Semi-rational solutions of the third-type Davey-Stewartson equation

TL;DR

This paper derives semi-rational solutions, including rogue waves, breathers, and solitons, for the third-type Davey-Stewartson equation using the bilinear method, revealing complex wave interactions and new phenomena.

Contribution

It introduces a bilinear method-based approach to generate semi-rational solutions for the DS-III equation, including rogue waves and hybrid wave patterns, with explicit determinant forms.

Findings

Fundamental rogue waves are line rogue waves.

Multi-rogue waves depict interactions of several fundamental rogue waves.

Lumps form on dark solitons and gradually separate from them.

Abstract

General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simplest) rogue waves are line rogue waves. It is also shown that the multi-rogue waves describe interactions of several fundamental rogue waves, which would generate interesting curvy wave patterns. The higher order rogue waves originate from a localized lump and retreat back to it. Several types of hybrid solutions composed of rogue waves, breathers and solitons have also been illustrated. Specifically, these semi-rational solutions have a new phenomenon: lumps form on dark solitons and gradual separation from the dark solitons is observed.