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# Scattering of linear waves on a soliton

A.S. Dmitriev<EMAIL_ADDRESS>Institute for Nuclear Research of
the Russian Academy of Sciences, Moscow 117312, Russia  E.A. Dmitrieva
<EMAIL_ADDRESS>Lomonosov Moscow State University, Faculty of Physics,
Moscow 119991, Russia  Institute for Nuclear Research of the Russian Academy
of Sciences, Moscow 117312, Russia  A.G. Panin<EMAIL_ADDRESS>Institute
for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia

###### Abstract

We have shown that the wave scattering by a soliton occurs in a peculiar way.
The nonlinear interaction leads to the generation of waves with frequencies
that are multiples of the frequency of the incident wave, minus the frequency
of the soliton, while the soliton experiences growth due to the absorption of
particles from the incoming wave. We propose an analytical approach based on
the Born approximation to calculate the amplitudes of scattered waves and,
consequently, the number of particles absorbed by the soliton. This approach
is validated through direct comparison with the results of numerical
simulation.

##  I Introduction

Solitons are stable bound states which exist on a classical as well as quantum
mechanical levels in nonlinear field theories. They have a finite stable shape
in space, which, unlike wave packets formed by superpositions of plane waves,
preserves over time and during free propagation. The dispersion in the
solitons is balanced by focusing nonlinear effects. Since their first
observation on the surface of water Russell (1844), solitons have been widely
studied both theoretically and experimentally in many branches of physics such
as optics Kivshar and Agrawal (2003), plasma physics Ichikawa (1979),
condensed matter physics Bishop  _et al._ (1980), cosmology Kusenko and
Shaposhnikov (1998) and other natural sciences.

There are several exactly integrable nonlinear theories, in which solitons can
be obtained by means of the inverse scattering transform Lax (1968); Zakharov
and Shabat (1972). These theories play an exceptional role because they
provide universal mathematical tools for studying some general physical
phenomena Novikov  _et al._ (1984); Drazin and Johnson (1989). In particular,
in exactly integrable case the interactions between colliding soliton and wave
packet or another soliton are purely elastic Zakharov and Shabat (1973). The
soliton recovers its exact initial shape and velocity after a collision.

In real physical problems, however, various perturbations usually occur that
violate the exact integrability. These not only modify the soliton properties,
but also, under certain conditions, lead to completely new phenomena Kivshar
and Malomed (1989). In particular, collisions between solitons under influence
of higher order nonlinear terms exhibit a variety of outcomes, including
fusion, fission, annihilation Gatz and Herrmann (1992); Tikhonenko  _et al._
(1996); Nguyen  _et al._ (2014) and emission of quasi-linear waves Kivshar and
Malomed (1989). Another example is the transmissions of solitons through
linear and nonlinear inhomogeneities which proceeds with increased rates under
certain conditions Garnier and Abdullaev (2006).

In this paper, we consider the scattering of linear high-frequency waves on
non-topological solitons in the model of a nonlinear non-integrable
Schrödinger equation. We show that, in contrast to the integrable case in
which the incoming wave causes a shift in the soliton’s position Haus  _et
al._ (1996), the presence of non-integrable nonlinear interaction leads to a
non-trivial scattering picture. Nonlinear interactions give rise to waves of
double, triple, and so on, frequencies, while the soliton grows by absorbing
particles from the incoming wave. We propose an analytical method for
describing the scattering pattern of high-frequency waves on a soliton. To
validate the method, we compare its predictions with the results of numerical
simulations and observe a strong agreement.

The paper is organized as follows. In Section II we present the general setup
and introduce the notations that will be used throughout the paper. An
explicit numerical example of scattering of a wide Gaussian wave packet on a
soliton is considered in Section III. In Section IV we address the computation
of the amplitudes of the waves transmitted through and reflected from the
soliton, and compare analytic predictions with the data of numerical
simulations. In Section V we discuss the results. Appendix A includes details
of method used in numerical simulations.

