Phase Dynamics of the Dysthe equation and the Bifurcation of Plane Waves

TL;DR

This paper studies how plane waves bifurcate into localized structures in the Dysthe equation, revealing the role of mean flow and wave steepening through phase modulation and KdV dynamics.

Contribution

It introduces a phase modulation approach to connect the Dysthe equation with KdV, elucidating the effects of mean flow and steepening on bifurcation and solitary wave formation.

Findings

Bifurcation of plane waves analyzed using phase modulation.

Solitary wave solutions serve as prototypes for bifurcating dynamics.

Higher order phase effects influence wave transformation under phase defects.

Abstract

The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such occurrences may be described using a Korteweg - de Vries (KdV) equation. The solitary wave solutions of this system form a qualitative prototype for the bifurcating dynamics, and the role of mean flow and steepening is then made clear through how they enter the amplitude and width of these solitary waves. Additionally, higher order phase dynamics are investigated, leading to increased nonlinear regimes which in turn have a more profound impact on how the plane waves transform under defects in the phase.