Rogue wave patterns associated with Adler--Moser polynomials featuring multiple roots in the nonlinear Schr\"odinger equation

TL;DR

This paper explores the asymptotic behaviors of high-order rogue waves in the nonlinear Schrödinger equation, revealing novel patterns linked to Adler--Moser polynomial roots and demonstrating controllable rogue wave structures.

Contribution

It establishes a novel connection between rogue wave patterns and the root structures of Adler--Moser polynomials with multiple roots, introducing new rogue wave configurations and control mechanisms.

Findings

Discovered new rogue wave patterns including claw-like and TTR-type.

Linked rogue wave structures to roots of Adler--Moser polynomials.

Showed controllability of rogue wave positions and structures.

Abstract

In this work, we analyze the asymptotic behaviors of high-order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including claw-like, OTR-type, TTR-type, semi-modified TTR-type, and their modified patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler--Moser polynomials with multiple roots. At the positions in the $(x,t)$-plane corresponding to single roots of the Adler--Moser polynomials, these high-order rogue wave patterns asymptotically approach first-order rogue waves. At the positions in the $(x,t)$-plane corresponding to multiple roots of the Adler--Moser polynomials, these rogue wave patterns asymptotically tend toward lower-order fundamental rogue waves, dispersed first-order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler--Moser polynomials with new free parameters, such as the Yablonskii--Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower-order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions.