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Hamiltonian aspects of the kinetic equation for soliton gas

Pierandrea Vergallo, Evgeny V. Ferapontov

Abstract

We investigate Hamiltonian aspects of the integro-differential kinetic equation for dense soliton gas which results as a thermodynamic limit of the Whitham equations. Under a delta-functional ansatz, the kinetic equation reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several $2\times 2$ Jordan blocks. We demonstrate that the resulting system possesses local Hamiltonian structures of differential-geometric type, for all standard two-soliton interaction kernels (KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, and separable cases). In the hard-rod case, we show that the continuum limit of these structures provides a local multi-Hamiltonian formulation of the full kinetic equation.

Hamiltonian aspects of the kinetic equation for soliton gas

Abstract

We investigate Hamiltonian aspects of the integro-differential kinetic equation for dense soliton gas which results as a thermodynamic limit of the Whitham equations. Under a delta-functional ansatz, the kinetic equation reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several Jordan blocks. We demonstrate that the resulting system possesses local Hamiltonian structures of differential-geometric type, for all standard two-soliton interaction kernels (KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, and separable cases). In the hard-rod case, we show that the continuum limit of these structures provides a local multi-Hamiltonian formulation of the full kinetic equation.
Paper Structure (13 sections, 2 theorems, 130 equations)

This paper contains 13 sections, 2 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian formulation of delta-functional reductions
  3. Hamiltonian operators of differential-geometric type, Tsarev's conditions
  4. Flatness of the metric tensor
  5. Local Hamiltonian formulation of the case $(\log G)_{\mu \eta}\ne 0$
  6. Local Hamiltonian formulation of the case $(\log G)_{\mu \eta}= 0$
  7. Casimirs and momenta
  8. Hamiltonian formulation of the full hard-rod kinetic equation
  9. Moment representation and linearisation
  10. Hamiltonian structures
  11. Monge-Ampère form of delta-functional reductions
  12. Concluding remarks
  13. Acknowledgements

Key Result

Theorem 1

The metric specified by (nm) is flat if and only if the functions $g_i(r^i, \eta^i)$ are quadratic in $r^i$, furthermore, the following conditions must be satisfied:

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2