The linearly damped nonlinear Schrödinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events

Abstract

We prove spatiotemporal algebraically decaying estimates for the density of the solutions of the linearly damped nonlinear Schrödinger equation with localized driving, when supplemented with vanishing boundary conditions. Their derivation is made via a scheme, which incorporates suitable weighted Sobolev spaces and a time-weighted energy method. Numerical simulations examining the dynamics (in the presence of physically relevant examples of driver types and driving amplitude/linear loss regimes), showcase that the suggested decaying rates, are proved relevant in describing the transient dynamics of the solutions, prior their decay: they support the emergence of waveforms possessing an algebraic space-time localization (reminiscent of the Peregrine soliton) as first events of the dynamics, but also effectively capture the space-time asymptotics of the numerical solutions.

The Hybrid Finite Difference and Moving Boundary Methods for Solving a Linear Damped Nonlinear Schrödinger Equation to Model Rogue Waves and Breathers in Plasma Physics

“…x ψ + Q|ψ| 2 ψ + iRψ � 0) [38][39][40][41][42][43][44][45][46][47] was used in place of the standard CNLSE for studying the impact of fractional forces on the modulational instability (MI) of the modulated structures and associated damping waves. Here, R refers to the coefficient of the linear damping term and both CNLSE and dCNLSE can be derived from the fluid plasma equations using a reductive perturbation method (RPM) (the derivative expansion method (DEM)) [22][23][24][25][41][42][43][44][45][46].…”