The meshless kernel-based method of lines for the numerical solution of the nonlinear Schrödinger equation

In this paper, the nonlinear Schrödinger equation is solved numerically by using the meshless kernel-based method of lines. Multiquadric, Gaussian and Wendland's compactly supported radial basis functions are used as the kernel basis functions. In the numerical examples, the single soliton solution, interaction of two colliding solitons, birth of standing soliton, birth of mobile soliton and bound state of solitons are simulated. The accuracy and efficiency of the used method are tested by computing the lowest two invariants and the relative change of invariants for all test problems. Error norms L2 and L∞L∞ are computed for single soliton motion whose exact solution is known. Numerical results and figures off wave motions for all test problems are presented. The numerical solutions of the nonlinear Scrödinger equation are compared with both the analytical solutions and numerical solutions of some earlier papers in the literature.