On the method of strained parameters for a KdV type of equation with exact dispersion property

TL;DR

This paper introduces a novel method using strained parameters to analyze three-wave interactions in a KdV-type equation with exact dispersion, effectively managing secular growth in solutions for gravity waves.

Contribution

It develops an alternative methodology employing strained parameters to control secular growth in a KdV-type equation with exact dispersion for gravity waves.

Findings

Method successfully prevents spurious secular growth.

Perturbation theory applied in wavenumber domain.

Nonlinear dispersion relations derived for specific wavenumber combinations.

Abstract

This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact dispersion property is adopted as the governing equation for unidirectional wave packet evolution. Following the idea from Zakharov's seminal paper (Zakharov, V. E. (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. \textit{Journal of Applied Mechanics and Technical Physics}, {\bf 9}, 190--194), the equation is transformed from the spatial--temporal domain to the wavenumber--temporal domain. The solution of the transformed equation is sought using the perturbation theory, for which the ansatz is expressed in the form of a regular expansion in the increasing order of a small parameter. After implementing the na\"{i}ve perturbation method, due to nonlinear mode generation and particular combinations of wavenumbers, the third-order solution contains spurious secular growth terms which appear as a consequence of resonant interaction and nonlinear mode generation. These spurious secular growth terms can be prevented by implementing the method of strained parameters for which nonlinear dispersion relation terms are produced for particular combination of wavenumbers.