Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation

TL;DR

This paper predicts and confirms the existence of partial-rogue waves in the Sasa-Satsuma equation, showing they originate from specific rational solutions and split into solitons over time, with detailed mathematical analysis and numerical validation.

Contribution

It introduces a new class of partial-rogue wave solutions in the Sasa-Satsuma equation linked to generalized Okamoto polynomials, expanding understanding of wave dynamics.

Findings

Partial-rogue waves arise from certain rational solutions.

These waves split into fundamental solitons at large positive times.

Numerical results agree well with asymptotic predictions.

Abstract

Partial-rogue waves, i.e., waves that ``come from nowhere but leave with a trace", are analytically predicted and numerically confirmed in the Sasa-Satsuma equation. We show that, among a class of rational solutions in this equation that can be expressed through determinants of 3-reduced Schur polynomials, partial-rogue waves would arise if these rational solutions are of certain orders, where the associated generalized Okamoto polynomials have real but not imaginary roots, or imaginary but not real roots. We further show that, at large negative time, these partial-rogue waves approach the constant-amplitude background, but at large positive time, they split into several fundamental rational solitons, whose numbers are determined by the number of real or imaginary roots in the underlying generalized Okamoto polynomial. Our asymptotic predictions are compared to true solutions, and excellent agreement is observed.