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

TL;DR

This paper investigates the extended KdV equation as a model for moderate amplitude water waves, comparing it with the more complex SSGGN system, and shows how the extended KdV-Whitham approximation effectively regularizes the model.

Contribution

It demonstrates that the extended KdV-Whitham approximation provides a suitable regularization for the eKdV equation at moderate amplitudes, with numerical validation against the SSGGN system.

Findings

The extended KdV-Whitham approximation captures key wave dynamics.

Numerical simulations show good agreement between models.

Resonant radiation effects are identified in the eKdV equation.

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.