Stability of line solitons for the KP-II equation in ℝ²

Source abstract

We prove nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as x → ∞ x\to \infty . We find that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward y = ± ∞ y=\pm \infty . The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.