KdV breathers on a cnoidal wave background

TL;DR

This paper constructs exact breather solutions of the KdV equation on a cnoidal wave background, revealing their propagation characteristics and connections to soliton interactions, with implications for recent experimental observations.

Contribution

It introduces new breather solutions for the KdV equation on a cnoidal background using Darboux transformation, including bright and dark types, and analyzes their properties.

Findings

Bright breathers propagate faster than the background.

Dark breathers propagate slower and are approximated by NLS dark solitons.

Two-soliton solutions are derived in the cnoidal wave degeneration limit.

Abstract

Using the Darboux transformation for the Korteweg-de Vries equation, we construct and analyze exact solutions describing the interaction of a solitary wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these wave patterns are referred to as breathers. Both elevation (bright) and depression (dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate that the bright (dark) breathers propagate faster (slower) than the background cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated by dark soliton solutions of the nonlinear Schr\"odinger equation. These results provide insight into recent experiments on soliton-dispersive shock wave interactions and soliton gases.