Extreme wave events for a nonlinear Schrödinger equation with linear damping and Gaussian driving (Q2207736)

The paper addresses a perturbed Nonlinear Schrödinger equation in one dimension, with a linear dissipative term, with coefficient \(\gamma > 0\), and a direct driving term, \(f(x,t)\), of the ``instanton'' (localized-impulse) type, which is localized as a Gaussian both in time and in the coordinate: \(iu_t +(1/2)u_{xx}+ |u|^2u=-i\gamma u + f(x,t)\). The objective is to study, by means of direct numerical simulations of the equation, the evolutions of various spatially localized inputs, taken with exponential (quasi-soliton) or rational spatially decaying tails. One of the numerically generated solutions is construed as a transient (gradually decaying) Peregrine-like rogue waveform. Unlike the usual situation in which rogue waves of the Peregrine type are studied, here such a waveform emerges on top of a gradually vanishing localized background. In the course of the eventual decay, further evolution of the solution may resemble a breather of the nonlinear Schrödinger equation. Effects of variations of the dissipative constant and parameters of the impulse drive on properties of the transient solutions are studied.