# Cherenkov radiation and scattering of external dispersive waves by two-color
solitons

Ivan Oreshnikov<EMAIL_ADDRESS>Max Planck Institute for
Intelligent Systems, Max-Planck-Ring 4, 72076 Tübingen, Germany  Oliver
Melchert  Cluster of Exellence PhoenixD, Welfengarten 1, 30167 Hannover,
Germany  Leibniz Universität Hannover, Institute of Quantum Optics,
Welfengarten 1, 30167 Hannover, Germany  Hannover Centre for Optical
Technologies, Nienburger Strasse 17, 30167 Hannover, Germany  Stephanie Willms
Cluster of Exellence PhoenixD, Welfengarten 1, 30167 Hannover, Germany
Leibniz Universität Hannover, Institute of Quantum Optics, Welfengarten 1,
30167 Hannover, Germany  Surajit Bose  Cluster of Exellence PhoenixD,
Welfengarten 1, 30167 Hannover, Germany  Leibniz Universität Hannover,
Institute of Photonics, Nienburger Strasse 17, 30167 Hannover, Germany  Ihar
Babushkin  Cluster of Exellence PhoenixD, Welfengarten 1, 30167 Hannover,
Germany  Leibniz Universität Hannover, Institute of Quantum Optics,
Welfengarten 1, 30167 Hannover, Germany  Ayhan Demircan  Cluster of Exellence
PhoenixD, Welfengarten 1, 30167 Hannover, Germany  Leibniz Universität
Hannover, Institute of Quantum Optics, Welfengarten 1, 30167 Hannover, Germany
Hannover Centre for Optical Technologies, Nienburger Strasse 17, 30167
Hannover, Germany  Uwe Morgner  Cluster of Exellence PhoenixD, Welfengarten 1,
30167 Hannover, Germany  Leibniz Universität Hannover, Institute of Quantum
Optics, Welfengarten 1, 30167 Hannover, Germany  Hannover Centre for Optical
Technologies, Nienburger Strasse 17, 30167 Hannover, Germany  Alexey Yulin
Department of Physics, ITMO University, Kronverskiy pr. 49, 19701 St.
Petersburg, Russia

###### Abstract

For waveguides with two separate regions of anomalous dispersion, it is
possible to create a quasi-stable two-color solitary wave. In this paper we
consider how those waves interact with dispersive radiation, both generation
of Cherenkov radiation and scattering of incident dispersive waves. We derive
the analytic resonance conditions and verify them through numeric experiments.
We also report incident radiation driving the internal oscillations of the
soliton during the scattering process in case of an intense incident
radiation. We generalize the resonance conditions for the case of an
oscillating soliton and demonstrate how one can use the scattering process to
probe and excite an internal mode of two-color soliton molecules.

##  I Introduction

Solitary waves or solitons are localized nonlinear excitations that preserve
their shapes during evolution. In the systems close to integrable ones
solitons are proven to be robust and in many cases different interactions
result only in the variation of the soliton parameters, such as its intensity
or frequency. In a certain sense solitons can be called “eigenmodes of the
nonlinear problem”, meaning that the dynamics of the system can be considered
as a set of solitons interacting with quasi-linear dispersive waves. Due to
their robustness, solitons are of fundamental as well as practical importance,
for example, in the context of optical supercontinuum generation Dudley  _et
al._ (2006); Skryabin and Gorbach (2010); Dudley and Taylor (2010) or soliton
fiber lasers Mollenauer and Stolen (1984); Kafka  _et al._ (1989); Matsas  _et
al._ (1992); Turitsyn  _et al._ (2016).

The mutual interaction of optical solitons in fibers can enable the formation
of bound states, often referred to as soliton molecules. They can be realized
via dispersion engineering in the framework of the standard nonlinear
Schrödinger equation (NSE) and consist of two pulses that maintain a fixed
separation in time Stratmann  _et al._ (2005); Hause  _et al._ (2008).
Further, optical soliton clusters have been discovered and studied in a large
variety of physical systems described by different equations such as the
generalized nonlinear Schrödinger equation Buryak and Akhmediev (1995);
Malomed (1993); Zavyalov  _et al._ (2009), coupled NLSs Kivshar and Malomed
(1989); Abdullaev  _et al._ (1989), the Ginzburg-Landau equation Akhmediev
_et al._ (1997); Haelterman  _et al._ (1997), Lugiato-Lefever equation Weng
_et al._ (2020) and many others Neshev  _et al._ (2003); Afanasjev and
Vysloukh (1994); Tsoy and Abdullaev (2003); Chen  _et al._ (2000); Kartashov
_et al._ (2002); Xu  _et al._ (2019); Liu  _et al._ (2018); Krupa  _et al._
(2017); Al Khawaja and Boudjemâa (2012); Grelu and Akhmediev (2012); Turmanov
_et al._ (2015). The aforementioned bound states of solitons have a single
central frequency and the whole spectrum is localized around this frequency.