##  II Setup

We start by considering the nonlinear Schrödinger equation in dimensionless
variables

$i\partial_{t}\psi=-\frac{1}{2}\partial^{2}_{x}\psi+V(|\psi|^{2})\psi\;,$ (1)

where the last term introduces nonlinearity. Evolution governed by Eq. (1)
conserves a number of quantities: the particle number (norm) $N$,

$N=\int dx\,|\psi|^{2}\;,$ (2)

the energy of the system $E$,

$E=\int
dx\,\left[\frac{1}{2}|\partial_{x}\psi|^{2}+\int\limits_{0}^{|\psi|^{2}}ds\,V(s)\right]\;,$
(3)

and the total momentum $P$,

$P=i\int
dx\,\left[\psi\partial_{x}\psi^{*}-\psi^{*}\partial_{x}\psi\right]\;.$ (4)

Suppose that Eq. (1) admits a soliton solution of the form

$\psi_{s}(t,x)=f(x)\mathrm{e}^{-i\gamma t}\;,$ (5)

where real function $f(x)$ gives the soliton profile. It satisfies
$f(-x)=f(x)$ and $f(x)\to 0$ for $|x|\to\infty$. The soliton frequency is
negative, $\gamma<0$. It determines the binding energy of nonrelativistic
particles in a soliton according to the relation

$dE=\gamma dN\;,$ (6)

which directly follows from Eqs. (1), (5). The solution is supposed to be
classically stable in according to the Vakhitov-Kolokolov stability criterion
Vakhitov and Kolokolov (1973).

Below we will consider a wave packet moving from large negative $x$ to the
right and scattering on a soliton (5) centered at $x=0$,

$\psi_{0}=A(t,x)\mathrm{e}^{-i\omega
t+ipx+i\phi_{0}}\;,\qquad\omega=\frac{p^{2}}{2}\;,$ (7)

where $A$ specifies a wave packet shape and $\phi_{0}$ is a constant phase. In
what follows, we will assume that $|V(A^{2})|\ll\omega$, so the wave packet
freely propagates to the region of interaction with the soliton. In addition,
we will consider wave packets with a width $\sigma$ much larger than the
wavelength, $\sigma\gg 2\pi/p$ so that the change in its shape during movement
can be neglected.

##  III Numerical illustration

To begin with, let us consider the scattering of a wave packet on a soliton
numerically. For this purpose, we select a potential

$V(|\psi|^{2})=-\lambda|\psi|^{2}+g|\psi|^{4}\;.$ (8)

In what follows we will use $\lambda=g=1$. Nontrivial couplings can be
restored by the redefinitions $\psi\to\psi\,\sqrt{\lambda/g}$, $x\to
x\,\lambda/\sqrt{g}$, $t\to t\,\lambda^{2}/g$.

Figure 1: The soliton profile (9) for different frequencies.

The nonlinear Schödinger equation with the potential (8) admits the soliton
solution (5), which exists for $-3/16<\gamma<0$. It is remarkable that the
soliton profiles are known analytically,

$f(x)=\frac{2\sqrt{-\gamma}}{\left(1+\sqrt{1+\frac{16}{3}\gamma}\,\mathrm{cosh\left(2\sqrt{-2\gamma}x\right)}\right)^{\frac{1}{2}}}\;.$
(9)

Function $f(x)$ for different $\gamma$ is plotted in the Fig. 1.

Now, let us consider the scattering of a wide Gaussian wave packets on the
soliton. We solve numerically the nonlinear Schrödinger equation (1) with the
potential (8) starting at $t=0$ with the initial wave function
$\psi=\psi_{0}+\psi_{s}$. Here $\psi_{s}$ is the wave function of the soliton
centered at $x=0$ and $\psi_{0}$ is an incident Gaussian wave packet, Eq. (7)
with

$A(0,x)=A\,\mathrm{e}^{-(x-x_{0})^{2}/2\sigma^{2}}\;,$ (10)

localized at large negative $x\simeq x_{0}$ far away from the soliton. The
details of the numerical method are presented in Appendix A. The results are
demonstrated in Fig. 2 and in the movie Dmitriev  _et al._ (2023).

