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

TL;DR

This paper discovers new rogue wave patterns in the nonlinear Schrödinger equation, characterized by diverse structures linked to Adler-Moser polynomials, expanding understanding of rogue wave phenomena.

Contribution

It introduces a novel connection between rogue wave patterns and Adler-Moser polynomials, revealing more diverse rogue wave structures than previously known.

Findings

New rogue wave patterns such as heart-shaped and fan-shaped structures

Analytical description of patterns via Adler-Moser polynomial root structures

Good agreement between analytical predictions and actual solutions

Abstract

We report new rogue wave patterns in the nonlinear Schr\"{o}dinger equation. These patterns include heart-shaped structures, fan-shaped sectors, and many others, that are formed by individual Peregrine waves. They appear when multiple internal parameters in the rogue wave solutions get large. Analytically, we show that these new patterns are described asymptotically by root structures of Adler-Moser polynomials through a dilation. Since Adler-Moser polynomials are generalizations of the Yablonskii-Vorob'ev polynomial hierarchy and contain free complex parameters, these new rogue patterns associated with Adler-Moser polynomials are much more diverse than previous rogue patterns associated with the Yablonskii-Vorob'ev polynomial hierarchy. We also compare analytical predictions of these patterns to true solutions and demonstrate good agreement between them.