# Transparent boundary conditions for the nonlocal nonlinear Schrödinger
equation:
A model for reflectionless propagation of PT-symmetric solitons

M.E. Akramov1, J.R. Yusupov2, M. Ehrhardt3, H. Susanto4 and D.U. Matrasulov5
1National University of Uzbekistan, 4 Universitet Str., 100174, Tashkent,
Uzbekistan
2Yeoju Technical Institute in Tashkent, 156 Usman Nasyr Str., 100121,
Tashkent, Uzbekistan
3Bergische Universität Wuppertal, Gaußstrasse 20, D-42119 Wuppertal, Germany
4Khalifa University, 127788, Abu Dhabi, United Arab Emirates
5Turin Polytechnic University in Tashkent, 17 Niyazov Str., 100095, Tashkent,
Uzbekistan

###### Abstract

We consider the problem of reflectionless propagation of PT-symmetric solitons
described by the nonlocal nonlinear Schrödinger equation on a line in the
framework of the concept of transparent boundary conditions for evolution
equations. Transparent boundary conditions for the nonlocal nonlinear
Schrödinger equation are derived. The absence of backscattering at the
artificial boundaries is confirmed by the numerical implementation of the
transparent boundary conditions.

##  I Introduction

Modeling of wave dynamics in various media is of practical importance in the
fields of optics, optoelectronics, fluid dynamics, acoustics and communication
technology. An important problem to be solved by such models is tunable wave
propagation, which means achieving ballistic and diffusive states or absence
of backscattering and reflection in a certain subdomains. In quantum
mechanics, such a goal can be achieved by constructing a suitable scattering
matrix that ensures the absence of reflection.

However, in the case of nonlinear wave propagation, one cannot use a
scattering matrix and must develop efficient mathematical tools to describe
reflectionless propagation. One of these tools can be based on the use of the
concept of so-called “transparent boundary conditions (TBCs)” (other names are
“artificial boundary conditions” and “absorbing boundary conditions”). The
concept was previously applied to the linear Ehrhardt1999 ; Ehrhardt2001 ;
Arnold2003 ; Ehrhardt2008  and nonlinear Antoine ; Han0 ; Matthias2008 ;
Antoine2008 ; Zhang  Schrödinger equations. Recently, the application of the
TBC concept to Dirac-Hammer2014  and Klein-Gordon equations Gander ; Antoine1
, including the nonlinear Klein-Gordon equation Zheng07 ; Han ; Li ; Antoine1
; TBCSGE  was shown. Extensions of TBCs for evolution equations on graphs can
be found in references Jambul ; Jambul02 ; Jambul1 ; Jambul2 ; Jambul4 .

The basic idea of the TBC concept can be formulated as follows: For a given
partial differential equation formulated as an initial value problem in a
finite domain, it is required that the solution in one domain should match
that in the entire space restricted to the finite domain. This can be achieved
by artificial boundary conditions, which have a rather complicated form (e.g.,
for Schrödinger or Dirac equations given in terms of fractional derivatives
are Ehrhardt1999 ; Hammer2014 ).

In this paper we extend the TBC concept for the nonlocal nonlinear Schrödinger
equation (NNLS) describing the dynamics of PT-symmetric solitons. The NNLS
equation was first introduced by Ablowitz and Musslimani AM2013 , who showed
the integrability of the problem and obtained their soliton solutions. Later
it was used in various contexts in Refs. AM2013 ; Stalin ; AM2014 ; AM2016 ;
AM20161 ; Sinha ; Yang ; Zhenya ; AM2018 ; AM2018_1 ; AM2019 ; Hadi2019 ;
Kanna2020 ; Panos2020 ; Mashrab2022 . The NNLS equation describes the dynamics
of solitons in media with self-induced PT-symmetric nonlinearity (such
nonlinearity may be present, for example, in an optical waveguide with self-
induced gain loss). Our proposed model accounts for reflectionless propagation
of solitons in such media. The transparent boundary conditions derived here
provide mathematical constraints that ensure that there is no backscattering.
In practical applications of boundary conditions, e.g., for branched
waveguides, they can provide physically acceptable constraints on the
equivalence of the usual weight continuity and Kirchhoff’s rules with the
transparent boundary conditions at the branching point (see, e.g., Refs.
Jambul ; Jambul02 ; Jambul1  for details).

