A new framework for studying the stability of genus-1 and genus-2 KP patterns

Abstract

The Kadomtsev-Petviashvili equation - or KP equation - is a model equation for waves that are weakly two-dimensional in a horizontal plane, and models water waves in shallow water with weak three-dimensionality. It has a vast array of interesting genus— k pattern solutions which can be obtained explicitly in terms of Riemann theta functions. However the linear or nonlinear stability of these patterns has not been studied. In this paper, we present a new formulation of the KP model as a Hamiltonian system on a multi-symplectic structure. While it is well-known that the KP model is Hamiltonian - as an evolution equation in time - multi-symplecticity assigns a distinct symplectic operator for each spatial direction as well, and is independent of the integrability of the equation. The multi-symplectic framework is then used to formulate the linear stability problem for genus-1 and genus-2 patterns of the KP equation; generalizations to genus— k with k > 2 are also discussed.