On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves

TL;DR

This paper introduces a homotopy-based multiple-variable method for analyzing nonlinear interactions of gravity waves, which does not rely on small parameters and reveals complex resonance phenomena in fully developed wave systems.

Contribution

It presents a novel homotopy multiple-variable method that generalizes traditional techniques and enables analysis of large-amplitude wave interactions and multiple resonances.

Findings

Amplitudes of all wave components remain finite at resonance.

Multiple resonant waves can exist with smaller amplitudes than primary waves.

A generalized resonance condition for many interacting waves is derived.

Abstract

The basic ideas of a homotopy-based multiple-variable method is proposed and applied to investigate the nonlinear interactions of periodic traveling waves. Mathematically, this method does not depend upon any small physical parameters at all and thus is more general than the traditional multiple-scale perturbation techniques. Physically, it is found that, for a fully developed wave system, the amplitudes of all wave components are finite even if the wave resonance condition given by Phillips (1960) is exactly satisfied. Besides, it is revealed that there exist multiple resonant waves, and that the amplitudes of resonant wave may be much smaller than those of primary waves so that the resonant waves sometimes contain rather small part of wave energy. Furthermore, a wave resonance condition for arbitrary numbers of traveling waves with large wave amplitudes is given, which logically contains Phillips' four-wave resonance condition but opens a way to investigate the strongly nonlinear interaction of more than four traveling waves with large amplitudes. This work also illustrates that the homotopy multiple-variable method is helpful to gain solutions with important physical meanings of nonlinear problems, if the multiple variables are properly defined with clear physical meanings.