This paper is organized as follows. In the next section, we briefly recall
soliton solutions and conserving quantities for the NNLS equation on a line.
In Section III, we derive the transparent boundary conditions for the NNLS
equation. In Section IV, we demonstrate our numerical implementation of such
boundary conditions and show the results of numerical experiments in Section
V. Finally, Section VI contains the concluding remarks.

##  II Soliton solutions of the nonlocal nonlinear Schrödinger equation

Here we briefly recall basic results on the NNLS equation on a line, following
Ref. AM2013 . The nonlocal nonlinear Schrödinger equation is given as AM2013

$\mathrm{i}\partial_{t}q(x,t)+\partial^{2}_{x}q(x,t)+2q(x,t)q^{*}(-x,t)q(x,t)=0,$
(1)

where $q^{*}$ denotes the complex conjugate of $q$ and the potential, which
can be defined as $V(x,t)=2\,q(x,t)\,q^{*}(-x,t)$, has the PT symmetric
property, i.e. $V(x,t)=V^{*}(-x,t)$. We note that the nonlocality of Eq. (1)
results from the fact that the evolution of the field $q(x,t)$ at coordinate
$x$ always requires information from the opposite point $-x$. For the above
NNLS equation, there are many different types of soliton solutions, namely
breathing, periodic, rational, and others. A single soliton solution can be
found by the inverse scattering method as given in Ref. AM2013 :

$q(x,t)=-\frac{2(\eta_{1}+\bar{\eta}_{1})\,e^{\mathrm{i}\bar{\theta}_{1}}\,e^{4\mathrm{i}\bar{\eta}^{2}_{1}t}\,e^{-2\bar{\eta}_{1}x}}{1+e^{\mathrm{i}(\theta_{1}+\bar{\theta}_{1})}\,e^{-4\mathrm{i}(\eta^{2}_{1}-\bar{\eta}^{2}_{1})t}\,e^{-2(\eta_{1}+\bar{\eta}_{1})x}},$
(2)

with $\eta_{1}$, $\bar{\eta}_{1}$, $\theta_{1}$, and $\bar{\theta}_{1}$ being
real constants. The traveling soliton solution of Eq. (1) can be written as
Stalin

$\displaystyle
q(x,t)=\frac{\alpha_{1}\,e^{-\Delta/2}\,e^{(\bar{\xi}_{1R}-\xi_{1R})+i(\bar{\xi}_{1I}-\xi_{1I})}}{2[\cosh(\chi_{1})\cos(\chi_{2})+\mathrm{i}\sinh(\chi_{1})\sin(\chi_{2})]},$
(3)

where $\xi_{1R}=-k_{1I}(x+2k_{1R}t)$,
$\xi_{1I}=k_{1R}x-(k_{1I}^{2}-k_{1R}^{2})t$,
$\bar{\xi}_{1R}=-\bar{k}_{1I}(x+2\bar{k}_{1R}t)$,
$\bar{\xi}_{1I}=\bar{k}_{1R}x-(\bar{k}_{1R}^{2}-\bar{k}_{1I}^{2})t$,
$\Delta_{R}=\log\left(\frac{|\alpha_{1}|^{2}|\beta_{1}|^{2}}{|k_{1}+\bar{k}_{1}|^{2}}\right)$,
$\Delta_{I}=-\frac{i}{2}\log\left(\frac{\alpha_{1}\beta_{1}(k_{1}^{*}+\bar{k}_{1}^{*})^{2}}{\alpha^{*}_{1}\beta^{*}_{1}(k_{1}+\bar{k}_{1})^{2}}\right)$,
$e^{\Delta}=-\frac{\alpha_{1}\beta_{1}}{(k_{1}+\bar{k}_{1})^{2}}$,
$\chi_{1}=(\xi_{1R}+\bar{\xi}_{1R}+\Delta_{R})/2$ and
$\chi_{2}=(\xi_{1I}+\bar{\xi}_{1I}+\Delta_{I})/2$.

