

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