Dr. Eugene Ferapontov | Delta-functional reductions of the kinetic equation for soliton gas

Speaker: Dr Eugene Ferapontov (Loughborough University)
Date: 1st Aug 2024
Title: Delta-functional reductions of the kinetic equation for soliton gas
Event: (PDWW01) Frontiers of Dispersive Hydrodynamics
Abstract: Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states [1, 2]. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type [3] whose matrix consists of several 2 × 2 Jordan blocks of the type studied in [4]. Following [5], we discuss the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach can be viewed as a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure. Following [6, 7], Hamiltonian formulation of the reduced system will also be discussed.
[1] G.A. El, The thermodynamic limit of the Whitham equations, Phys. Lett. A 311 374-383(2003).[2] G.A. El, A.M. Kamchatnov, Kinetic equation for a dense soliton gas, Phys. Rev. Lett., 95204101 (2005).[3] G.A. El, A.M. Kamchatnov, M.V. Pavlov, S.A. Zykov, Kinetic equation for a soliton gasand its hydrodynamic reductions, J. Nonlinear Sci. 21 151-191 (2011).[4] Lingling Xue, E.V. Ferapontov, Quasilinear systems of Jordan block type and the mKPhierarchy, J. Phys. A: Math. Theor. 53 (2020) 205202.[5] E.V. Ferapontov, M.V. Pavlov, Kinetic equation for soliton gas: integrable reductions, J.Nonlinear Sci. 32 (2022).[6] P. Vergallo, E.V. Ferapontov, Hamiltonian systems of Jordan block type: delta-functionalreductions of the kinetic equation for soliton gas, J. Math. Phys. 64, 103505 (2023).[7] P. Vergallo, E.V. Ferapontov, Hamiltonian aspects of the kinetic equation for soliton gas,arXiv:2403.20162.

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