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

Abstract

We perform a numerical study of the initial-boundary value problem, with vanishing boundary conditions, of a driven nonlinear Schrödinger equation (NLS) with linear damping and a Gaussian driver. We identify Peregrine-like rogue waveforms, excited by two different types of vanishing initial data decaying at an algebraic or exponential rate. The observed extreme events emerge on top of a decaying support. Depending on the spatial/temporal scales of the driver, the transient dynamics -prior to the eventual decay of the solutions -may resemble the one in the semiclassical limit of the integrable NLS, or may, e.g., lead to large-amplitude breather-like patterns. The effects of the damping strength and driving amplitude in suppressing or enhancing respectively the relevant features, as well as of the phase of the driver in the construction of a diverse array of spatiotemporal patterns, are numerically analyzed.

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

“…In [1], we considered the case of a Gaussian driver f = f (x, t), and for the problem supplemented with vanishing boundary conditions, we found that the PRW-structures emerged on the top of a decaying support. Particularly, we found that pending on the spatial/temporal scales of the driver, the transient dynamics prior the ultimate decay of the solutions, resembles the one in the semiclassical limit of the integrable NLS [4], [5], [6] (or may lead to large-amplitude breather-like patterns).…”

“…In the present study, we aim to move a step forward to the analytical considerations for the problem discussed in [1]: For equation (1), when endowed with vanishing boundary and initial conditions lim |x|→∞ u(x, t) = 0, for all t ≥ 0,…”

“…Second, additional implications are caused by the absence of higher order dissipation, which may impose further restrictions to the chosen weight (e.g. on the boundedness of its derivative); these implications may be enhanced, by the necessity to derive some uniform in time estimates (as guided by the numerical results of [1]). Third, although the unbounded weights have been introduced in [15] (for the study of global attractors of localized solutions for reaction-difussion equations), or have been used at an auxiliary step to deduce the compactness of the attractor for the NLS (1) in standard Sobolev spaces (through a suitable decomposition of the semigroup) [13], [14], here, we aim to highlight the usefulness of the approach in deriving point-wise estimates for the density of its solutions; up to our knowledge this approach is novel itself in order to study the spatial asymptotics of the solutions of Eq.…”

“…Third, although the unbounded weights have been introduced in [15] (for the study of global attractors of localized solutions for reaction-difussion equations), or have been used at an auxiliary step to deduce the compactness of the attractor for the NLS (1) in standard Sobolev spaces (through a suitable decomposition of the semigroup) [13], [14], here, we aim to highlight the usefulness of the approach in deriving point-wise estimates for the density of its solutions; up to our knowledge this approach is novel itself in order to study the spatial asymptotics of the solutions of Eq. (1) in the presence of the localized driver.…”

“…Such a regime could describe the impact of strong external forcing Γ, against small fluid viscosity γ (see [27], [28]). Recalling some of the numerical results of [1] for the Gaussian driver, we proceed to the study of the dynamics in the presence of the weakly localized forcing. In the present study, we consider both cases, of algebraically and exponentially decaying initial conditions (2).…”