Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

Abstract

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, Forest and McLaughlin (1980) and Lax and Levermore (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 behaviours associated with integrable turbulence.

40 pages, 20 figures

References in corpus (4)

• 100 years of Weyl's law
• Generalized hydrodynamics of the KdV soliton gas
• Recent developments in spectral theory of the focusing NLS soliton and breather gases: the thermodynamic limit of average densities, fluxes and certain meromorphic differentials; periodic gases
• Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation

Cited by in corpus (6)

• KdV breathers on a cnoidal wave background
• Partial degeneration of finite gap solutions to the Korteweg-de Vries equation: soliton gas and scattering on elliptic background
• Soliton gas of the integrable Boussinesq equation and its generalised hydrodynamics
• Hamiltonian formulation and aspects of integrability of generalised hydrodynamics
• Genus two KdV soliton gases and their long-time asymptotics
• Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case