Higher-dimensional extended shallow water equations and resonant soliton radiation

Abstract

Higher order equations governing long surface waves in shallow water beyond the standard Korteweg-de Vries and Kadomtsev-Petviashvili equations are derived from the full water wave equations. The higher order dispersive and nonlinear terms in these equations lead to resonance between nonlinear wave structures and dispersive radiation. This resonance between solitary waves and dispersive shock waves (undular bores) is studied numerically and analytically using exponential asymptotic theory. A key result is that there is a near node in the resonant wave amplitude for the higher order equations with the water wave coefficients.