Figure 2: Results of numerical simulation of scattering of a Gaussian wave
packet ($A=0.01$, $x_{0}=-1500$, $\sigma=100$ and $p=1$) on a soliton
($\gamma=-0.1874375$). Frames (a)-(c) displays $|\psi|$ at different times.

The incident wave packet strikes the soliton and then separates into a
reflected and a transmitted parts. Each part consists of several wave packets
moving with different velocities (see Fig. 2c). In particular, the wave
packets with maximal amplitudes, transmitted $\psi_{1}^{(tr)}$ and reflected
$\psi_{1}^{(re)}$, move at the velocity of the incident wave packet
$v_{1}=\sqrt{2\omega}$. They are the result of quantum-mechanical-like
scattering on a potential barrier produced by a soliton. The wave packets
denoted by $\psi_{2}$ and $\psi_{3}$ in Fig. 2c have frequencies
$\omega_{2}=2\omega-\gamma$, $\omega_{3}=3\omega-\gamma$ and velocities
$v_{2}=\sqrt{4\omega-2\gamma}$, $v_{3}=\sqrt{6\omega-2\gamma}$ respectively
and appear due to nonlinear interaction of an incident wave packet and a
soliton. We have numerically verified that for $A\lesssim 0.01$ the amplitude
of $\psi_{1}^{(re)}$ scales as $A$, while the amplitudes of $\psi_{2}$ and
$\psi_{3}$ are proportional to $A^{2}$ and $A^{3}$ respectively. This fact
will be used in the next section to construct a perturbation theory with
respect to $A$.

Let us demonstrate that in the scattering picture described above a certain
number of particles from the incident wave packet get stuck in the soliton
region. Initially, the total number of particles is the sum of the number of
particles in the incident wave packet $N_{1}^{in}$ and in the soliton
$N_{s}^{in}$. After scattering the number of particles in the wave packets
$\psi^{(re)}_{2}$ and $\psi_{3}^{(tr)}$ is exponentially small and can be
neglected, see Fig. 2c. Thus, we have

$N_{1}+N_{2}^{(tr)}+N_{s}=N_{1}^{in}+N_{s}^{in}\;,$ (11)

where $N_{1}$ and $N_{2}^{(tr)}$ are the number of particles in the wave
packets $\psi_{1}^{(tr)}+\psi_{1}^{(re)}$ and $\psi_{2}^{(tr)}$ respectively
and $N_{s}$ takes into account particles in the soliton, including localized
states that can be excited as a result of the scattering. Similarly, for the
total energy one can write

$E_{1}+E_{2}^{(tr)}+E_{s}=E^{in}_{1}+E^{in}_{s}\;,$ (12)

where $E_{s}$ also includes the kinetic energy of the soliton after scattering
and the energy of the localized modes. The energy of the wave packets,
considering that $|V(A^{2})|\ll\omega$, equals to the product of the frequency
by the number of particles, $E=\omega N$. Then from Eqs. (11), (12) we can
obtain

$(\omega-\gamma)\Delta N_{1}+(2\omega-\gamma)\Delta N_{s}=\Delta E_{s}\;,$
(13)

where $\Delta N_{1,s}=N_{1,s}-N^{in}_{1,s}$ and $\Delta
E_{s}=E_{s}-E^{in}_{s}$. Let us show that Eq. (13) has no solution with
$\Delta N_{s}=0$. Indeed, in this case the number of particles in the wave
packets $\psi_{1}$ decreases by $N_{2}^{(tr)}$ after scattering (see Eq.
(11)), thus the first term in the right-hand side of Eq. (13) is negative. On
the other hand, a soliton with constant number of particles can only increase
its energy, for example, by acquiring kinetic energy. Thus one concludes that
$\Delta E_{s}$ must be positive, which contradicts Eq. (13).