The integrability of the problem was proved in AM2013 , which means that the
NNLS equation has many conservation laws. In particular, two important
conservation quantities, the norm and the energy, were derived in AM2013  and
can be written as

$\displaystyle N(t)=\underset{-\infty}{\overset{+\infty}{\int}}q(x,t)q^{*}($
$\displaystyle-x,t)\,dx,$ $\displaystyle
E(t)=\underset{-\infty}{\overset{+\infty}{\int}}\Big{[}\frac{\partial}{\partial
x}q(x,t)$ $\displaystyle\cdot\frac{\partial}{\partial x}q^{*}(-x,t)$
$\displaystyle+q^{2}(x,t)\cdot q^{*2}(-x,t)\Big{]}\,dx.$ (4)

The above soliton solutions of Eq. (1) are obtained assuming asymptotic
boundary conditions at infinity, i.e. $q(x,t)\to 0$ at $x\to\pm\infty$. In the
next section, we impose additional (artificial) boundary conditions for a
given finite interval $[-L,L]$ that allow an almost reflection-free
transmission of a soliton through the points $\pm L$.

##  III Transparent boundary conditions for the nonlocal nonlinear Schrödinger
equation

Here we consider the problem of transparent boundary conditions for the NNLS
equation (1). To derive TBCs for the nonlocal nonlinear Schrödinger equation,
we use the so-called potential approach, which was proposed earlier in Antoine
and used to derive TBCs for nonlinear Schrödinger equations Jambul3  and sine-
Gordon equations TBCSGE . Within the framework of this approach, the NNLS
equation can be formally reduced to the linear Schrödinger equation

$\mathrm{i}\partial_{t}q(x,t)+\partial^{2}_{x}q(x,t)+V(x,t)q(x,t)=0,$ (5)

with the potential is $V(x,t)=2q(x,t)q^{*}(-x,t)$. In the remainder of this
section, we invoke the same procedure and derive TBCs at the end.

To do so, we introduce a new unknown $Q(x,t)$, which is given by the relation

$Q(x,t)=e^{-\mathrm{i}\mathcal{V}(x,t)}\,q(x,t),$ (6)

where

$\mathcal{V}(x,t)=\int_{0}^{t}V(x,s)\,ds.$ (7)

The temporal and spatial derivatives of $q$ can be written as derivatives of
$Q$ as

$\partial_{t}q=e^{\mathrm{i}\mathcal{V}}(\partial_{t}+\mathrm{i}V)Q,$ (8)

and

$\partial^{2}_{x}q=\mathrm{i}e^{\mathrm{i}\mathcal{V}}(\partial^{2}_{x}Q+2\mathrm{i}\partial_{x}\mathcal{V}\partial_{x}Q+\mathrm{i}Q\partial^{2}_{x}\mathcal{V}-(\partial_{x}\mathcal{V})^{2}Q).$
(9)

As a result, we obtain the Schrödinger equation in terms of $Q(x,t)$ as

$L(x,t,\partial_{x},\partial_{t})Q=\mathrm{i}\partial_{t}Q+\partial_{x}^{2}Q+A\partial_{x}Q+BQ=0,$
(10)

where $A=2\mathrm{i}\partial_{x}\mathcal{V}$ and
$B=(\mathrm{i}\partial^{2}_{x}\mathcal{V}-(\partial_{x}\mathcal{V})^{2})$.
Linearizing Eq. (10) using the pseudo-differential operator calculus, we
obtain

$L=(\partial_{x}+\mathrm{i}\Lambda^{-})(\partial_{x}+\mathrm{i}\Lambda^{+})\\\
=\partial_{x}^{2}+\mathrm{i}(\Lambda^{+}+\Lambda_{-})\partial_{x}+\mathrm{i}\operatorname{Op}(\partial_{x}\lambda^{+})-\Lambda^{+}\Lambda^{-},$
(11)

where $\lambda^{+}$ denotes the principal symbol of the operator $\Lambda^{+}$
and $\operatorname{Op}(p)$ denotes the associated operator of a symbol $p$.
From the Eqs. (10) and (11) we obtain the system of operators

