Breather Solutions to the Cubic Whitham Equation

TL;DR

This paper provides numerical evidence that breather solutions, previously known in integrable models like mKdV, may also exist in the non-integrable cubic Whitham equation, a water-wave model combining nonlinearity and non-local effects.

Contribution

The study demonstrates, through numerical methods, the potential existence of breather solutions in the non-integrable cubic Whitham equation, extending the understanding of localized wave structures.

Findings

Numerical evidence of breather solutions in the cubic Whitham equation.

Breather solutions resemble those in integrable models like mKdV.

Indications that breather-like structures can exist in non-integrable water-wave models.

Abstract

We are concerned with numerical approximations of breather solutions for the cubic Whitham equation which arises as a water-wave model for interfacial waves. The model combines strong nonlinearity with the non-local character of the water-wave problem. The equation is non-integrable as suggested by the inelastic interaction of solitary waves. As a non local model, it generalizes, in the low frequency limit, the well known modified KdV (mKdV) equation which is a completely-integrable model. The mKdV equation has breather solutions, i.e. periodic in time and localized in space biparametric solutions. It was recently shown that these breather solutions appear naturally as ground states of invariant integrals, suggesting that such structures may also exist in non-integrable models, at least in an approximate sense. In this work, we present numerical evidence that in the non-integrable case of the cubic Whitham equation, breather solutions may also exist.