Wave evolution within the Cubic Vortical Whitham equation

TL;DR

This paper investigates wave evolution in the Cubic Vortical Whitham equation, revealing breather structures and differences in wave breaking behavior depending on the sign of cubic nonlinearity, relevant for shear flow wave processes.

Contribution

It provides the first detailed analysis of wave dynamics in the CV-Whitham equation, highlighting effects of cubic nonlinearity sign on wave structures and breaking.

Findings

Breather-type structures form from depression disturbances with positive nonlinearity.

CV-Whitham and Gardner equations are similar for negative nonlinearity, with minor phase differences.

Positive nonlinearity leads to sharper waves and potential wave breaking.

Abstract

In this work, we study the evolution of disturbances within the framework of the Cubic Vortical Whitham (CV-Whitham) equation, considering both positive and negative cubic nonlinearities. This equation plays important role for description of the wave processes in the presence of shear flows. We find well-formed breather-type structures arising from the evolution of depression disturbances with positive cubic nonlinearity. For elevation disturbances, the results are two-fold. When the cubic nonlinearity is negative, we show that the CV-Whitham equation and the Gardner equation are qualitatively similar, differing only by a small phase lag due to differences in the dispersion term. However, with positive cubic nonlinearity, the differences between the solutions become more pronounced, with the CV-Whitham equation producing sharper waves that suggest the onset of wave breaking.