Zero-gain branch selection of higher-order rational solitons in a fifth-order nonlinear Schrödinger equation

Abstract

We investigate a fifth-order nonlinear Schrödinger equation, the zero- and nonzero-gain conditions on the modulation-instability resonance line acts as a branch selection of rational solutions. The modulational instability suppressing factor dictates from the generalized Darboux transformation: rogue waves with nonzero-gain, and W-shaped rational solitons with zero-gain. This zero-gain branch generates the higher-order rogue wave and the rational solitons. Asymptotic expressions confirm elastic interactions for second-order solitons and a third-order three-channel structure. Residuals validate solutions, proving modulational instability zero-gain selects rational soliton branches in higher-order models.

Graphical abstract

High-order rational soliton branch selection based on the zero-gain condition of modulation instability

This study reveals the zero-gain condition on the modulation instability (MI) resonance lines in the quintic nonlinear Schrödinger equation as a key mechanism for selecting rational wave solution branches. As shown in the figure, MI gain analysis identifies distinct stability regions:

1. 1.

   Non-zero gain branch: a rogue wave structure with dual temporal and spatial localization.

2. 2.

   Zero-gain branch: The selected propagating W-shaped rational soliton exhibits a distinctive peak-valley structure on the background wave and demonstrates channel characteristics of elastic interactions in higher-order scenarios.

This discovery establishes the MI zero-gain condition as an algebraic selection criterion for transitioning from linear analysis to precise nonlinear solution structures.