The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discusse...

The Kadomtsev-Petviashvili equation describes nonlinear dispersive waves which travel mainly in one direction, generalizing the Korteweg de Vries equation for purely uni-directional waves. In this paper we derive an improved KP-equation that has exact dispersion in the main propagation direction and that is accurate in second order of the wave height. Moreover, different from the KP-equation, t...

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the...

We introduce an integrable time-discretized version of the classical CalogeroMoser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a discretized version of the Kadomtsev-Petviashvili equation, leading to a finite-dimensional symplectic mapping. Lax pair, symplectic structure and sufficient set of invariants of the discrete Cal...

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is ...

We show that the fifth-order Kadomtsev-Petviashvili II equation is globally well-posed in an anisotropic Gevrey space, which complements earlier results on well-posedness of this Sobolev spaces.