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 versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation / Girotti, M.; Grava, T.; Jenkins, R.; Mclaughlin, K. T. R.; Minakov, A.. - In: COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS. - ISSN 0010-3640. - 76:11(2023), pp. 3233-3299. [10.1002/cpa.22106]

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

Girotti M.
;
Grava T.;Jenkins R.;
2023-01-01

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.
2023
76
11
3233
3299
10.1002/cpa.22106
https://arxiv.org/abs/2205.02601
Girotti, M.; Grava, T.; Jenkins, R.; Mclaughlin, K. T. R.; Minakov, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/138271
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