Another possibility to observe soliton molecules is to provide interaction
between solitons having their carrier frequencies well detuned from each
other. To make the interaction efficient the velocities of the solitons must
be close. In the case of scalar solitons this means that we need higher order
dispersion with two different frequency ranges where the solitons can be
accommodated. These solitons were recently reported in Melchert  _et al._
(2019); Tam  _et al._ (2020); Melchert and Demircan (2021); Melchert  _et al._
(2021) for pulses propagating in conservative fibers with higher order
dispersion. Similar solitons were discovered also in mode-locked cavity lasers
Lourdesamy  _et al._ (2022) and in coherently pumped ring resonators Moille
_et al._ (2018); Melchert  _et al._ (2020).

In the case of nonintegrable systems, solitons can interact with dispersive
waves (DWs) of low intensity and this interaction leads to interesting
physical phenomena, as the efficient generation of new frequencies Yulin  _et
al._ (2004); Skryabin and Yulin (2005); Efimov  _et al._ (2004, 2005, 2006).
It should be noted that this effect is closely related to the optical push
broom effect de Sterke (1992). Such resonant scattering can be cascaded,
enriching optical supercontinuum spectra significantly Gorbach and Skryabin
(2007). Similar effects were studied in Refs. Demircan  _et al._ (2013, 2012);
Philbin  _et al._ (2008), where the term “optical event horizon” was coined.
It was also established that resonant dispersive waves affect the parameters
of the solitons and can lead to dispersive wave mediated acceleration of
solitons Skryabin and Yulin (2005); Demircan  _et al._ (2011, 2013); Yulin
_et al._ (2013); Demircan  _et al._ (2014); Wang  _et al._ (2015); Yulin
(2018). It should be mentioned that resonant scattering of dispersive waves is
also studied for dark solitons Oreshnikov  _et al._ (2015a); Marest  _et al._
(2018); Deng  _et al._ (2018); Rong  _et al._ (2021), and oscillating solitons
Kodama  _et al._ (1994); Conforti  _et al._ (2015); Driben  _et al._ (2015);
Wright  _et al._ (2015); Krupa  _et al._ (2016); Oreshnikov  _et al._ (2017).

The present paper aims to study in detail the interaction of dispersive waves
with two-color soliton molecules, consisting of two bound solitons having well
separated frequencies. In the prior work Melchert  _et al._ (2019) we have
demonstrated how to create quasi-stable configurations of two tightly-coupled
pulses in a dispersion landscape $\beta(\omega)$ with two regions of anomalous
dispersion, separated by a region of normal dispersion. Each of these
subpulses propagates on its own carrier frequency. This coupled state,
especially in the process of initial evolution, sheds dispersive waves that
resemble Cherenkov radiation that is observed for other types of solitary
waves in a vast variety of settings Akhmediev and Karlsson (1995); Afanasjev
_et al._ (1996); Driben  _et al._ (2015); Conforti  _et al._ (2015); Wright
_et al._ (2015); Oreshnikov  _et al._ (2017). And since DW generation is
demonstrated, it is reasonable to anticipate that two-color soliton molecules
colliding with DWs will produce resonant emission in the same way as it occurs
in the case of conventional solitons. Thanks to a more complex structure of
the two-color solitons, one can expect the scattering dynamics to be much
richer compared to the conventional single soliton case. For instance, it is
possible that the internal degrees of freedom can be excited due to the
interaction with a DW and this should strongly affect the dynamics of the
system. Subsequently we confirm this influence, providing a combined
analytical and numerical view of the process.

The paper is structured as follows. In Sect. II we introduce a mathematical
model for the light propagating in a nonlinear fiber with higher order
dispersion and derive the condition of resonant four-wave mixing of the
soliton molecules with DWs. In Sect. III we report the results of the
numerical simulations of the different propagation regimes of bichromatic
soliton molecules. The results of the simulations are compared against
analytical resonance conditions. Section IV discusses the weakly nonlinear
case where the intensity of the dispersive waves is sufficiently large to
modify the interaction between the soliton molecules and the waves. The
excitation of the internal mode is also discussed in this section. The paper
concludes with a summary in Sect. V.

