Elliptic-rogue waves and modulational instability in nonlinear soliton equations

TL;DR

This paper introduces elliptic-rogue wave solutions for integrable nonlinear equations on elliptic backgrounds, linking their emergence to modulational instability and providing a method to construct higher-order solutions.

Contribution

It refines the modified squared wave function method with Darboux-Bäcklund transformation to connect elliptic-rogue waves with modulational instability and derives higher-order solutions.

Findings

Elliptic-rogue waves emerge on elliptic backgrounds unlike plane wave solutions.

Modulational instability triggers elliptic-rogue waves, while stability leads to elliptic solitons or breathers.

The method enables construction of higher-order elliptic-rogue waves.

Abstract

We present elliptic-rogue wave solutions for integrable nonlinear soliton equations in theta functions. Unlike solutions generated on the plane wave background, these solutions depict rogue waves emerging on elliptic function backgrounds. By refining the modified squared wave function method in tandem with the Darboux-B\"acklund transformation, we establish a quantitative correspondence between elliptic-rogue waves and the modulational instability. This connection reveals that the modulational instability of elliptic function solutions triggers rational-form solutions displaying elliptic-rogue waves, whereas the modulational stability of elliptic function solutions results in the rational-form solutions exhibiting the elliptic-solitons or elliptic-breathers. Moreover, this approach enables the derivation of higher-order elliptic-rogue waves, offering a versatile framework for constructing elliptic-rogue waves and exploring modulational stability in other integrable equations.