$\displaystyle i(\Lambda^{+}+\Lambda^{-})=A$ $\displaystyle
i\operatorname{Op}(\partial_{x}\lambda^{+})-\Lambda^{+}\Lambda^{-}=i\partial_{t}+B,$
(12)

which yields the symbolic system of equations

$\displaystyle\mathrm{i}(\lambda^{+}+\lambda^{-})=a$
$\displaystyle\mathrm{i}\partial_{x}\lambda^{+}-\sum_{\alpha=0}^{+\infty}\frac{(-1)^{\alpha}}{\alpha!}\partial_{\tau}^{\alpha}\lambda^{-}\partial_{t}^{\alpha}\lambda^{+}=-\tau+b,$
(13)

where $\operatorname{Op}(a)=A$ and $\operatorname{Op}(b)=B$ can be set as
$a=A$ and $b=B$, since these two functions correspond to zero-order operators.
An asymptotic evolution in the inhomogeneous symbols is defined as

$\lambda^{\pm}\sim\sum_{j=0}^{+\infty}\lambda_{1/2-j/2}^{\pm}.$ (14)

Substituting the expansion (14) into Eq. (13), one can identify the terms of
order $1/2$ in the first relation of the system (13):

$\lambda_{1/2}^{-}=-\lambda_{1/2}^{+},\quad\lambda_{1/2}^{+}=\pm\sqrt{-\tau}.$
(15)

The Dirichlet-to-Neumann operator corresponds to the choice
$\lambda_{1/2}^{+}=\pm\sqrt{-\tau}$. For the zeroth-order terms we obtain

$\displaystyle\lambda_{0}^{-}=-\lambda_{0}^{+}-\mathrm{i}a,$
$\displaystyle\mathrm{i}\partial_{x}\lambda_{1/2}^{+}-(\lambda_{0}^{-}\lambda_{1/2}^{+}+\lambda_{0}^{+}\lambda_{1/2}^{-})=0.$
(16)

From Eq. (16) we get

$\displaystyle\lambda_{0}^{+}=-\mathrm{i}\frac{a}{2}=\frac{1}{2}\partial_{x}\mathcal{V},$
$\displaystyle\lambda_{0}^{-}=-\lambda_{0}^{+}-\mathrm{i}a=\frac{1}{2}\partial_{x}\mathcal{V}.$
(17)

For the terms of order $-1/2$ we obtain

$\displaystyle\mathrm{i}(\lambda_{-1/2}^{+}+\lambda_{-1/2}^{-})=0,$
$\displaystyle\mathrm{i}\partial_{x}\lambda_{0}^{+}-(\lambda_{-1/2}^{-}\lambda_{1/2}^{+}+\lambda_{0}^{+}\lambda_{0}^{-}+\lambda_{-1/2}^{+}\lambda_{1/2}^{-})=b,$
(18)

since
$\partial_{t}^{\alpha}\lambda_{-1/2}^{\pm}=\partial_{\tau}^{\alpha}\lambda_{0}^{\pm}=0$,
$\alpha\in N$. From Eq. (III) we get

$\lambda_{-1/2}^{\pm}=0.$ (19)

Furthermore, the terms of the next order can be obtained as

$\lambda_{-1}^{-}=-\lambda_{-1}^{+},\quad\lambda_{-1}^{+}=\mathrm{i}\frac{\partial_{x}V}{4\tau}.$
(20)

Thus, the first-order approximation is

$\displaystyle\partial_{x}q|_{x=-L}-e^{-\mathrm{i}\frac{\pi}{4}}e^{\mathrm{i}\mathcal{V}}\partial^{1/2}_{t}(e^{-\mathrm{i}\mathcal{V}}q)\big{|}_{x=-L}=0,$
(21a)
$\displaystyle\partial_{x}q|_{x=L}+e^{-\mathrm{i}\frac{\pi}{4}}e^{\mathrm{i}\mathcal{V}}\partial^{1/2}_{t}(e^{-\mathrm{i}\mathcal{V}}q)\big{|}_{x=L}=0.$
(21b)

The second-order approximation reads

$\displaystyle\partial_{x}q|_{x=-L}-e^{-\mathrm{i}\frac{\pi}{4}}e^{\mathrm{i}\mathcal{V}}\partial^{1/2}_{t}(e^{-\mathrm{i}\mathcal{V}}q)\big{|}_{x=-L}$
$\displaystyle-\mathrm{i}\frac{\partial_{x}V}{4}e^{\mathrm{i}\mathcal{V}}I_{t}(e^{-\mathrm{i}\mathcal{V}}q)\big{|}_{x=-L}=0,$
(22a)
$\displaystyle\partial_{x}q|_{x=L}+e^{-\mathrm{i}\frac{\pi}{4}}e^{\mathrm{i}\mathcal{V}}\partial^{1/2}_{t}(e^{-\mathrm{i}\mathcal{V}}q)\big{|}_{x=L}$
$\displaystyle+\mathrm{i}\frac{\partial_{x}V}{4}e^{\mathrm{i}\mathcal{V}}I_{t}(e^{-\mathrm{i}\mathcal{V}}q)\big{|}_{x=L}=0,$
(22b)

