Solitary waves of moderate amplitude in the SSGGN equations: the extended KdV-Whitham approximation

Abstract

We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves. In the slow time formulation this equation generates fast propagating resonant radiation due to the non-convexity of its linear dispersion curve, which is not present in the strongly nonlinear Serre-Su-Gardner-Green-Naghdi (SSGGN) parent system. We show that the extended KdV-Whitham approximation and the slow space formulation of the eKdV equation are suitable regularisations of the eKdV equation in several cases of interest, and even for moderate amplitudes. Numerical comparisons are made between the SSGGN system and the respective reduced models, where simulations are initiated with an approximate soliton solution of the eKdV equation, constructed by use of Kodama-Fokas-Liu near-identity transformation to the KdV equation.