Strang splitting for a semilinear Schrödinger equation with damping and forcing

Abstract

We propose and analyze a Strang splitting method for a cubic semilinear Schrödinger equation with forcing and damping terms. The nonlinear part is solved analytically, whereas the linear part -space derivatives, damping and forcing -is approximated by the exponential trapezoidal rule. The necessary operator exponentials and φ-functions can be computed efficiently by fast Fourier transforms if space is discretized by spectral collocation. We show wellposedness of the problem and H 4 (T) regularity of the solution for initial data in H 4 (T) and sufficiently smooth forcing. Under these regularity assumptions, we prove a first-order error bound in H 1 (T) and a second-order error bound in L 2 (T) on bounded time-intervals.

A Priori Bounds and Global Bifurcation Results for Frequency Combs Modeled by the Lugiato--Lefever Equation

“…With these choices we numerically integrated equation (5.1) starting from initial data given by the spatially constant steady-state solution at ζ = −5 perturbed by a random function of size 10 −14 . The numerical scheme is a Strang-splitting 1 in time as suggested in [13], and a pseudo-spectral method in space.…”

“…Further mathematical results. A rigorous study of the time-dependent problem (1.7) both from the analytical and from the numerical point of view was recently given in [13]. Applying Theorem 2.1 of [13] to the function a(x, t)e iζt one obtains that for d = 1 and initial data lying in H 4 per ([0, 2π]; C) the initial value problem associated to (1.7) admits a unique solution…”

The Lugiato–Lefever Equation with Nonlinear Damping Caused by Two Photon Absorption

“…Finally, still in the case κ = 0 we mention some results about the time-dependent equation (1). In [11] it was proved that the initial value problem is globally well-posed in…”

“…In the corresponding model with an additional third order dispersion effect well-posedness results and even the existence of a global attractor were proved in [21]. Convergence results for the numerical Strang-splitting scheme can be found in [11]. Finally, the orbital asymptotic stability of 2π-periodic solutions was investigated in [29] (Theorem 1) with the aid of the Gearhart-Prüss-Theorem, see also [18,20].…”

“…A second threshold may be obtained by studying the time-dependent Lugiato-Lefever equation (2). Modifying slightly the proof by Jahnke, Mikl and Schnaubelt [11] for (1) we first derive the global well-posedness of the initial value problem for (2) with initial data a(0) = φ ∈ H 4 (T). In [11] the corresponding well-posedness result for κ = 0 is based on the observation that the flow remains bounded in L 2 (T) and that the H 1 (T)-norm grows at most like √ t as t → ∞.…”