##  II Analytical model

In this section we derive a perturbation theory that explains the resonance
conditions that were proposed before in Melchert  _et al._ (2019), demonstrate
additional Cherenkov radiation mechanisms due to four-wave mixing (FWM)
between the frequency components of the soliton, and then extend this theory
to describe the process of scattering of external waves on the soliton.

We start by considering a non-envelope version of a nonlinear Schrödinger
equation Amiranashvili and Demircan (2010)

$i\partial_{z}\tilde{u}+\beta(\omega)\tilde{u}+\gamma(\omega)\mathcal{F}\left\\{\left|u\right|^{2}u\right\\}_{(\omega>0)}=0.$
(1)

Here and further, $\tilde{u}\equiv\tilde{u}(z,\omega)$ indicates the Fourier
image of the field $u\equiv u(z,t)$ and
$\mathcal{F}\left\\{\cdots\right\\}_{(\omega>0)}$ is the explicit Fourier
transform taken for the positive frequencies only.

To build the perturbation theory let us introduce an ansatz that represents
the solution as a sum of two single-frequency solitary waves $U_{1,2}(z,t)$
and a small residue radiation $\psi(z,t)$

$\displaystyle u(z,t)=U_{1}(z,t)+U_{2}(z,t)+\psi(z,t),$ (2)
$\displaystyle\left|\psi\right|\ll\left|U_{1}\right|\sim\left|U_{2}\right|.$

For the solitary waves $U_{1,2}(z,t)$ we assume that they satisfy a pair of
coupled nonlinear Schrödinger equations below

$i\partial_{z}U_{n}+\beta_{n}(i\partial_{t})U_{n}(z,t)+\gamma(\omega_{n})\left(\left|U_{n}\right|^{2}U_{n}+2\left|U_{m}\right|^{2}U_{n}\right)=0,$
(3)

where $n=1,2$ and $m=2,1\neq n$. Essentially, this is an assumption that the
two-color soliton molecule consists of two pulses, which are incoherently
coupled by inducing a refractive-index potential on each other Agrawal (2013).
Each subpulse exists in some sort of truncated dispersion landscape, defined
by the operator $\beta_{n}(i\partial_{t})$. Under specific conditions, the
subpulses are given by fundamental solitons of a modified NLS Melchert and
Demircan (2021). A reasonable guess for the truncated operator is a parabolic
approximation close to the carrier frequency

$\beta_{n}(i\partial_{t})=\beta(\omega_{n})+i\beta^{\prime}(\omega_{n})\partial_{t}-\frac{1}{2}\,\beta^{\prime\prime}(\omega_{n})\partial_{t}^{2}.$
(4)

We additionally suppose that in the soliton’s frame of reference the envelope
evolves with wavenumber $k_{n}(\omega)$ which we approximate by the wavenumber
of the fundamental soliton in the ordinary nonlinear Schrödinger equation and
a correction from a secondary soliton as

$k_{n}(\omega)\approx\frac{\gamma(\omega_{n})A_{n}^{2}}{2}+\beta(\omega_{n})+\beta^{\prime}(\omega_{n})(\omega-\omega_{n})+\gamma(\omega_{n})A_{m}^{2},$
(5)

where $n=1,2$, $m=2,1\neq n$ and $A_{n}$ is the soliton amplitude.

By substituting ansatz (2) into Eq. (1), linearizing with respect to
perturbation $\psi(z,\omega)$, and discarding the terms corresponding to the
soliton Eqs. (3), we get the equation for $\tilde{\psi}$

$i\partial_{z}\tilde{\psi}+\beta(\omega)\tilde{\psi}+\gamma(\omega)\mathcal{F}\left\\{2\left|U_{1}+U_{2}\right|^{2}\psi+\left(U_{1}+U_{2}\right)^{2}\psi^{*}\right\\}_{(\omega>0)}=\\\
-\left[\beta(\omega)-\beta_{1}(\omega)\right]\tilde{U}_{1}-\left[\beta(\omega)-\beta_{2}(\omega)\right]\tilde{U}_{2}-\gamma(\omega)\mathcal{F}\left\\{U_{2}^{2}U_{1}^{*}+U_{1}^{2}U_{2}^{*}\right\\}_{(\omega>0)}.$
(6)