Figure 3: The change in the number of particles of the soliton $\Delta N_{s}$
(red solid lines) with $\gamma=-0.125$ (a) and with $\gamma=-0.1874375$ (b)
and the number of particles in the $\psi_{2}^{(tr)}$ wave packet
$N_{2}^{(tr)}$ (blue dashed lines) as a function of the incident wave packet
frequency. The green dotted lines represent the estimation for the norm
$\psi_{2}^{(tr)}$ derived from Eq. (30) (see Sec. IV). The other parameters
are the same as in Fig. 2.

The above argument indicates that the scattering of the wave packet is
accompanied by an increase in the number of particles in the soliton. This is
confirmed by numerical simulations. The change in the norm of the soliton as a
function of the incident wave packet frequency is presented in Fig. 3 by solid
line. Surprisingly that this change coincides with high accuracy with the
number of particles in the transmitted wave packet $\psi_{2}^{(tr)}$ (dahsed
line in Fig 3),

$\Delta N_{s}=N_{2}^{(tr)}.$ (14)

Considering that $\psi_{2}^{(tr)}\propto A^{2}$ the number of particles
captured by the soliton is parametrically small, $\Delta N_{s}\propto A^{4}$.

Substituting Eq. (14) into Eq. (11) we obtain $\Delta N_{1}=-2\Delta N_{s}$.
This equation together with Eq. (13) leads to $\Delta E_{s}=\gamma\Delta
N_{s}$ (cf Eq. (6)). This relation implies that the particles captured by the
soliton settle in the ground state with energy $\gamma$, while the exited
soliton modes remain unoccupied.

Let us note that Eq. (14) is remarkable in the following respect. Instead of
calculating the change in the soliton’s number of particles one can find the
norm of the wave packet $\psi_{2}^{(tr)}$ and use Eq. (14). The calculation of
the norm of $\psi_{2}^{(tr)}$ in turn can be performed analytically, as we
will show in the next section.

##  IV Perturbation theory

Now, we construct a perturbation theory based on the small amplitude $A$ of a
wave packet scattered on a soliton. First, we consider $A\ll f(x)$. Second,
for simplicity we will neglect the dependence of $A$ on $x$, assuming that the
width of the wave packet is much larger than the size of the soliton. In this
case, the wave packet can be considered as a plane wave. Third, we assume that
$|V(A^{2})|\ll\omega$, so the wave freely propagates to the region of
interaction with the soliton.

Thus, the solution of the equation of motion (1) can be found in the form of a
series

$\psi=\psi_{s}+\psi_{1}+\psi_{2}+\dots\;,$ (15)

where $\psi_{n}\propto A^{n}$. Consider each of these perturbations
separately.

###  IV.1 First order

Substituting Eq. (15) into Eq. (1) and combining the terms of order of $A$ we
arrive to the liner equation

$i\partial_{t}\psi_{1}+\frac{1}{2}\partial^{2}_{x}\psi_{1}=\left(V^{0}+V^{(1)}|\psi_{s}|^{2}\right)\psi_{1}+V^{(1)}\psi_{s}^{2}\psi_{1}^{*}\;,$
(16)

where we expand the potential in a Taylor series and introduce the notations

$\begin{split}&V^{(0)}(x)=V(|\psi_{s}|^{2})\;,\\\
&V^{(i)}(x)=\left.\frac{d^{i}V(s)}{ds^{i}}\right|_{s=|\psi_{s}|^{2}}\;.\end{split}$
(17)

For the scattering problem under consideration the wave function $\psi_{1}$
satisfies the boundary condition (7) at $x\to-\infty$ with
$A(t,x)=\mathrm{const}$. Eq. (16) describes the incident wave scattering on
the soliton in the leading order. It can be solved analytically for several
specific models. In the general case $\psi_{1}$ can be found for
$\omega\gg-\gamma$. In this limit the last term in Eq. (16) oscillates rapidly
compared to the others, which can be seen by multiplying the equation by
$\mathrm{e}^{i\omega t}$, and can be neglected. As a result, we arrive to the
linear Schödinger equation for the wave function $\psi_{1}$ in the effective
potential $V^{eff}_{1}(x)$, where

$V^{eff}_{1}(x)=V^{(0)}+V^{(1)}f^{2}\;.$ (18)

