Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used ...for integrable models where solitons interact elastically. However, soliton turbulence can also be a part of non-integrable dynamics, where long-lasting solutions in the form of almost solitons may exist. In the present paper, the complex dynamics of ensembles of solitary waves is studied within the Schamel equation using direct numerical simulations. Some important statistical characteristics (distribution functions, moments, etc.) are calculated numerically for unipolar and bipolar soliton gases. Comparison of results with integrable Korteweg–de Vries (KdV) and modified KdV (mKdV) models are given qualitatively. Our results agree well with the predictions of the KdV equation in the case of unipolar solitons. However, in the bipolar case, we observed a notable departure from the mKdV model, particularly in the behavior of kurtosis. The observed increase in kurtosis signifies the amplification of distribution function tails, which, in turn, corresponds to the presence of high-amplitude waves.
•Schamel equation.•Soliton gas.•Soliton collisions.•Non-integrable equations.
Soliton turbulence is studied within the framework of Gardner equation (generalized Korteweg–de Vries equation including quadratic and cubic nonlinear terms) by virtue of the direct numerical ...simulation of the ensemble dynamics. This equation allows the different soliton polarities to exist which make possible waves with extreme amplitudes to occur. Though the pairwise soliton collisions happen more frequently in the soliton gas, multiple soliton collisions have been identified as well involving up to five solitons. The emergence of abnormally large waves (rogue waves) of “unexpected” polarity is demonstrated. Different statistical properties of soliton turbulence (statistical moments, distribution functions) are analyzed.
•Dynamics of soliton ensembles is studied numerically within the Gardner equation.•Abnormally large wave emergences of “unexpected” polarity are demonstrated.•Multiple soliton collisions occur in a soliton gas.•The tail of the negative wave amplitude distribution function extends with time.
Dynamics of random multi-soliton fields within the framework of the modified Korteweg–de Vries equation is considered. Statistical characteristics of a soliton gas (distribution functions and ...moments) are calculated. It is demonstrated that the results sufficiently depend on the soliton gas properties, i.e., whether it is unipolar or bipolar. It is shown that the properties of a unipolar gas are qualitatively similar to the properties of a KdV gas Dutykh and Pelinovsky (2014) 1: nonlinear interaction leads to an increase in the part of small-amplitude waves and decrease in the third and fourth statistical moments. The dynamics of bipolar soliton fields is more interesting. In this case, kurtosis (the fourth moment) and the part of large-amplitude waves increase during the interaction. It is demonstrated that the freak wave appearance in a soliton gas is possible due to the attraction of large bipolar solitons.
•The dynamics of multi-soliton field within the framework of the modified Korteweg–de Vries equation is considered.•It is shown that nonlinear interaction modifies the distribution function of the wave amplitudes and statistical moments.•It is demonstrated that freak waves can appear in a bipolar soliton gas.
Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. ...Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several
2
×
2
Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.
We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg–de Vries (KdV) equation in the special “condensate” limit. We prove that in this limit the integro-differential ...kinetic equation for the spectral density of states reduces to the
N
-phase KdV–Whitham modulation equations derived by Flaschka et al. (Commun Pure Appl Math 33(6):739–784, 1980) and Lax and Levermore (Commun Pure Appl Math 36(5):571–593, 1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for “diluted” soliton condensates exhibiting rich incoherent behaviors associated with integrable turbulence.
We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These ...equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of
N
-component ‘cold-gas’ hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary
N
which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the ‘cold-gas’ component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.
The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations ...(KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in 1, within the KdV equation. In the recent paper 2, an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.
We study the breather collisions in a weakly discrete sine-Gordon equation and reveal a new mechanism of fractal soliton scattering. We demonstrate how this new mechanism manifests itself in the ...dynamics of cold soliton gas.
PDF
