An asymptotic approach to solitary wave instability and critical collapse in long-wave KdV-type evolution equations

Instability development and critical collapse of solitary waves are considered in the framework of generalized Korteweg-de Vries (KdV) equations in one and two dimensions. An analytical theory of the solitary wave dynamics and generation of radiation is constructed for the critical case when the solitary waves are weakly unstable. Characteristic types of the global, essentially nonlinear evolution of the unstable solitary waves are analyzed for some typical generalized KdV equations. The scalling laws of the self-similar wave field transformation are found analytically for the power-like KdV equation in the critical case p = 4. The asymptotic approach is also developed for the modified Zakharov-Kuznetsov equation in two dimensions and the rate of the singularity formation is found to be smaller than in one dimension due to diffractive wave effects.