On the right-hand side (RHS) of Eq. (6) we see two types of driving terms and
each of those terms can be in resonance with the linear DWs that exist in the
system if the particular wavenumber $k(\omega_{*})$ of the driving term is
equal to the wavenumber of a DW $\beta(\omega_{*})$ at some frequency
$\omega_{*}$ Akhmediev and Karlsson (1995); Yulin  _et al._ (2004). The first
type is given by
$\left[\beta(\omega)-\beta_{n}(\omega)\right]\tilde{U}_{n}(z,\omega)$ which
drives the generation of Cherenkov radiation by an individual soliton $U_{n}$
if the following resonance condition is satisfied

$\beta(\omega)=k_{n}(\omega).$ (7)

The second type of the terms are
$\gamma(\omega)\mathcal{F}\left\\{U_{2}^{2}U_{1}^{*}\right\\}$ and
$\gamma(\omega)\mathcal{F}\left\\{U_{1}^{2}U_{2}^{*}\right\\}$ and they
correspond to the process of four-wave mixing that in our case results in the
generation of dispersive radiation at some frequency where

$\beta(\omega)=2k_{n}(\omega)-k_{m}(\omega).$ (8)

Let us move on to the problem of external dispersive wave scattering. For
that, we split the perturbation into the incident and the scattered parts

$\psi(z,t)=\psi_{\text{inc}}(z,t)+\psi_{\text{sc}}(z,t).$ (9)

We explicitly define $\psi_{\text{inc}}(z,t)$ as a linear wave that is
propagating in a soliton-free medium

$i\partial_{z}\tilde{\psi}_{\text{inc}}(z,t)+\beta(\omega)\tilde{\psi}_{\text{inc}}(z,t)=0.$
(10)

Substituting Eq. (9) into (6) and eliminating the terms corresponding to Eq.
(10) we are left with the equation for the scattered component

$i\partial_{z}\tilde{\psi}_{\text{sc}}+\beta(\omega)\tilde{\psi}_{\text{sc}}+\gamma(\omega)\mathcal{F}\left\\{2\left|U_{1}+U_{2}\right|^{2}\psi_{\text{sc}}+\left(U_{1}+U_{2}\right)^{2}\psi_{\text{sc}}^{*}\right\\}_{(\omega>0)}=\text{\ldots(omitted
is the RHS of Eq.~{}\eqref{eq:PerturbationEquation})}\\\
-\gamma(\omega)\mathcal{F}\left\\{2\left(\left|U_{1}\right|^{2}+U_{1}U_{2}^{*}+U_{2}U_{1}^{*}+\left|U_{2}\right|^{2}\right)\psi_{\text{inc}}+\left(U_{1}^{2}+2\,U_{1}U_{2}+U_{2}^{2}\right)\psi_{\text{inc}}^{*}\right\\}_{(\omega>0)}.$
(11)

In addition to the resonance terms already discussed in Eq. (6), we see terms
that arise due to interaction between the incident radiation and the soliton.
Here, six new types of resonance behavior are possible. The first one is due
to terms $\left|U_{1}\right|^{2}\psi_{\text{inc}}$ and
$\left|U_{2}\right|^{2}\psi_{\text{inc}}$, both with the resonance condition

$\beta(\omega_{\text{sc}})=\beta(\omega_{\text{inc}}).$ (12)

The next two are due to mixed terms $U_{1}U_{2}^{*}\psi_{\text{inc}}$ and
$U_{2}U_{1}^{*}\psi_{\text{inc}}$, with the corresponding resonance condition
being

$\beta(\omega_{\text{sc}})=\pm k_{1}\mp k_{2}+\beta(\omega_{\text{inc}}).$
(13)

Another two are due to $U_{n}^{2}\psi_{\text{inc}}^{*}$ and the resonance
condition is

$\beta(\omega_{\text{sc}})=2k_{n}(\omega_{\text{sc}})-\beta(\omega_{\text{inc}}),~{}n=1,2.$
(14)

The final one is due to $2U_{1}U_{2}\psi_{\text{inc}}^{*}$ with the resonances
at

$\beta(\omega_{\text{sc}})=k_{1}(\omega_{\text{sc}})+k_{2}(\omega_{\text{sc}})-\beta(\omega_{\text{inc}}).$
(15)

To verify the predictions given by resonance conditions (7), (8), (12), (13),
(14), and (15) we proceed to numeric experiments.

