The infinite-genus limit of the KdV–Whitham equations is derived. The limit involves special scaling for the associated spectral surface such that the integrated density of states remains finite as N →∞ (the thermodynamic type limit). The limiting integro-differential system describes slow evolution of the density of states and can be regarded as the kinetic equation for soliton gas.  2003 Els...

The existence of traveling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves of finite depth. It is found that there exist small-amplitude periodic traveling waves with sub-critical speeds. As the period of these traveling waves tends to infinity, their velocities ap...

The author’s recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter ǫ. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dy...