Here the operator $\partial^{1/2}_{t}$ denotes the fractional time derivative
operator of half order given as

$\partial_{t}^{1/2}f(t)=\frac{1}{\sqrt{\pi}}\partial_{t}\int_{0}^{t}\frac{f(s)}{\sqrt{t-s}}\,ds,$

and the operator $I_{t}(f)$ is given as

$I_{t}f(t)=\int_{0}^{t}f(s)\,ds.$

##  IV A discretization scheme for the transparent boundary conditions

In this section, we present a numerical scheme for Eq. (1) and the numerical
implementation of the transparent boundary conditions (21) and (22). We have
chosen the finite difference scheme of Duran–Sánz-Serna Duran , a second order
scheme based on the implicit midpoint rule

$\mathrm{i}\frac{q_{j}^{n}-q_{j}^{n-1}}{\Delta
t}+D^{2}_{x}\frac{q_{j}^{n}+q_{j}^{n-1}}{2}\\\
+2\Bigl{(}\frac{q_{j}^{n}+q_{j}^{n-1}}{2}\Bigr{)}^{2}\frac{\overline{q_{J-j}^{n}}+\overline{q_{J-j}^{n-1}}}{2}=0,$
(23)

with the standard second-order difference quotient

$D_{x}^{2}q_{j}^{n}=\frac{1}{\Delta
x^{2}}(q_{j-1}^{n}-2q_{j}^{n}+q_{j+1}^{n}),$ (24)

where $\Delta x$ and $\Delta t$ are the spatial and temporal discretization
steps, respectively. Here $\overline{q^{n}_{j}}$ denotes the complex conjugate
of $q^{n}_{j}$, $J$ is the number of discretized spatial steps, and
$q^{n}_{j}$ denotes the approximate value of $q$ at spatial coordinate $x_{j}$
and time $t_{n}$.

We now give an effective discretization scheme for the TBC, which is
implemented together with (23). We give the scheme only for $x=L$, saying that
the implementation for the left-hand side (at $x=-L$) can be done in the same
way. The approximation of the fractional differential operator is given by the
numerical quadrature formula Antoine

$\partial_{t}^{1/2}f(t_{n})\approx\sqrt{\frac{2}{\Delta
t}}\sum_{k=0}^{n}\beta_{k}f^{n-k},$ (25)

where $\\{f_{n}\\}_{n\in N}$ is a sequence of complex values approximating
$\\{f(t_{n})\\}_{n\in N}$ and $(\beta_{k})_{k\in N}$ denotes the sequences
defined by

$(\beta_{0},\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5},\dots)=\\\
\Bigl{(}1,-1,\frac{1}{2},-\frac{1}{2},\frac{1\cdot 3}{2\cdot 4},-\frac{1\cdot
3}{2\cdot 4},\dots\Bigr{)}.$ (26)

The function $\mathcal{V}(x,t)$ given by (7) can be discretized using the
trapezoidal rule as

$\mathcal{V}_{j}^{n}=\Delta
t\biggl{[}\,\sum_{k=1}^{n-1}V_{j}^{k}+\frac{1}{2}(V_{j}^{0}+V_{j}^{n})\biggr{]}\quad\text{for}\;n\geq
2,$ (27)

with $\mathcal{V}_{j}^{0}=0$ and
$\mathcal{V}_{j}^{1}=\frac{1}{2}(V_{j}^{0}+V_{j}^{1})$, where $V_{j}^{n}$ is
defined as

$V_{j}^{n}=2q_{j}^{n}\overline{q_{J-j}^{n}}.$ (28)