##  III Numerical experiments

To numerically integrate Eq. (1) we use the integrating factor method and
transform the equation into a non-stiff version for a modified spectrum Dudley
and Taylor (2010). The modified equation can be handled by any standard ODE
solver; we use a scipy interface to ZVODE solver from ODEPACK Hindmarsch
(1983); Virtanen  _et al._ (2020) (the code necessary to reproduce the results
in the paper can be found in sou ). All the computations are performed in a
frame of reference co-moving with the soliton, which is achieved by a
transformation

$t\to t-\beta^{\prime}(\omega_{1})z,$

which, in turn, results in

$\displaystyle\beta(\omega)$
$\displaystyle\to\beta(\omega)-\beta(\omega_{1})-\beta^{\prime}(\omega_{1})(\omega-\omega_{1})$
$\displaystyle k_{1}(\omega)$
$\displaystyle\to\frac{\gamma(\omega_{1})A_{1}^{2}}{2}+\gamma(\omega_{1})A_{2}^{2}$
$\displaystyle k_{2}(\omega)$
$\displaystyle\to\frac{\gamma(\omega_{2})A_{2}^{2}}{2}+\gamma(\omega_{1})A_{1}^{2}+\beta(\omega_{2})-\beta(\omega_{1}).$

Description of the specific dispersion profile model $\beta(\omega)$ used in
the simulations can be found in the Supplemental Material sup .

To study Cherenkov radiation of two color solitons we chain two separate
simulations. First, following the prior work Melchert  _et al._ (2019), we
produce a two-color soliton by integrating an initial condition that is given
by a sum of two fundamental solitons of standard NLSE

$u_{0}(t)=A_{1}\operatorname{sech}(t/T_{1})e^{-i\omega_{1}t}+A_{2}\operatorname{sech}(t/T_{2})e^{-i\omega_{2}t},$
(16)

where frequencies $\omega_{1}$ and $\omega_{2}$ are both lying in the regions
of anomalous dispersion. Frequency $\omega_{1}$ is otherwise arbitrary;
$\omega_{2}$ is chosen so that group velocities of both the frequency
components match

$\beta^{\prime}(\omega_{1})=\beta^{\prime}(\omega_{2}).$

In most of the simulations presented subsequently, we fix $T_{1}=T_{2}=20$ fs
and the amplitudes $A_{1}$ and $A_{2}$ are chosen as the fundamental soliton
amplitudes at the corresponding frequencies. This configuration sheds a
significant amount of radiation and relaxes to a quasi-stable solitary wave.
We propagate up to $z=10~{}\text{cm}$, take the output field of this seed
simulation, and suppress the radiation tails by multiplying it by a super-
Gaussian temporal window, centered on the peak of the soliton molecule. This
isolated soliton molecule serves as an input to the second simulation that is
carried with the same parameters as the original one.

Once we suppress the radiation that is shed by the seed solitons during the
initial relaxation process the isolated two-color soliton molecule propagates
generating only a narrow-spectrum Cherenkov radiation. An example is shown in
Fig. 1. Figure 1(b) shows the normalized spectral densities at the input and
output, i.e. $z=0\,\mathrm{cm}$ and $10~{}\mathrm{cm}$, respectively. For
clarity, the difference between both spectra is shown on a logarithmic scale
in Fig. 1(c). It is clearly evident that on top of the input spectrum [thin
black line in Fig. 1(b)], two additional spectral lines appear in the output
spectrum [gray line in Fig. 1(b)]. To clarify the origin of these pronounced
spectral lines, Fig. 2(d) demonstrates resonance conditions (7) and (8): the
black curve corresponds to $\beta(\omega)$, i.e. the left-hand side of both
equations; the horizontal lines correspond to the right-hand sides.
Intersections between the dispersive curve and the horizontal lines that
contribute to Cherenkov radiation are marked separately: 1 labels radiation
due to the second component of the soliton as predicted by Eq. (7); 2 labels
the location of frequencies due to FWM between the frequency components,
resulting in radiation with wavenumber $2k_{2}-k_{1}$, as predicted by Eq.
(8).

Figure 1: (color online) Cherenkov radiation of an isolated two-color soliton
molecule, produced by an initial condition with first-component carrier
frequency $\omega_{1}=1.010$ rad/fs. (a) Time-domain view, (b) input and
output spectra, (c) difference between both spectra on a logarithmic scale,
and (d) diagram of the resonance conditions.

