The linearly damped nonlinear Schrödinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events
arXiv:1910.08425 · doi:10.1007/s00033-019-1223-y
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.
References in corpus (8)
- Universal nonlinear stage of the locally induced modulational instability in fiber optics
- Approximate rogue wave solutions of the forced and damped Nonlinear Schrödinger equation for water waves
- Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model
- Nonlinear fast growth of water waves under wind forcing
- Wave amplification in the framework of forced nonlinear Schrodinger equation: the rogue wave context
- One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: Remarks on duality and flows
- Collapse dynamics for the discrete nonlinear Schrödinger equation with gain and loss
- Excitation of Peregrine-type waveforms from vanishing initial conditions in the presence of periodic forcing