Efficient numerical methods for nonlinear Schringer equations

Abstract

The nonlinear Schringer equations are widely used to model a number of important physical phenomena, including solitary wave propagations in optical fibers, deep water turbulence, laser beam transmissions, and the Bose-Einstein condensation, just to mention a few. In the field of optics and photonics, the systems of nonlinear Schringer equations can be used to model multi-component solitons and the interaction of self-focusing laser beams. In three spatial dimensions, the nonlinear Schringer equation is known as the Gross-Pitaevskii equation, which models the soliton in a low-cost graded-index fiber. Recently, research on nonlinear space fractional Schringer equations, which capture the self-similarity in the fractional environment, has become prevalent. Our study includes the systems of multi-dimensional nonlinear space fractional Schringer equations.

To solve the systems of multi-dimensional nonlinear Schringer equations efficiently, several novel numerical methods are presented. The central difference and quartic spline approximation based exponential time differencing Crank-Nicolson method is introduced for solving systems of one- and two-dimensional nonlinear Schringer equations. A local extrapolation is employed to achieve fourth-order accuracy in time. The numerical examples include the transmission of a self-focusing laser beam. The local discontinuous Galerkin methods combined with the fourth-order exponential time differencing Runge-Kutta time discretization are studied for solving the systems of nonlinear Schringer equations with hyperbolic terms, which are critical in modeling optical solitons in the birefringent fibers. The local discontinuous Galerkin method is able to achieve any order of accuracy in space, thanks to the usage of piecewise polynomial spaces. The exponential time differencing methods are employed to deal with the coupled nonlinearities for the reason that there is no need to solve nonlinear systems at every time step, while the approach achieves expected accuracy.

The fourth-order exponential time dierencing Runge-Kutta method is combined with the fourth-order compact scheme in space to solve the space fractional coupled nonlinear Schrodinger equations, involving Riesz derivatives. The system of four space fractional equations models the interaction of four water waves moving in the Lvy motion. A locally extrapolated exponential operator splitting scheme is applied to multi-dimensional nonlinear space fractional Schringer equations. The scheme achieves second-order accuracy in time for both two-dimensional and three-dimensional problems, compared to the second-order ADI method, whose application is constrained to two-dimensional problems. The Gross-Pitaevskii equation containing space fractional derivatives is demonstrated to indicate the usage of the scheme.

Theoretical and numerical study of stability and convergence of the numerical methods have been discussed. Extensive numerical examples are provided to illustrate the accuracy, efficiency, and reliability of the proposed methods.