Next, the term $e^{\mathrm{i}\mathcal{V}}$ is written as

$\displaystyle P_{j}^{n}:=$ $\displaystyle\exp(\mathrm{i}\mathcal{V}_{j}^{n})$
$\displaystyle=$ $\displaystyle\exp\left\\{\mathrm{i}\Delta
t\left[\,\sum_{k=1}^{n-1}V_{j}^{k}+\frac{1}{2}(V_{j}^{0}+V_{j}^{n})\right]\right\\}.$
(29)

One can rewrite Eq. (IV) as a recurrence formula

$P_{j}^{n}=P_{j}^{n-1}\exp\Bigl{(}\frac{\mathrm{i}\Delta
t}{2}(V_{j}^{n-1}+V_{j}^{n})\Bigr{)}.$ (30)

The TBC operator of the first order approximation (21) on the left and right
boundaries $j=0$ and $j=J$ can be approximated by the discrete convolutions

$\Lambda_{1}^{n}=e^{-\mathrm{i}\pi/4}\sqrt{\frac{2}{\Delta
t}}P_{j}^{n}\sum_{k=0}^{n}\beta_{k}\frac{q_{j}^{n-k}}{P_{j}^{n-k}}.$ (31)

Then the values of the wave function at the boundaries (together with other
intermediate values) can be obtained by solving the system of nonlinear
equations as

$\displaystyle\frac{q_{1}^{n}-q_{0}^{n}}{\Delta
x}-e^{-\mathrm{i}\pi/4}\sqrt{\frac{2}{\Delta
t}}P_{0}^{n}\sum_{k=0}^{n}\Bigl{(}\beta_{k}\frac{q_{0}^{n-k}}{P_{0}^{n-k}}\Bigr{)}$
$\displaystyle=0,$ (32a) $\displaystyle\frac{q_{J}^{n}-q_{J-1}^{n}}{\Delta
x}+e^{-\mathrm{i}\pi/4}\sqrt{\frac{2}{\Delta
t}}P_{J}^{n}\sum_{k=0}^{n}\Bigl{(}\beta_{k}\frac{q_{J}^{n-k}}{P_{J}^{n-k}}\Bigr{)}$
$\displaystyle=0,$ (32b)

and the Eq. (23) with respect to $q_{j}^{n}$. Note that the values of
$q_{j}^{n}$ to be found also exist in $P_{j}^{n}$ and here we have written
Eqs. (32) in short form. Using the same approach, we can proceed with the
discretization of the second-order approximation. We approximate the integral
term as

$I_{j}^{n}=\Delta
t\biggl{[}\,\sum_{k=1}^{n-1}\frac{q_{j}^{k}}{P_{j}^{k}}+\frac{1}{2}\Bigl{(}\frac{q_{j}^{0}}{P_{j}^{0}}+\frac{q_{j}^{n}}{P_{j}^{n}}\Bigr{)}\biggr{]}.$
(33)

In the same way we construct a recurrence formula for $I_{j}^{n}$

$I_{j}^{n}=I_{j}^{n-1}+\frac{\Delta
t}{2}\Bigl{(}\frac{q_{j}^{n-1}}{P_{j}^{n-1}}+\frac{q_{j}^{n}}{P_{j}^{n}}\Bigr{)},$
(34)

with $I_{j}^{0}=0$. Then the TBC operator of the second-order approximation
(22) takes the form

$\Lambda_{2}^{n}=\Lambda_{1}^{n}+\mathrm{i}\frac{dV_{j}^{n}}{4}P_{j}^{n}I_{j}^{n},$
(35)

where

$dV_{j}^{n}=\frac{2}{\Delta
x}(q_{j+1}^{n}\overline{q_{J-j}^{n}}-2q_{j}^{n}\overline{q_{J-j}^{n}}+q_{j}^{n}\overline{q_{J-j-1}^{n}}).$

Again, the values of the wave function at the boundaries can be obtained by
solving the system of nonlinear equations with respect to $q_{j}^{n}$, given
as

