​​Soliton Gas Interactions in Dispersive Hydrodynamics​

Abstract

The understanding and quantification of the process of turbulence represents a fundamental problem at the forefront of applied mathematics and physics. Consideration of this problem extends beyond the discipline of fluid mechanics, involving a broader consideration in terms of statistical mechanics of complex nonlinear systems far from equilibrium. In particular, the wave turbulence theory developed in the 1960s by V. E. Zakharov is concerned with the evolution of Fourier spectra of large ensembles of random weakly nonlinear dispersive waves described by non-integrable equations.

More recently, a concept of integrable turbulence has been put forward (Zakharov 2009), which is principally concerned with the description of random nonlinear dispersive waves described by integrable equations such as the Korteweg-deVries (KdV) and focusing nonlinear Schrodinger (fNLS) equation. The integrable turbulence framework encompasses both weakly and strongly nonlinear waves, including coherent structures like solitons and breathers.

In this PhD project we study a specialised class of integrable turbulence dominated by solitons, termed soliton gas, introduced, again, by Zakharov as early as in 1971. The original rarefied soliton gas model was introduced as a large ensemble of well-separated KdV solitons randomly distributed on the line and having certain distribution over amplitudes. Zakharov’s model has been substantially extended in 2000s (El 2003, El & Tovbis 2020) using the tools of the spectral finitegap theory and Whitham modulation theory to derive the kinetic equation for dense soliton gases of the KdV and fNLS equations. Recent years have seen rapidly growing interest in the theory of soliton gas and its applications since soliton gas dynamics have been found to underpin many fundamental nonlinear wave phenomena including modulational instability and the formation of rogue waves.

In the first part of the project we use the Riemann problem for soliton gas as a benchmark for a detailed numerical verification of the spectral kinetic theory for the KdV and fNLS equations. We construct weak, discontinuous, solutions to the kinetic equation for soliton gas describing collision of two dense “polychromatic” soliton gases composed of a finite number of “monochromatic” components, each consisting of solitons with nearly identical spectral parameters of the scattering operator in the Lax pair. We then use the solutions of the spectral kinetic equation to evaluate macroscopic physical observables in KdV and fNLS soliton gases and compare them with the respective ensemble averages extracted from the “exact” soliton gas numerical solutions of the KdV and fNLS equations. To numerically synthesise dense polychromatic soliton gases we develop a new method which combines recent advances in the spectral theory of the so-called soliton condensates and the effective algorithms for the numerical realisation of n-soliton solutions with large n. The very good agreement observed between analytical predictions of the kinetic theory and the direct numerical simulations over a broad range of soliton gas parameters provides a much needed confirmation of the validity of the spectral kinetic equation as robust analytical model for non-equilibrium soliton gas dynamics.

In the second part of the project the analytical framework of solitonic dispersive hydrodynamics (Maiden et. al. 2018) is used to analyse the interaction of a rarefied monochromatic soliton gas with macroscopic dispersive hydrodynamic structures: rarefaction and dispersive shock waves. We re-interpret the results of solitonic hydrodynamics in terms of soliton gas-mean field interaction problem and then implement the problem numerically using the algorithms developed in the first part of the project. The results of numerical simulations are shown to be in a very good agreement with analytical predictions.

Date of Award	24 Oct 2024
Original language	English
Awarding Institution	Northumbria University
Supervisor	Gennady El (Supervisor) & Thibault Congy (Supervisor)