Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation

Abstract

We analyse the case of a dense modified Korteweg–de Vries (mKdV) soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann–Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally, we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.

Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

“…Such oscillations were recently described analytically for the modified KdV equation using rigorous asymptotic theory in Ref. [28]. As the soliton passes through the DSW region, its average amplitude increases, while its velocity decreases.…”

“…We note recent, related work in which a mKdV soliton interacts propagates into an oscillatory wave train that asymptotes to a cnoidal wave solution as $$x \rightarrow \infty$$ 28 . The initial value problem is solved using Riemann–Hilbert methods.…”

“…The problem of a soliton interacting with a nonlinear wave train that asymptotes to a cnoidal wave in the mKdV equation was studied in Ref. [28]. This soliton–mean field problem is equivalent to a test soliton propagating through a soliton condensate, a special kind of soliton gas, 29 linking soliton–mean interaction to another rapidly growing field of nonlinear wave research.…”