A Meshfree Splitting Method for Soliton Dynamics in Nonlinear Schrödinger Equations
A new method for the numerical simulation of the so-called soliton dynamics arising in a nonlinear Schrödinger equation in the semi-classical regime is proposed. For the time discretization a classical fourth-order splitting method is used. For the spatial discretization, however, a meshfree method is employed in contrast to the usual choice of (pseudo) spectral methods. This approach allows one to keep the degrees of freedom almost constant as the semi-classical parameter \(\epsilon \) becomes small. This behavior is confirmed by numerical experiments.
KeywordsMeshfree discretization Splitting methods Nonlinear Schrödinger equations Soliton dynamics Semi-classical regime
The work of Stefan Rainer was partially supported by the Tiroler Wissenschaftsfond grant UNI-0404/880.
- 13.M. Caliari, A. Ostermann, S. Rainer, Meshfree Exponential Integrators, to appear in SIAM J. Sci. Comput. (2011)Google Scholar
- 14.M. Caliari, A. Ostermann, S. Rainer, Meshfree integrators. Oberwolfach Rep. 8, 883–885 (2011)Google Scholar
- 18.R. Schaback, Creating surfaces from scattered data using radial basis functions, in Mathematical Methods for Curves and Surfaces, ed. by M. Dæhlen, T. Lyche, L.L. Schumaker (Vanderbilt University Press, Nashville, 1995), pp. 477–496Google Scholar