On nonlocal structure of the kinetic equation for a soliton gas
نویسندگان
چکیده
We investigate the structure of the nonlocal closure relation in the kinetic equation for a soliton gas. This kinetic equation represents an integro-differential nonlinear system which has been recently shown to possess a number of remarkable properties and seems to be a representative of an entirely new class of integrable systems. In this paper, we identify the nonlocal kinetic closure relation as the Fredholm integral equation of the second kind. For the typical case of a singular logarithmic kernel, this integral equation can be expressed in terms of the Hilbert transform, which opens a new perspective for the study of the integrability properties of the full kinetic equation. We then briefly discuss admissible hydrodynamic reductions obtained by the so-called ‘cold-gas’ delta-function ansatz for the distribution function.
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