To study the scattering processes we perform the seed simulation and isolate
the two-color soliton. To accomplish that we add an incident DW in the form of
a Gaussian pulse

$\psi_{\rm{inc}}(t)=A_{\text{inc}}\exp\left(-\frac{(t-t_{0})^{2}}{T_{\text{inc}}^{2}}-i\omega_{\text{inc}}t\right).$

Below we set $A_{\text{inc}}$ to 1% of the maximum amplitude of the isolated
soliton and fix the width to $T_{\rm{inc}}=300~{}\mathrm{fs}$. The initial
pulse delay $t_{0}$ is chosen as $\pm 1000$ fs from the soliton center, with
the sign depending on the relative group velocity between the soliton and the
DW and chosen so that both pulses engage in a collision. Upon propagation the
incident radiation interacts with the two-color soliton to produce scattered
radiation. This process can evolve in several different ways depending on the
frequencies of the soliton and the incident radiation. Below we consider three
concrete examples. In each of them the incident radiation is split between
three different components.

The first configuration, shown in Fig. 2, is very close to the degenerate case
where the wavenumbers of the individual soliton components $k_{1}$ and $k_{2}$
coincide. Equation (13) then turns into Eq. (12). This case resembles the case
of fundamental single-component solitons Yulin  _et al._ (2004), however, due
to the shape of the dispersive curve $\beta(\omega)$ there is now more than
one nontrivial solution to Eq. (12) and a single incident frequency yields up
to four possible resonances. In practice, only some of them contribute to the
scattered radiation. Those solutions are separately marked on a diagram on
panel 2(d): $i$ corresponds both to the incident and the partially transmitted
radiation, 1 is the reflected component and 2 is the additional transmitted
component.

Figure 2: (color online) Scattering of a weak DW with carrier frequency
$\omega_{\text{inc}}=1.650$ rad/fs on a two-color soliton with first-component
carrier frequency $\omega_{1}=1.070$ rad/fs. (a) Time-domain view, (b) input
and output spectra, (c) difference between both spectra on a logarithmic
scale, and (d) diagram of the resonance conditions.

The second configuration, shown in Fig. 3, is a case with a significant
difference between $k_{1}$ and $k_{2}$. Since Eq. (13) is no longer
degenerate, the resonance diagram in Fig. 3(d) is more complex. However, as
one can notice, in that case only the solution marked with 2 , corresponding
to the upper (i.e. “$-,+$”) branch of Eq. (13), contributes to the scattered
radiation. Component 1 , corresponding to the scattered radiation, comes from
resonance condition (12). The unmarked spectral line close to $\omega\approx
2.4\,\mathrm{rad/fs}$ is the Cherenkov radiation of the soliton itself.

Figure 3: (color online) Scattering of a weak DW with carrier frequency
$\omega_{\text{inc}}=1.500$ rad/fs on a two-color soliton with first-component
carrier frequency $\omega_{1}=1.150$ rad/fs. (a) Time-domain view, (b) input
and output spectra, (c) difference between both spectra on a logarithmic
scale, and (d) diagram of the resonance conditions.

The third configuration, shown in Fig. 4, is in a sense symmetric to the
previous case. Again, the difference between $k_{1}$ and $k_{2}$ is
significant and Eq. (13) is far from being degenerate, but now the resonances
from the lower (i.e. “$+,-$”) branch of the resonance condition (13) play a
significant role. This is achieved by choosing the incident frequency greater
than the carrier frequency of the second soliton component and enhanced by
using an asymmetric seed soliton with $T_{1}=30$ fs and $T_{2}=10$ fs. This
creates a solitary wave with the amplitude of the second component almost
equal to the amplitude of the first one.

Figure 4: (color online) Scattering of a weak DW with carrier frequency
$\omega_{\text{inc}}=3.500$ rad/fs on a two-color soliton with first-component
carrier frequency $\omega_{1}=1.150$ rad/fs. (a) Time-domain view, (b) input
and output spectra, (c) difference between both spectra on a logarithmic
scale, and (d) diagram of the resonance conditions.

Overall, in all of our experiments, only the scattered components
corresponding to Eqs. (12) and (13) turn out to be significant, while the
components predicted by Eqs. (14) and (15) do not seem to contribute to the
resulting radiation. In other words, the terms proportional to
$\tilde{\psi}_{\rm{inc}}^{*}$ on the RHS of Eq. (11) can be safely neglected.

##  IV Nonlinear effects

When deriving resonance conditions (12) – (15) we assumed that the parameters
of the solitons themselves stay constant throughout propagation and scattering
processes. This can be considered a reasonable approximation when the
amplitude of the incident wave is negligible compared to the amplitudes of the
individual soliton components. However, in the general case of more intensive
incident radiation this assumption does not hold. In this section we will
briefly discuss two specific examples, where the process of scattering
noticeably affects the parameters of the soliton. The general setup of the
numerical experiments remains as in the preceding section, i.e. we consider
scattering of a Gaussian pulse on an isolated two-color soliton, but this time
we increase the amplitude of the DW to 5% of the soliton’s maximum amplitude.
This change might seem subtle, but it is sufficient to make the scattering
dynamics much more involved.

