Universal patterns of rogue waves

TL;DR

This paper reveals that rogue wave patterns in integrable systems like the NLS equation exhibit universal geometric structures, which are analytically linked to the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, especially for large free parameters.

Contribution

It establishes a universal connection between rogue wave patterns and the root structures of Yablonskii-Vorob'ev polynomials across different integrable equations.

Findings

Rogue wave patterns form geometric structures such as triangles and polygons.

Patterns are determined by the root structures of Yablonskii-Vorob'ev polynomials.

Orientation of patterns is controlled by the phase of free parameters.

Abstract

Rogue wave patterns in the nonlinear Schr\"{o}dinger (NLS) equation and the derivative NLS equation are analytically studied. It is shown that when the free parameters in the analytical expressions of these rogue waves are large, these waves would exhibit the same patterns, comprising fundamental rogue waves forming clear geometric structures such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are controlled by the phase of the large free parameter. This connection of rogue wave patterns to the root structures of the Yablonskii-Vorob'ev polynomial hierarchy goes beyond the NLS and derivative NLS equations, and it gives rise to universal rogue wave patterns in integrable systems.