Spectral stability of resonant Y-solitons for the KP-II equation

Abstract

The KP-II equation was introduced by Kadomtsev and Petviashvili to explain the transverse stability of line solitary waves in shallow water. Using Darboux transformations, we prove the spectral stability of Y-shaped line solitons in an exponentially weighted space that captures the amount of waves receding behind the line solitons. We show that the spectrum of the linearized operator consists of the continuous spectrum of the zero background together with three residual spectral domains whose boundaries contain continuous spectra associated with the constituent line solitons. Exploiting the analytic dependence of the pairings with adjoint eigenfunctions on the spectral parameters, we prove that the orthogonality conditions associated with all adjoint eigenfunctions can be reduced to orthogonality conditions against three families of adjoint continuous eigenfunctions. Moreover, we investigate linear stability of Y-solitons in a subspace satisfying orthogonality conditions against adjoint continuous eigenfunctions associated with outgoing modulations along each line soliton constituting the Y-soliton.