On transverse stability of periodic waves in the Kadomtsev-Petviashvili equation

Abstract

This thesis is devoted to the proof of linear stability of the one-dimensional periodic waves in the Kadomtsev-Petviashvili (KP-II) equation with respect to two-dimensional bounded perturbations. The method of the proof is based on the construction of a self-adjoint operator K such that the operators JL and JK commute, expresses a symplectic structure for the KP-II equation and L is a self-adjoint Hessian operator of the energy function at the periodic wave. In the situation when K is strictly positive except for a nite-dimensional kernel included in the kernel of L, the operator JL has no unstable eigenvalues and the associated time evolution is globally bounded.