Abstract
AbstractPlane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.
Highlights
We consider the cubic nonlinear Schrodinger equation i ∂ ∂t u = −∆u +λ|u|2u, u = u(x, t), (1)in the defocusing (λ = +1) or focusing (λ = −1) case
We impose periodic boundary conditions in arbitrary spatial dimension d 1: the spatial variable x belongs to the d-dimensional torus Td = Rd/(2π Z)d. This nonlinear Schrodinger equation has a class of simple solutions, the plane wave solutions u(x, t ) = ρ ei( ·x−ωt)
The nonresonance condition of Assumption 2 on the frequencies is crucial for the proof of our long-time result
Summary
That we do not (and cannot) impose this nonresonance condition on the completely resonant frequencies of the nonlinear Schrodinger equation (1) and its discretization by the split-step Fourier method, but only on the frequencies ω j of (10) for the linearization around a plane wave. Theorem 3 states in particular that this implies linear stability of (numerical) plane wave solutions ((8) in Assumption 1) under the step-size restriction (12). For nonzero but small , the condition of linear stability in Assumption 1 can still be expected to hold under a step-size restriction similar to (12) In this case, the frequencies ω j differ from those for = 0 only for large j (we have n( j) =. This leads us to consider the equation for w j together with the one for w− j , wnj +1 wn−+j 1
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