A generating mechanism for higher order rogue waves

TL;DR

This paper presents a new mechanism for generating higher order rogue waves in the nonlinear Schrödinger equation through the fusion and fission of breathers, enabling control over wave patterns and confirming conjectures about their structure.

Contribution

It introduces a novel mechanism involving breather fusion and fission for creating higher order rogue waves and proves related structural conjectures.

Findings

Mechanism based on breather fusion and fission generates HRWs.

Adjusting breather phases alters HRW types.

Proof of conjectures on peak count and circular pattern decomposition.

Abstract

We introduce a mechanism for generating higher order rogue waves (HRWs) of the nonlinear Schr\"odinger(NLS) equation: the progressive fusion and fission of $n$ degenerate breathers associated with a critical eigenvalue $\lambda_0$, creates an order $n$ HRW. By adjusting the relative phase of the breathers at the interacting area, it is possible to obtain different types of HRWs. The value $\lambda_0$ is a zero point of the eigenfunction of the Lax pair of the NLS equation and it corresponds to the limit of the period of the breather tending to infinity. By employing this mechanism we prove two conjectures regarding the total number of peaks, as well as a decomposition rule in the circular pattern of an order $n$ HRW.