Higher-order vector Peregrine solitons and asymptotic estimates for the multi-component nonlinear Schr\"odinger equations

TL;DR

This paper introduces higher-order vector Peregrine solitons for multi-component nonlinear Schrödinger equations, providing explicit solutions, symmetry properties, and asymptotic analysis methods to understand their structure and maximal amplitudes.

Contribution

It presents the first explicit construction of higher-order vector Peregrine solitons using loop group theory and develops a systematic approach for their asymptotic analysis.

Findings

Explicit higher-order vector Peregrine solitons are derived.

Vector rogue waves exhibit parity-time symmetry under certain conditions.

Asymptotic behavior and maximal amplitudes of rogue waves are characterized.

Abstract

We first report the first- and higher-order vector Peregrine solitons (alias rational rogue waves) for the any multi-component NLS equations based on the loop group theory, an explicit (n + 1)-multiple eigenvalue of a characteristic polynomial of degree (n + 1) related to the condition of Benjamin-Feir instability, and inverse functions. Particularly, these vector rational rogue waves are parity-time symmetric for some parameter constraints. A systematic and effective approach is proposed to study the asymptotic behaviors of these vector rogue waves such that the decompositions of rogue waves are related to the so-called governing polynomials, which pave a powerful way in the study of vector rogue wave structures of the multi-component integrable systems. The vector rogue waves with maximal amplitudes can be determined via the parameter vectors, which is interesting and useful in the multi-component physical systems.