A corresponding relationship between nonlinear Hermitian systems and linear non-Hermitian models

TL;DR

This paper explores the deep connections between nonlinear Hermitian systems and linear non-Hermitian models, revealing how spectral degeneracies relate to nonlinear localized wave phenomena like solitons and rogue waves.

Contribution

It demonstrates a quantitative relationship between nonlinear Hermitian dynamics and eigenvalue degeneracies in linear non-Hermitian matrices using integrable systems.

Findings

Degeneracies correspond to rogue waves and dark solitons.

Exceptional points relate to modulational instability and stability.

Provides new insights into the connection between nonlinear and linear spectral properties.

Abstract

We note that the non-orthogonality of states and their coincidence at the degeneracy point are both admitted by nonlinear Hermitian systems and linear non-Hermitian systems. These striking characteristics motivate us to re-investigate the localized waves of nonlinear Hermitian systems and the eigenvalue degeneracies of linear non-Hermitian models, based on several well-known Lax integrable systems that have wide applications in nonlinear optics. We choose nonlinear Schrodinger equation integrability hierarchy to demonstrate the quantitative relations between dynamics of nonlinear Hermitian systems and eigenvalue degeneracies of linear non-Hermitian models. Specifically, the degeneracies of the real or imaginary spectrum of the linear non-Hermitian matrices are uncovered to clarify several essential characteristics of nonlinear localized waves, such as breathers, rogue waves, and solitons. We find that the exceptional points generally correspond to rogue waves for modulational instability cases and dark solitons with maximum velocity for the modulational stability cases. These insights provide another interesting perspective for understanding nonlinear localized waves, and hint that there are closer relations between nonlinear Hermitian systems and linear non-Hermitian systems.