$\displaystyle\frac{q_{1}^{n}-q_{0}^{n}}{\Delta x}$
$\displaystyle-e^{-\mathrm{i}\pi/4}\sqrt{\frac{2}{\Delta
t}}P_{0}^{n}\sum_{k=0}^{n}\Bigl{(}\beta_{k}\frac{q_{0}^{n-k}}{P_{0}^{n-k}}\Bigr{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\mathrm{i}\frac{dV_{0}^{n}}{4}P_{0}^{n}I_{0}^{n},$
(36a) $\displaystyle\frac{q_{J}^{n}-q_{J-1}^{n}}{\Delta x}$
$\displaystyle+e^{-\mathrm{i}\pi/4}\sqrt{\frac{2}{\Delta
t}}P_{J}^{n}\sum_{k=0}^{n}\Bigl{(}\beta_{k}\frac{q_{J}^{n-k}}{P_{J}^{n-k}}\Bigr{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\mathrm{i}\frac{dV_{J}^{n}}{4}P_{J}^{n}I_{J}^{n}.$
(36b)

Figure 1: Evolution of the soliton profile with TBCs imposed at both (left and
right) end points. Figure 2: The plot of the energy versus time in the finite
domain $x\in[-L,L]$.

##  V Numerical Experiment

We solve the nonlocal nonlinear Schrödinger equation given by Eq. (1) on the
finite interval $[-L,L]$, and impose the TBC with the first-order
approximation (21) on the left ($x=-L$) and right ($x=L$) boundaries. For the
initial condition, we choose the sum of two analytic solutions (3) that are
symmetric about the origin of the spatial coordinate ($x=0$)

$q(x,0)=\frac{\alpha_{1}}{2}\sqrt{\frac{3\mathrm{i}}{\alpha_{1}\beta_{1}}}\bigg{(}G_{+}(x)+G_{-}(x)\bigg{)},$
(37)

where

$G_{\pm}(x)=\frac{e^{\mp 5\mathrm{i}(x\pm
L/2)}}{g_{r}^{\pm}+\mathrm{i}g_{i}^{\pm}},$
$g_{r}^{\pm}(x)=\cosh\Bigl{(}\frac{-3(x\pm
L/2)-\Delta_{R}}{2}\Bigr{)}\cos\Bigl{(}\frac{\Delta_{I}}{2}\Bigr{)},$
$g_{i}^{\pm}(x)=\sinh\Bigl{(}\frac{-3(x\pm
L/2)-\Delta_{R}}{2}\Bigr{)}\sin\Bigl{(}\frac{\Delta_{I}}{2}\Bigr{)}$

with parameters $\alpha_{1}=1.13+1.13\mathrm{i}$,
$\beta_{1}=1.13-1.13\mathrm{i}$,
$\Delta_{R}=\log\big{(}\frac{|\alpha_{1}|^{2}|\beta_{1}|^{2}}{9}\big{)}$ and
$\Delta_{I}=-\frac{\mathrm{i}}{2}\log\big{(}\frac{\alpha_{1}\beta_{1}}{\alpha_{1}^{*}\beta_{1}^{*}}\big{)}$.
This choice of double solitons is made to avoid a vanishing of the norm and
energy quantities given by Eq. (II).

In our experiments, we chose the following system parameters: $L=6$, the
discretization parameters $\Delta x=0.014$ and $\Delta t=0.002$. The evolution
of the left and right traveling solitons is shown in Fig. 1, from which it can
be seen that the solitons leave the computational domain almost without any
reflection.

Following our simulation, we plot the time dependence of the energy of the
solitons restricted to the computational domain. For this purpose, we
discretize the energy in Eq. (II) by

$E_{n}=\frac{1}{2\Delta
x}\sum_{j=1}^{J-1}\bigg{[}(q_{j+1}^{n}-q_{j-1}^{n})(\overline{q_{J-j-1}^{n}}-\overline{q_{J-j+1}^{n}})\\\
+\Delta x^{2}(q_{j}^{n}\overline{q_{J-j}^{n}})^{2}\bigg{]}.$ (38)

Figure 3: The plot of the absolute error $||\mathcal{E}||_{2}^{2}$ versus
time. The considered time interval is the period of interest, when solitons
encounter boundaries.

The time dependence of the energy in the computational domain is shown in Fig.
2. This plot shows that the energy vanishes, which means that there are almost
no (reflected) waves as time passes.

