Thibault Congy; Department of Mathematics, Physics, and Electrical Engineering; University of Northumbria; Newcastle, UK
Bidirectional soliton gas
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 gas involves the description of the gas dynamics as well as the corresponding modulation of 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 soliton gases of two bidirectional integrable dynamics: 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 both integrable systems. We notably show that the structure of the kinetic equation depends on the "isotropic" or "anisotropic" nature of 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 the relevant exact solutions of the kinetic equations is observed.