Gas dynamics of solitons in integrable systems
Gennady El
Department of Mathematical Sciences, Loughborough University
Date and time:
Tuesday, September 22, 2015 - 4:00pm
Location:
ECOT 226
Abstract:
The collective dynamics of a large number of random solitons in an integrable system are described by the soliton gas kinetic equation. This equation must be complemented by the expressions for the moments of the nonlinear wave field associated with the soliton gas — the soliton turbulence. This particle-wave duality of soliton gas makes it a particularly rich and interesting mathematical and physical object. On the one hand one might be interested in the "gas-dynamics" properties of the soliton gas: its temperature, pressure, sound speed etc. On the other hand, there are intriguing questions about Fourier power spectra of soliton turbulence or the probability of the rogue wave occurrence in random wave fields. Integrability of the underlying (KdV or NLS) equation holds a promise for exact analytic results.
In this talk I will present recent results on the comparison of the analytical predictions of the KdV soliton gas kinetics with the direct "particle dynamics" numerical simulations of a large random ensembles of KdV solitons. Two test problems will be considered: (i) the propagation of a "trial" soliton through a cold ("monochromatic") soliton gas; and (ii) soliton gas shock tube problem. In both cases excellent agreement with the relevant exact solutions of the kinetic equation was observed. Next I will try to answer the pertinent question: "What is a dense soliton gas?'' by deriving an explicit expression for the critical density of the KdV soliton gas with gaussian distribution over the IST spectrum. If time permits I will outline an integrable Landau-Hopf scenario of trasition to soliton turbulence in a modulationally unstable system described by the focusing NLS equation and discuss the associated mechanism of the rogue wave formation.