Existence and stability of solitons for fully discrete approximations of the nonlinear Schrödinger equation

Dario Bambusi (1) , Erwan Faou (2, 3) , Benoît Grébert (4)

1 Dipartimento de Matematica [Milano]
2 IPSO - Invariant Preserving SOlvers
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 LMJL - Laboratoire de Mathématiques Jean Leray

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical soliton which is close in energy norm to the continuous soliton. Such result is valid under a CFL condition between the time and space stepsizes. Furthermore we prove that if the initial datum is symmetric and close to the continuous soliton, then the associated numerical solution remains close to the orbit of the continuous soliton for very long times.

Mots clés: Discrete nonlinear Schrödinger equation; Numerical soliton; Stability; Backward error analysis; Modified Hamiltonian

Domaines: Analyse numérique [math.NA]

Dario Bambusi, Erwan Faou, Benoît Grébert. Existence and stability of solitons for fully discrete approximations of the nonlinear Schrödinger equation. Numerische Mathematik, 2013, 123, pp.461-492. ⟨10.1007/s00211-012-0491-7⟩. ⟨hal-00681730⟩