Existence and stability of solitons for fully discrete approximations of the nonlinear Schrödinger equation
Provides theoretical guarantees for numerical methods used in simulating solitons, relevant to computational physics and applied mathematics.
The paper proves existence and long-time stability of numerical solitons for a fully discrete approximation of the cubic nonlinear Schrödinger equation under a CFL condition, showing the numerical solution stays close to the continuous soliton orbit for long times.
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.