In the first example, shown in Fig. 5, we consider scattering of a DW with
incident frequency $\omega_{i}=2.100$ rad/fs on a two-color soliton with
$\omega_{1}=1.010$ rad/fs. From panel 5(a) we can immediately notice, that
interaction with the DW significantly decelerates the two-color soliton. This
is connected to the change of the soliton’s carrier frequency, as demonstrated
in panel 5(b.2). This effect has been demonstrated before for the conventional
solitons of nonlinear Schrödinger equation Demircan  _et al._ (2011); Tartara
(2015). What is remarkable about this interaction in case of a two-color
soliton is the fact that the soliton appears to be stable during this process.
Granted, the frequency offset gained by the soliton during the scattering is
not especially prominent [panel 5(b.2)], but at the same time the amplitude
difference in both the frequency components stays under 1% [panel 5(b.1)], so
almost no power loss occurs.

Figure 5: (color online) Scattering of an intensive DW with carrier frequency
$\omega_{\text{inc}}=2.100$ rad/fs on a two-color soliton with first-component
carrier frequency $\omega_{1}=1.010$ rad/fs. (a) Time-domain view. (b.1)
Oscillation of the two-color soliton subpulse amplitudes, and, (b.2)
oscillation of the corresponding center frequencies. (c) Output spectrum.

In the second example, shown in Fig. 6, we consider the scattering of a DW
with the incident frequency $\omega_{i}=1.100$ rad/fs on a two-color soliton
with $\omega_{1}=1.010$ rad/fs. In the plots in panels 6(b.1) and 6(b.2) one
can notice how the interaction with the DW generates oscillations in the
soliton parameters; this is especially prominent in 6(b.2) that displays the
absolute change of the soliton’s central frequencies $\omega_{1}$ and
$\omega_{2}$. From the latter plot the period of those oscillations can be
estimated as $Z_{0}=2~{}\text{mm}$. The interaction of freely oscillating
nonlinear waves, both the radiation and the scattering processes, is a well
studied problem Driben  _et al._ (2015); Oreshnikov  _et al._ (2015b); Wright
_et al._ (2015); Conforti  _et al._ (2015); Oreshnikov  _et al._ (2017) and
the common trait in this setting, independent of the nature of the soliton
oscillations, is that the dispersive radiation produced (generated or
scattered) by the soliton is polychromatic, i.e. it consists of several
isolated spectral components. This is indeed what we see in the output
spectrum on panel 6(c).

To explain this behavior let us return to equations (6) and (11). In the
oscillating case the individual solitons $U_{1}$ and $U_{2}$ are no longer
represented by a single spatial frequency $\propto e^{ik_{1}z}$ and $\propto
e^{ik_{2}z}$ and they rather correspond to Fourier series’

$U_{n}(z,t)=\sum\limits_{N\in\mathbb{Z}}C_{nN}(t)\exp\left(ik_{1}z+i\frac{2\pi
N}{Z_{0}}z\right),$

where $Z_{0}$ is the oscillation period. This leads to a split in resonance
conditions, and for the oscillating case equations (7), (8), (12) and (13)
read

$\displaystyle\beta(\omega)=k_{n}(\omega)+\frac{2\pi N}{Z_{0}}$ (7*)
$\displaystyle\beta(\omega)=2k_{n}(\omega)-k_{m}(\omega)+\frac{2\pi N}{Z_{0}}$
(8*)
$\displaystyle\beta(\omega_{\text{sc}})=\beta(\omega_{\text{inc}})+\frac{2\pi
N}{Z_{0}}$ (12*) $\displaystyle\beta(\omega_{\text{sc}})=\pm k_{1}\mp
k_{2}+\beta(\omega_{\text{inc}})+\frac{2\pi N}{Z_{0}},$ (13*)

with $N\in\mathbb{Z}$ being the spatial harmonic number.