This equation can be solved by means of semiclassical approach, which is
applicable in the considered high-frequency limit. Introducing the classical
momentum,

$p_{\omega}(x)=\sqrt{2(\omega-V^{eff}_{1}(x))}\;,$ (19)

for the transmitted wave one writes

$\begin{split}&\psi_{1}^{(tr)}=A_{1}^{(tr)}(x)\cdot\mathrm{e}^{-i\omega
t+iS_{\omega}(x)}\;,\\\
&A_{1}^{(tr)}(x)=\frac{A\sqrt{p}}{\sqrt{p_{\omega}(x)}}\;,\end{split}$ (20)

where

$S_{\omega}(x)=\int^{x}p_{\omega}(y)dy$ (21)

is the time-independent part of the classical action and the amplitude is
normalized in accordance with Eq. (7).

Figure 4: The amplitude of the left moving wave $\psi_{1}^{(re)}$ created in
the scattering on the soliton with $\gamma=-0.125$ (a) and with
$\gamma=-0.1874375$ (b) as a function of the incident wave frequency. Solid
lines represent theoretical results, see Eq. (25), circles show the results of
numerical simulations.

The reflected wave also can be found by semiclassical method. However, in this
paper we apply another approach base on the Born approximation following Ref.
Chan  _et al._ (2022), which is simple and sufficiently accurate. Let us
substitute $\psi_{1}=\psi_{0}+\delta\psi_{1}$, where $\psi_{0}$ is a freely
propagating plane wave (7), into the linear Schrödinger equation. The solution
for $\delta\psi_{1}$ can be written as

$\delta\psi_{1}=\int
dt^{\prime}dx^{\prime}G(t-t^{\prime},x-x^{\prime})V^{eff}_{1}(x^{\prime})\psi_{0}(t^{\prime},x^{\prime})\;,$
(22)

where $G$ is the retarded Green’s function. As we discussed above, the
evolution of a high-frequency waves in potential $V_{1}^{eff}$ proceeds
semiclassically. In this regime for the Green’s function one writes

$G(t,x,x^{\prime})=\frac{1}{(2\pi)^{2}}\int d\Omega
dk\frac{\psi_{\omega_{k}}(x)\psi^{*}_{\omega_{k}}(x^{\prime})}{\Omega-\omega_{k}+i0}\mathrm{e}^{-i\Omega
t}\;,$ (23)

where $\psi_{\omega_{k}}$ is the time-independent semiclassical wavefunction,

$\psi_{\omega_{k}}(x)=\frac{\sqrt{k}}{\sqrt{p_{\omega_{k}}(x)}}\mathrm{e}^{iS_{\omega_{k}}(x)}\;,\qquad\omega_{k}=\frac{k^{2}}{2}\;.$
(24)

Performing integration with respect to $t^{\prime}$, $\Omega$ and $k$ in Eq.
(22) for the reflected wave we obtain

$\begin{split}&\psi_{1}^{(re)}=A_{1}^{(re)}(x)\cdot\mathrm{e}^{-i\omega
t-iS_{\omega}(x)}\;,\\\
&A_{1}^{(re)}(x)=\frac{-iA}{\sqrt{p_{\omega}(x)}}\int_{x}^{\infty}dx^{\prime}\frac{V^{eff}_{1}(x^{\prime})\,\mathrm{e}^{ipx^{\prime}+iS_{\omega}(x^{\prime})}}{\sqrt{p_{\omega}(x^{\prime})}}\;.\end{split}$
(25)

Note, that the integrand in Eq. (25) oscillates rapidly with changing
$x^{\prime}$ so that the amplitude of the reflected wave is expected to be
exponentially small111The integration in this case can be done by the
stationary phase method..

In Fig. 4 we compare predictions of Eq. (25) for the ampitude of
$\psi_{1}^{(re)}$ at large negative $x$ with numerical results in the model
(8) presented in Sec. III. For this purpose the integral in Eq. (25) was
computed numerically. One observes that the analytic predictions (lines in the
figure) coincide with the numerical results (circles in the figure) in the
high frequency limit $\omega\gg-\gamma$ but deviate from them at small
$\omega$.

