Line-solitons, line-shocks, and conservation laws of a universal KP-like equation in 2+1 dimensions

Abstract

A universal KP-like equation in 2+1 dimensions, which models general nonlinear wave phenomena exhibiting p-power nonlinearity, dispersion, and small transver-sality, is studied. Special cases include the integrable KP (Kadomtsev-Petviashvili) equation and its modified version, as well as their p-power generalizations. Two main results are obtained. First, all low-order conservation laws are derived, includ-ing ones that arise for special powers p. The conservation laws comprise momenta, energy, and Galilean-type quantities, as well as topological charges. Their physical meaning and properties are discussed. The topological charges are shown to give rise to integral constraints on initial data for the Cauchy problem. Second, all line-soliton solutions are obtained in an explicit form. A parameterization is given using the speed and the direction angle of the line-soliton, and the allowed kinematic region is determined in terms of these parameters. Basic kinematical properties of the line-solitons are also discussed. These properties differ significantly compared to those for KP line-solitons and their p-power generalizations. A line-shock solution is shown to emerge when a special limiting case of the kinematic region is considered.