We study soliton collisions in the Dyachenko–Zakharov equation for the envelope of gravity waves in deep water. The numerical simulations of the soliton interactions revealed several fundamentally different effects when compared to analytical two-soliton solutions of the nonlinear Schrodinger equation. The relative phase of the solitons is shown to be the key parameter determining the dynamics of the interaction. We find that the maximum of the wave field can significantly exceed the sum of the soliton amplitudes. The solitons lose up to a few percent of their energy during the collisions due to radiation of incoherent waves and in addition exchange energy with each other. The level of the energy loss increases with certain synchronization of soliton phases. Each of the solitons can gain or lose the energy after collision, resulting in increase or decrease in the amplitude. The magnitude of the space shifts that solitons acquire after collisions depends on the relative phase and can be either positive or negative.

The existence and interactions of coherent structures like solitons and breathers on the surface of a deep water are a remarkably rich and fascinating subject for both experimental and theoretical studies. The exact mathematical model describing gravity waves in the ocean is the Euler equation, yet it is often rather complicated to study it by analytic or numerical means. Instead, various reduced models for water waves have demonstrated good agreement with the experimental data and have been widely adopted in the fluid dynamics and geophysics communities.

The most prominent and widely used model for weakly nonlinear surface waves
in deep water is the nonlinear Schrödinger (NLS) equation. It describes time evolution of the
envelope of a quasi-monochromatic wave train
(

By means of the IST one can find NLS soliton solutions and track their
evolution in time until their collision and beyond analytically. The
collision of the NLS solitons is perfectly elastic; that is, no loss of the
energy occurs. The equations which are not integrable by the IST may have
exact stationary solitary solutions interacting inelastically. For example,
the Dysthe equation is known to admit solitary solutions whose existence has
been demonstrated by other approaches unrelated to the IST (see

Both the NLS and Dysthe equations are formulated to describe the evolution
of the envelope function. They require that the steepness of the wave train
is small and that it is modulated weakly, i.e., that there are sufficiently
many carrier wavelengths in the characteristic scale of the modulation. In
terms of the Fourier transform of the surface elevation this is equivalent to
having a sufficiently narrow band concentrated in the vicinity of the carrier
wave number. The DZ equation is formulated for the wave train itself and is
free from the assumptions of the weak nonlinearity and narrow
bandness (

In the work

The study of soliton (or breather) interactions in the reduced deep water
models is an important step in the understanding of the surface waves'
dynamics and the fundamental properties of the Euler equation. In this work
we focus on the DZ equation in the form suggested
by

The soliton interactions in the NLS equation depend drastically on their
relative complex phases; e.g., the maximum amplification of the amplitude in
a collision is determined by the synchronization of the phases of the
solitons. The phase synchronization plays an important role in the formation
of the waves of extreme amplitude, the rogue waves, and has been studied in
water wave theory (

In the present work we study soliton interactions in the DZe equation and their dependence on the phases of interacting solitons. We demonstrate how the amplitude amplification, the energy exchange between the solitons, the energy loss to emission of incoherent radiation and the space shift of the solitons after collisions reveal fundamental differences from the NLS equation.

A one-dimensional potential flow of an ideal fluid of infinite depth in the
presence of gravity is a Hamiltonian system. The surface elevation

Equation (

Recently

In this work we study only the model (

We consider solitons in the frame moving with the velocity

In this work we focus on the interactions of the NLS solitons and the DZe
solitons of equal amplitude

The dynamics of DZe soliton collisions can be investigated only by numerical
simulations. We study interactions of the DZe solitons of the same amplitudes

Comparison of DZe solitons and NLS solitons with
amplitude

We fix a carrier wave number

The NLS equation is a completely integrable model and an exact multisoliton
solution is available (see the work by

Collision of NLS solitons with amplitudes

The maximal amplitude

In the NLS model the amplitude amplification function decreases when the

The maximum amplification

The numerical simulations of the soliton interactions in the DZe equation
were carried out in a periodic domain

Our numerical simulations show that the dependences

The maximum amplification

We found that the maximum value of amplitude amplification

Collision of DZe solitons with the wave steepness

Collision of DZe solitons with the wave
steepness

The interactions of solitons (or breathers) in the DZ model are inelastic
(

We quantitatively study the dependence of soliton energy losses

Figure

The total energy losses

In this paragraph we describe the individual changes in DZe solitons after
collision. We measure the energy changes in soliton 1 and soliton 2 relative
to their individual energies:

We have found that solitons of the DZe equation exchange energy with each
other. Each of the solitons can gain or lose the energy after collision in
dependence on the relative phase

The individual energy change (in percent –
see formula

Comparison of the DZe solitons after mutual
collision and the same solitons propagated without interaction. The wave
steepness of the solitons

Space shifts of the DZe solitons depending on the
relative phase

The energy exchange and energy losses result in the increase or decrease in
the soliton amplitudes, which is demonstrated by Fig.

In addition, Fig.

In this work we have studied how the relative phase of solitons in the DZe
model affects the key properties of their interaction. All results presented
here for solitons of the DZe equation are valid also for breathers of the DZ
equation since solutions of these two models are linked by the
transformation (

Interactions of the breathers in the DZ equation at a certain phase synchronization can lead to the formation of extreme amplitude waves. It is well known that the maximum value of a wave field as a result of soliton interactions in the NLS model is equal to the sum of the soliton amplitudes. In this work we have found that in the DZe equation the maximum amplification can be higher than the sum of amplitudes of the solitons. Interestingly, at large values of the wave field steepness this effect is enhanced, which can be a valuable complement in extreme amplitude wave studies.

We have also studied the phenomena of the energy exchange between the
colliding solitons. This energy exchange is caused by inelasticity of the
soliton interactions. The universal long-term consequences of this process
were studied in different nonintegrable
models (

Furthermore, we have studied space shifts that solitons acquire after the
collision. Solitons of the NLS equation always acquire a positive constant
shift

The inelasticity of soliton collisions in nonintegrable models may destroy
the initially coherent wave groups. However, as we have demonstrated here
the total energy loss for interactions described by Eq. (

Pairwise collisions of solitons (or breathers) is an important elementary
process that can be observed in the wave dynamics of an arbitrarily disturbed
fluid surface. For example, the recent numerical simulations of the DZe
equation demonstrate that an ensemble of interacting solitons can appear as a
result of modulation instability driven by random perturbations of an
unstable plane wave (

The data used to generate figures in the article are accessible upon request to Dmitry Kachulin.

Both authors (DK and AG) proposed key ideas, performed numerical simulations and contributed equally to this work.

The authors declare that they have no conflict of interest.

The authors thank Alexander I. Dyachenko and Sergey A. Dyachenko for the helpful discussions. The theoretical part of the work was performed with support from the Russian Science Foundation (grant no. 14-22-00174). The study reported in Sects. 4.1 and 4.2 (results of numerical simulations) was funded by RFBR and the government of the Novosibirsk region according to research project no. 17-41-543347. Simulations were performed at the Novosibirsk Supercomputer Center of Novosibirsk State University. Edited by: Roger Grimshaw Reviewed by: three anonymous referees