###  IV.2 Second order

Now let us consider the scattering of the wave on the solitons in the second
order in its amplitude. Substituting Eq. (15) into Eq. (1) and collecting the
term of order of $A^{2}$ we obtain

$\begin{split}&i\partial_{t}\psi_{2}+\frac{1}{2}\partial^{2}_{x}\psi_{2}-V^{eff}_{1}\psi_{2}-V^{(1)}\psi_{s}^{2}\psi^{*}_{2}\\\
&=\left(V^{(1)}+\frac{1}{2}V^{(2)}|\psi_{s}|^{2}\right)\psi_{1}^{2}\psi_{s}^{*}\\\
&+\left(2V^{(1)}+V^{(2)}|\psi_{s}|^{2}\right)|\psi_{1}|^{2}\psi_{s}\\\
&+\frac{1}{2}V^{(2)}(\psi_{1}^{*})^{2}\psi_{s}^{3}\;.\end{split}$ (26)

Due to the source in the right-hand side, this equation describes the creation
of a second-order wave function $\psi_{2}$ as a result of the interaction of
an incident wave $\psi_{1}$ with a soliton. There are three terms in the
source that oscillate at different frequencies. Each term generates its own
distinct contribution to $\psi_{2}$ and can be analyzed separately.

Figure 5: The amplitude of the right moving wave $\psi_{2}^{(tr)}$ created by
nonlinear interaction between the soliton with $\gamma=-0.125$ (a) and with
$\gamma=-0.1874375$ (b) and the incident wave as a function of its frequency.
Solid lines represent theoretical results, see Eq. (30), circles show the
results of numerical simulations.

The time-dependence of the first term of the source in Eq. (26) is given by
$\psi_{1}^{2}\psi_{s}^{*}$ multiplier. As a result it supports creation of the
wave $\psi_{2}$ with frequency $\omega_{2}=2\omega-\gamma$ propagating from
the soliton. This process can be described analitically in high-frequency
regime $\omega\gg-\gamma$. In this case the last term in the left-hand side of
Eq. (26) is highly-oscillating thus can be neglected. As a result, we obtain
the linear Schrödinger equation

$i\partial_{t}\psi_{2}+\frac{1}{2}\partial^{2}_{x}\psi_{2}-V^{eff}_{1}\psi_{2}=V^{eff}_{2}\psi_{1}^{2}\mathrm{e}^{i\gamma
t}\;,$ (27)

where we introduce

$V^{eff}_{2}(x)=V^{(1)}f+\frac{1}{2}V^{(2)}f^{3}\;.$ (28)

Its solution for high-frequency waves can be obtained by using semiclassical
Green’s function (23),

$\psi_{2}=\int
dt^{\prime}dx^{\prime}G(t-t^{\prime},x,x^{\prime})V^{eff}_{2}(x^{\prime})\psi^{2}_{1}(t^{\prime},x^{\prime})\mathrm{e}^{i\gamma
t^{\prime}}\;.$ (29)

Here $\psi_{1}=\psi_{1}^{(tr)}+\psi_{1}^{(re)}$. For the wave propagating to
the right the main contribution comes from the transmitted wave (20).
Integrating over $t^{\prime}$, $\Omega$ and $k$ we get

$\begin{split}&\psi_{2}^{(tr)}=A_{2}^{tr}(x)\cdot\mathrm{e}^{-i\omega_{2}t+iS_{\omega_{2}}(x)}\;,\\\
&A_{2}^{tr}(x)=\frac{-iA^{2}\,p}{\sqrt{p_{\omega_{2}}(x)}}\int_{-\infty}^{x}dx^{\prime}\,\frac{V_{2}^{eff}(x^{\prime})\,\mathrm{e}^{2iS_{\omega}(x^{\prime})-iS_{\omega_{2}}(x^{\prime})}}{p_{\omega}(x^{\prime})\sqrt{p_{\omega_{2}}(x^{\prime})}}\;.\end{split}$
(30)

