Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

Abstract

We study here the initial value problem for a two-dimensional Korteweg-de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient T 0 (y, k), k ∈ C on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp-discontinuity. Finally, we give conditions under which the evolution u(x, y, t) is reduced to the classical KdV.