Coexistence of two distinct rogue wave patterns in the coupled nonlinear Schr\"odinger equation

TL;DR

This paper explores high-order vector rogue wave solutions of the coupled nonlinear Schrödinger equation, revealing new patterns with two distinct regions of coexisting rogue waves linked to Adler--Moser polynomial roots.

Contribution

It introduces novel high-order rogue wave patterns in the CNLS system, connecting their structure to Adler--Moser polynomial roots and demonstrating parameter tuning for pattern control.

Findings

Identified new rogue wave patterns including double-sector, double-heart, and mixed configurations.

Showed each pattern contains two regions with different rogue wave types linked to Adler--Moser polynomial roots.

Demonstrated parameter tuning allows shifting rogue wave regions arbitrarily in space-time.

Abstract

This paper investigates the asymptotic behavior of high-order vector rogue wave (RW) solutions of the coupled nonlinear Schr\"odinger (CNLS) equation in the presence of multiple large internal parameters. We report several new high-order RW patterns in the CNLS system, including double-sector, double-heart, and mixed sector-heart configurations. The main novelty is that each RW pattern contains two distinct regions in which two different fundamental first-order RWs coexist simultaneously, potentially appearing as bright (eye-shaped) versus four-petaled or dark (anti-eye-shaped) forms. These two regions are respectively associated with the simple root structures of two different Adler--Moser polynomials: each region consists of well-separated first-order RWs in one-to-one correspondence with the simple roots of the associated polynomial. In addition, by tuning certain free parameters, the two regions of the RW pattern can be shifted to arbitrary locations in the $ (x,t) $-plane. This flexibility, together with the rich simple-root structures of Adler--Moser polynomials, enables the systematic generation of a much broader family of structured RW patterns in the CNLS equation.