Similarly, from Eq. (29) for reflected wave we get

$\begin{split}&\psi_{2}^{(re)}=A_{2}^{re}(x)\cdot\mathrm{e}^{-i\omega_{2}t-iS_{\omega_{2}}(x)}\;,\\\
&A_{2}^{re}(x)=\frac{iA^{2}}{\sqrt{p_{\omega_{2}}(x)}}\int_{x}^{\infty}dx^{\prime}\,V_{2}^{eff}(x^{\prime})\,\frac{\mathrm{e}^{iS_{\omega_{2}}(x^{\prime})}}{\sqrt{p_{\omega_{2}}(x^{\prime})}}\\\
&\times\left(A_{1}^{(tr)}(x^{\prime})\mathrm{e}^{iS_{\omega}(x^{\prime})}+A_{1}^{(re)}(x^{\prime})\mathrm{e}^{-iS_{\omega}(x^{\prime})}\right)^{2}\;,\end{split}$
(31)

where the contributions from the transmitted and reflected waves of the first
order, given by Eqs. (20), (25), have been taken into account.

Figure 6: The amplitude of the left moving wave $\psi_{2}^{(re)}$ created by
nonlinear interaction between the soliton with $\gamma=-0.125$ (a) and with
$\gamma=-0.1874375$ (b) and the incident wave as a function of its frequency.
Solid lines represent theoretical results, see Eq. (31), circles show the
results of numerical simulations.

The amplitudes of $\psi_{2}^{(tr)}$ and $\psi_{2}^{(re)}$ obtained by
numerical integration in Eqs. (30), (31) for the model (8) in comparison with
the results of numerical simulations are presented in Fig. 5 and Fig. 6
respectively. As one can see from the figures, the analytical results are in
good agreement with the numerical data.

As we discussed in the previous section, the norm of $\psi_{2}^{(tr)}$ equals
the number of particles absorbed by the soliton. As one can see from Fig. 2,
the width of the wave packets leaving the interaction region after scattering
on the soliton coincides with the width of the incident wave packet. Thus,
$\psi_{2}^{(tr)}$ is a Gaussian wave packet, see Eq. (10), with the amplitude
estimated by Eq. (30). Calculating its norm, we obtain an estimate for the
number of particles absorbed by the soliton, which is shown (by green dotted
lines) in Fig. 3.

Now let us discuss the second and third terms of the source in the right-hand
side of Eq. (26). These terms oscillate with negative frequencies, $\gamma$
and $3\gamma-2\omega$ respectively, and may contribute to the emergence of
localized modes associated with changes in the number of particles in the
soliton. However, the analytical search for non-propagating solutions is very
challenging. In this case the last term of the operator in the left-hand side
of Eq. (26) cannot be neglected and its inversion appears to be rather
complicated (see, however, Kovtun (2022)). Given that the change in the number
of particles in the soliton $\propto A^{4}$, following this approach we will
have to solve equations up to the fourth order of perturbation theory. In this
way, the method outlined above for calculation of the soliton norm change is
simpler and more convenient.

##  V Discussion

In this study, we have demonstrated that wave scattering by a soliton is
accompanied by the generation of waves with frequencies that are multiples of
the incident wave frequency, minus the soliton frequency. Additionally, we
have observed that the soliton undergoes growth by absorbing particles from
the incident wave into its ground state, while the exited states remain
unoccupied. This fact can be understood by adopting the methods of quantum
field theory222It is important to emphasize that quantum physics is not
essential for the soliton-wave interaction. The nonlinear self-interaction
term $|\psi|^{4}$ is responsible for scattering of two particles from the
incident wave packet into two other states. The occupation number of the
soliton ground state is large, thus the transition to this state is strongly
Bose-enhanced. Based on the Born approximation, a wave packet with a frequency
of $2\omega-\gamma$ is also produced in this process. According to the
conservation laws of total energy and total number of particles, it follows
that the norm of this wave packet should be equal to the number of particles
absorbed by the soliton.

