Growth rate of modulation instability driven by superregular breathers

TL;DR

This paper establishes an exact relationship between super-regular breathers and modulation instability growth rate in nonlinear Schrödinger equations, supported by numerical simulations demonstrating robustness and potential for controlled excitations.

Contribution

It reveals a precise link between SR breathers' group velocity differences and MI growth rate across a broad class of equations, advancing understanding of MI dynamics.

Findings

The absolute difference of SR breathers' group velocities equals the linear MI growth rate.

The MI growth rate of SR breathers matches that of individual quasi-Akhmediev breathers in certain cases.

Numerical simulations confirm the robustness of SR breathers from various initial conditions.

Abstract

We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schr\"{o}dinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.