Generalized hydrodynamics of the KdV soliton gas Bonnemain, Thibault; Doyon, Benjamin; El, Gennady
Journal of physics. A, Mathematical and theoretical,
09/2022, Letnik:
55, Številka:
37
Journal Article
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Abstract
We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory ...for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg–de Vries equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport.
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.
Abstract
We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states
u
(
z
) in the spectral theory of soliton gases for the one ...dimensional integrable focusing nonlinear Schrödinger equation (fNLS) and for the Korteweg–de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimising (positive) measure for a certain energy functional is absolutely continuous and its density
u
(
z
) ⩾ 0 solves the required integral equation. In a similar fashion we show that
v
(
z
), the temporal analog of
u
(
z
), is the difference of densities of two absolutely continuous measures. Together, the integral equations for
u
,
v
represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis (2020
Phys. Rev. E
101
052207). It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems.
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.
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