Statistical limit in a completely integrable system with deterministic initial conditions

Gurevich, A.; Mazur, N. G.; Zybin, K. P.

doi:10.1134/1.559155

Cited by 9 publications

(18 citation statements)

References 11 publications

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“…Since ρ(x, t) = 1 0 f dη is the density of solitons the quantity s(η, x, t) can naturally be interpreted as the velocity of the soliton gas (or, more precisely, the velocity of a 'trial' soliton with the spectral parameter λ = −η 2 -see [23]). One can see from (26) that this velocity differs from the velocity 4η 2 of the free soliton with the same spectral parameter.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

et al. 2010

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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 Ncomponent '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 (quasi-periodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed the 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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Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

et al. 2010

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“…This is in line with the proposition made in the beginning of this section that the long-time asymptotic behaviour of the semiclassical NLS should be generally described in statistical terms. In this connection we note that the statistical description of the long-time asymptotic solution of the small-dispersion KdV equation with deterministic initial conditions defined on the entire x-axis was considered in [80], [81].…”

Section: Long-time Behaviourmentioning

confidence: 99%

Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves

2016

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Abstract. We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrödinger (NLS) equation with the initial condition in the form of a rectangular barrier (a "box"). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains -the dispersive dam break flows -generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated largeamplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain spacetime domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.

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“…The integral equations (43), (44) have been derived in [9] in connection with the establishing the thermodynamic limit for the stochastic processes generated by the finite-gap potentials (stochastic soliton lattices). We note that these integral equations also appear in [16] where they determine the Lax-Levermore type minimizer for the N-soliton solution with randomly distributed soliton phases as N → ∞. It is clear that there should be a direct connection between the 'stochastic' version of the Lax-Levermore variational problem and the thermodynamic limit of the rotation number.…”

Section: The Thermodynamic Limit For the Rotation Numbers: Basic Intementioning

confidence: 99%