The scattering of a wide, low-amplitude soliton on a high-amplitude soliton
proceeds similarly. We have verified that in this case a reflected wave packet
is produced, as well as wave packets with frequencies that are integer
multiples of the incident soliton frequency, minus the frequency of the
stationary soliton. The shape of the wave packets precisely matches the shape
of the incident soliton. Their amplitudes can be estimated by the method
developed in Section IV.

It should also be noted, that we expect similar processes in the scattering of
a plane wave on a soliton in 3+1 dimensions, and our analytical calculation
method can be easily generalized to this case.

###### Acknowledgements.

We thank M. Smolyakov and E. Nugaev for numerous discussions and interest in
the work. The work was supported by the grant RSF 22-12-00215 and, in its
numerical part, by the “BASIS” foundation. Numerical calculations were
performed on the Computational Cluster of the Theoretical Division of INR RAS.

##  Appendix A Numerical scheme

We solve numerically the nonlinear Schrödinger equation (1) using 4-th order
pseudo-spectral operator-splitting method Yoshida (1990); McLachlan (1993).
This method is unitary, stable, symplectic and therefore exceptionally
accurate for computing small-valued high-order effects. The wave function at
the next time step is given by

$\psi(t+\Delta t)=\hat{\cal U}\,\psi(t)\;,$ (32)

where the evolution operator is replaced by discrete formula

$\hat{\cal U}=\prod_{\alpha=1}^{4}\mathrm{e}^{-id_{\alpha}\Delta
t\,V_{\alpha}(x)}\mathrm{e}^{-ic_{\alpha}\Delta t\,\hat{p}^{2}/2}\;+O(\Delta
t^{5}).$ (33)

Here the product is ordered right-to-left, its parameters $c_{\alpha}$ and
$d_{\alpha}$ are given in Yoshida (1990); McLachlan (1993), and the
”potentials” $V_{\beta}$ are computed using the “current” field $\psi(t)$
(multiplied by all operators with $\alpha\leq\beta$), i.e.
$V_{\beta}=V(|\psi|^{2})$. Eq. (33) breaks the time interval $\Delta t$ into
$c$\- and $d$-sub-intervals, on which the ”kinetic” part and the ”potential”
part of the evolution operator are used respectively.

The numerical application of the formula (33) is as follows. We introduce
uniform lattice with $N$ sites in the finite box of size $L$. The field values
$\psi(t,x_{j})$ are stored on the lattices sites $x_{j}=-L/2+j\Delta x$,
$0\leq j<N$, where $\Delta x=L/N$. Typically, we use $L=12000$ with $N=2^{15}$
and switch to another values for resolution tests.

Time evolution of $\psi(x_{j})$ is calculated by sequentially acting with
operators in Eq. (33). The action of the ”kinetic” parts of the evolution
operator $\mathrm{e}^{-ic_{\alpha}\Delta t\,\hat{p}^{2}/2}$ is performed as
follows. First, we Fourier-transform the field,
$\psi(x_{j})=\sum_{k}\tilde{\psi}(p_{k})\mathrm{e}^{-ip_{k}x_{j}}$, using Fast
Fourier transform algorithm, where $\tilde{\psi}(p_{k})$ is the image at
discrete momenta $p_{k}=2\pi k/L$, $-N/2<k\leq N/2$. After that multiplication
by ”kinetic” part of the evolution operator corresponds to phase rotation
$\tilde{\psi}(p_{k})\to\tilde{\psi}(p_{k})\,\mathrm{e}^{-ic_{1}\Delta
t\,p_{k}^{2}/2}$. Then we return to the coordinate representation for the
$\psi$ with the inverse Fourier transform and multimply it by the ”potential”
part of the evolution operator. These actions are repeated the required number
of times according to Eq. (33).

Each step of the algorithm is reduced to multiplying the wave function by a
phase, so it exectly (up to truncation errors) preserves the norm. The total
energy is not conserved. However, we have chosen a time step such that the
relative non-conservation of energy in our solutions has never exceeded
$10^{-12}$. This accuracy allows us to observe the appearance of wave packets
with an amplitude up to $10^{-10}$.

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