# Two-dimensional rogue waves on zero background of the Davey-Stewartson II equation

## TL;DR

This paper constructs two-dimensional rogue wave solutions for the Davey-Stewartson II equation using a novel eigenfunction hierarchy and Darboux transformation, providing models for oceanic rogue waves and 2D fluid phenomena.

## Contribution

It introduces a new analytical method to generate 2D rogue waves in the DS II equation via eigenfunction hierarchy and Darboux transformation, extending rogue wave theory to two dimensions.

## Key findings

- Derived localized 2D rogue wave solutions for DS II
- These solutions are analogues of the Peregrine soliton in 2D
- Potential applications in modeling oceanic rogue waves

## Abstract

A prototypical example of a rogue wave structure in a two-dimensional model is presented in the context of the Davey-Stewartson~II (DS~II) equation arising in water waves. The analytical methodology involves a Taylor expansion of an eigenfunctionof the model's Lax pair which is used to form a hierarchy of infinitely many new eigenfunctions. These are used for the construction of two-dimensional (2D) rogue waves (RWs) of the DS~II equation by the even-fold Darboux transformation (DT). The obtained 2D RWs, which are localized in both space and time, can be viewed as a 2D analogue of the Peregrine soliton and are thus natural candidates to describe oceanic RW phenomena,as well as ones in 2D fluid systems and water tanks.