The resonance conditions for the scattering process in the last simulation are
displayed on panel 6(d). In this process only the last two equations, i.e.
Eqs. (12*) and (13*) are relevant. The solid black lines correspond to
equation (12) with harmonics $N=-3,\dots,0$. This harmonic split not only
leads to the reflected part of the radiation 1 becoming polychromatic, but
also has the same effect on the incident component i . Resonance condition
(13) is represented by the gray dashed lines for harmonics $N=-3,\dots,+3$. It
contributes to the scattered radiation twice: with a wide band 2 that
otherwise would degenerate into a sharp spectral line in a non-oscillating
case, and with three wide spectral lines in the vicinity of 3 which is the
contribution of the lower harmonics $N=-3,-2,-1$. One can notice that there
are only two vertical lines corresponding to the numeric solution of Eq. (13)
displayed around 3 . This is merely due to a small numeric error in estimating
the soliton wavenumber $k_{2}$ in the plotting procedure, which causes
harmonic $N=-1$ to touch the dispersive rather than intersect it.

Figure 6: (color online) Scattering of an intensive DW with carrier frequency
$\omega_{\text{inc}}=1.100$ rad/fs on a two-color soliton with first-component
carrier frequency $\omega_{1}=1.010$ rad/fs. (a) Time-domain view. (b.1)
Oscillation of the two-color soliton subpulse amplitudes, and, (b.2)
oscillation of the corresponding center frequencies. (c) Output spectrum. (d)
Diagram of the resonance conditions.

The specific oscillation mode we see in Fig. 6 appears to be heavily damped,
since launching the initial soliton with an additional frequency detuning does
not lead to free frequency oscillations during the propagation. Such internal
dynamics, reminiscent of molecular vibrations, were reported also previously
Melchert  _et al._ (2019, 2021), and the emission of DWs by the two-color
soliton was found to explain the dampening of the oscillation mode Willms  _et
al._ (2022). Here, the incident radiation drives this oscillation, and one
could expect to observe a resonance behavior with respect to the frequency of
the incident DW. And indeed, as demonstrated by the parameter sweep in Fig.
7(a), by adjusting the incident frequency $\omega_{i}$ one can affect, to a
certain degree, the amplitude of the frequency oscillations of the second
component $\Delta\omega_{2}$. For this specific mode, where the frequency
oscillations dominate, it is straightforward to get an estimate for the period
$Z_{0}$ of the mode and the corresponding wavenumber $K_{0}$ in terms of a
variational approach (see Supplemental Material sup ). As shown in Fig. 7(b),
superimposing the resulting resonance wavenumber $K_{0}$ on the dispersion
curve allows to graphically estimate the frequency of the incident DW that
corresponds to the internal oscillation mode. The vertical dashed lines in
Figs. 7(a,b) indicate this resonance frequency, confirming the excellent
agreement of the numerical simulations and the approximate analytic estimate.

Figure 7: Resonance behavior of the internal mode of the two-color soliton
molecule during intensive scattering. (a) Oscillation of the center frequency
of the second component of the two-color soliton as function of the frequency
of the incident DW. (b) Diagram of the resonance condition for the internal
mode.

##  V Conclusion

In this paper, we considered the resonant interaction of two-color soliton
molecules with DWs. The resonance condition for Cherenkov radiation is derived
and analyzed. The comparison with the results of numerical simulations shows
that these conditions correctly predict the positions of the Cherenkov
radiation. We also study the process of the collision of the two-color
solitons with the incident DWs. The resonance conditions predict well all
newly generated frequencies observed in the numerical experiments. Some of the
predicted resonances are not seen in the numerically calculated radiation
spectra. This can be explained by the low efficiency of the corresponding
scattering processes. The theory developed for a simpler case in Skryabin and
Yulin (2005) shows that indeed for some parameters the generated radiation can
be extremely weak. The theory giving the analytical estimations for four waves
mixing between the DWs and two-color solitons is of interest but is out of the
scope of the present paper. We also studied how the intensity of the incident
dispersive waves affects the scattering. In particular, it is shown that the
scattering can affect the soliton trajectory without affecting its integrity.
Another important finding is that in the nonlinear case the DWs can resonantly
excite internal oscillations of the soliton. This results in polychromatic
emission of DWs by oscillating two-color solitons.

### Acknowledgements

OM, SW, IB, AD, and UM acknowledge financial support from Deutsche
Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy within the
Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering—Innovation
Across Disciplines) (EXC 2122, Project No. 390833453). SB acknowledges
financial support from DFG under Germany’s Excellence Strategy within the
Cluster of Excellence QuantumFrontiers (EXC 2123, Project No. 390837967). IB
also acknowledges support from DFG (Project No. BA4156/4-2). UM also
acknowledges support from DFG (Project No. MO 850-20/1).

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