Finally, we show that the inclusion of the second-order approximation term in
the TBC can slightly improve the results. To this end, we calculate the
absolute error $\mathcal{E}$, i.e., the difference between the numerical
solution with TBC imposed at $x=\pm L$ and the numerical solution for the
extended interval $[-2L,2L]$ (such that boundaries are not reached within the
considered time frame) restricted to $[-L,L]$, measured with the $L^{2}$-norm
(discretized by the trapezoidal rule)

$||\mathcal{E}||_{2}^{2}=\Delta x\Bigg{[}\sum_{j=1}^{J-1}\Delta
q_{j}^{n}\Delta\overline{q_{J-j}^{n}}+\\\ \frac{1}{2}(\Delta
q_{0}^{n}\Delta\overline{q_{J}^{n}}+\Delta
q_{J}^{n}\Delta\overline{q_{0}^{n}})\Bigg{]},$ (39)

where $\Delta q_{j}^{n}=q_{j}^{n}-Q_{j}^{n}$ and $Q_{j}^{n}$ is the numerical
solution for the extended interval. This is done to exclude the discretization
error caused by the finite-difference scheme and to compare only errors caused
by the approximations of TBCs. The plot of this error versus time for the time
period of solitons’ leaving is shown in Fig. 3.

Figure 4: Evolution of the asymmetric soliton profile given by Eq. (40) with
TBCs imposed at both (left and right) end points.

Here we also consider the asymmetric case, i.e. the case of asymmetric
solitons. To do this, we use variational solutions in the numerical solution
of the NNLS equation with TBC as initial conditions, which are described in
Ref. Hadi2019 :

$\displaystyle q(x,0)=\sum_{j=1}^{2}q_{j}(x,0),$ $\displaystyle
q_{j}(x,0)=A_{j}\exp(\mathrm{i}B_{j})\,\operatorname{sech}[C_{j}(x-X_{j})]$
$\displaystyle\times\exp\bigl{[}\mathrm{i}D_{j}(x-X_{j})^{2}+\mathrm{i}E_{j}(x-X_{j})\bigr{]},$
(40)

where $A_{1}=A_{2}=1$, $B_{1}=0$, $B_{2}=0.1$, $C_{1}=C_{2}=1$,
$D_{1}=D_{2}=0$, $E_{1}=-E_{2}=2$, $X_{1}=-X_{2}=5$. The evolution of the
traveling asymmetric soliton is shown in Fig. 4. It shows similar dynamics as
the symmetric counterpart, i.e. no reflection is visible. As an additional
confirmation of the reflectionless propagation of the asymmetric soliton, we
have plotted in Fig. 5 the time dependence of the energy confined to the
finite interval $[-L,L]$. The plot shows that TBC also works in these cases.

Figure 5: Energy vs. time in the finite domain $x\in[-L,L]$ for the case of an
asymmetric soliton evolution.

##  VI Conclusions

In this work, we have derived transparent boundary conditions (TBCs) for the
nonlocal nonlinear Schrödinger equation using the so-called potential
approach. Such boundary conditions allow to obtain the solution of an initial
value problem given in an interval, which is equal to the solution of the
problem for the whole space confined in this interval. The discretization of
the derived TBCs and their numerical implementation were presented in detail.

The confirmation of the nearly reflectionless transition by artificial
boundaries was achieved by the simulation of traveling solitons. The time
dependence of the energy in the computational domain was calculated to verify
the obtained results. Although the first-order approximation of the TBC shows
good results, the additional second-order terms are also considered to show
that the absolute error decreases in this case (which is natural). Although
TBCs are commonly used for numerical simulations, they can also be explained
from a physical point of view: The incoming wave does not “feel” the boundary
where the TBC is imposed, which ensures that there is no or minimal loss in
the transmission of waves from one domain to another.

The above model can be used for the development and design of PT-symmetric
optical waveguides that allow quasi-reflectionless propagation of solitons.
The practical application of such functional materials in optoelectronic
devices would allow to save resources and improve performance by reducing
signal losses. Finally, we note that the above consideration, similar to
Jambul1 , can be directly extended to optical waveguide networks by
determining physically relevant conditions for the transparency of the
branching points of the network. Such structures are even more attractive from
the point of view of optoelectronic applications. A corresponding study is
currently in progress.

###### Acknowledgements.

The work is supported by the grant of the Ministry for Innovation Development
of Uzbekistan (Ref. No. F-2021-440). One of the authors (DM) thanks the
Associates Program of the Abdus Salam ICTP for his hospitality during his
visit.

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