Bidirectional soliton gas

Abstract: The soliton structure plays a fundamental role in many physical systems due to its fundamental feature: its shape remains unchanged after the collision with another soliton in the case of integrable dynamics. Such particle-like behaviour has been at the origin of a new mathematical object: the soliton gas, consisting of an incoherent collection of solitons for which phases (positions) and spectral parameters (e.g. amplitudes) are randomly distributed. The study of soliton gases involves the description of the gas dynamics as well as the corresponding modulation of the nonlinear wave field statistics, which makes the soliton gas a particularly interesting embodiment of the particle-wave duality of solitons.

Motivated by the recent realisation of bidirectional soliton gases in a shallow water experiment, we investigate two integrable models of bidirectional wave: the nonlinear Schrödinger equation and the Kaup-Boussinesq equation. Using a physical approach, we derive the so-called kinetic equation that governs the gas dynamics for the two integrable systems. We notably show that the structure of the kinetic equation depends on the "isotropic" or the "anisotropic" nature of the solitons interaction. Additionally we derive expressions for statistical moments of the physical fields (e.g. mean water level). As an illustration of the theory, we solve numerically the gas shock tube problem describing the collision of two "cold" soliton gases. An excellent agreement with exact solutions of the kinetic equations is observed.

mathematical physicsanalysis of PDEsdynamical systems

Audience: researchers in the topic

Waves in One World (WOW) series

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