Nonlinearly dispersive KP equations with new compacton solutions

Abstract

An exhaustive classification of compacton solutions is carried out for a generalization of the Kadomtsev-Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are given on the nonlinearity powers in this equation under which a travelling wave can be cut off to obtain a compacton. It is shown that there are no compactons which are classical (strong) solutions. Instead, the compactons consist of pointwise distributional solutions as well as weak solutions of an integrated from of the ODE for travelling waves. Weak-compacton solutions constitute a new type of solution which has not been studied previously. An explicit example is obtained in which the compacton profile is a power of an expression that is linear in the travelling wave variable and its sine. Explicit compactons with profiles given by powers of a cosine, a sine, Jacobi sn and cn functions, and a quadratic function are also derived. In comparison, explicit solitary waves of the generalized KP equation are found to have profiles given by a power of a sech and a reciprocal quadratic function. Kinematic properties of all of the different types of compactons and solitary waves are discussed, along with conservation laws of the generalized KP equation.