Universal rogue wave patterns associated with the Yablonskii-Vorob'ev polynomial hierarchy

TL;DR

This paper reveals that universal rogue wave patterns in integrable systems are governed by the root structures of Yablonskii-Vorob'ev polynomials, manifesting as geometric arrangements like triangles and pentagons in space-time.

Contribution

It establishes a novel connection between rogue wave patterns and Yablonskii-Vorob'ev polynomial root structures, providing explicit examples across multiple integrable equations.

Findings

Rogue wave patterns correspond to root structures of Yablonskii-Vorob'ev polynomials.

Patterns include geometric shapes like triangles, pentagons, heptagons.

Patterns are influenced by internal parameters and transformations.

Abstract

We show that universal rogue wave patterns exist in integrable systems. These rogue patterns comprise fundamental rogue waves arranged in shapes such as triangle, pentagon and heptagon, with a possible lower-order rogue wave at the center. These patterns appear when one of the internal parameters in bilinear expressions of rogue waves gets large. Analytically, these patterns are determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy through a linear transformation. Thus, the induced rogue patterns in the space-time plane are simply the root structures of Yablonskii-Vorob'ev polynomials under actions such as dilation, rotation, stretch, shear and translation. As examples, these universal rogue patterns are explicitly determined and graphically illustrated for the generalized derivative nonlinear Schr\"odinger equations, the Boussinesq equation, and the Manakov system. Similarities and differences between these rogue patterns and those reported earlier in the nonlinear Schr\"odinger equation are discussed.