Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

Abstract

The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger equation, prominent in modeling rogue waves, are unstable. In this paper we numerically investigate the routes to stability of the SPBs and related rogue wave activity in the framework of the nonlinear damped higher order nonlinear Schrödinger (NLD-HONLS) equation. The NLD-HONLS solutions are initialized with single-mode and two-mode SPB data at different stages of their development. The Floquet spectral theory of the NLS equation is applied to interpret and provide a characterization of the perturbed dynamics in terms of nearby solutions of the NLS equation. A broad categorization of the routes to stability of the SPBs is determined. Novel behavior related to the effects of nonlinear damping is obtained: tiny bands of complex spectrum develop in the Floquet decomposition of the NLD-HONLS data, indicating the breakup of the SPB into either a one or two “soliton-like” structure. For solutions initialized in the early to middle stage of the development of the modulational instability, we find that all the rogue waves in the NLD-HONLS flow occur when the spectrum is in a one or two soliton-like configuration. When the solutions are initialized as the modulational instability is saturating, rogue waves also occur when the spectrum is not in a soliton-like state. Another distinctive feature of the nonlinear damped dynamics is that the growth of instabilities can be delayed and expressed at higher order due to permanent frequency downshifting.