Abstract
The non-linear Schrödinger (NLS) equation has often been used as a model equation in the study of quantum states of physical systems. Numerical solution of NLS equation is obtained using cubic B-spline Galerkin method. We have applied the Crank–Nicolson scheme for time discretization and the cubic B-spline basis function for space discretization. Three numerical problems, including single soliton, interaction of two solitons and birth of standing soliton, are demonstrated to evaluate to the performance and accuracy of the method. The error norms and conservation laws are determined and found to be in good agreement with the published results. The obtained results show that the approach is feasible and accurate. The proposed method has almost second order convergence. The linear stability of the method is performed using the Von Neumann method.
Highlights
The non-linear Schrödinger (NLS) equation describes how the behavior of quantum states of a physical system changes in time and space
The NLS equation can be used to describe the propagation of optical pulses and waves in water and plasmas, among other things
A new algorithm was developed for the numerical solution of differential equations. This algorithm was obtained by utilizing exponential B-spline functions for the Galerkin finite element method
Summary
The non-linear Schrödinger (NLS) equation describes how the behavior of quantum states of a physical system changes in time and space. Dag [1] presented the quadratic and cubic B-spline Galerkin finite element method for solving the Burger’s equation. A new algorithm was developed for the numerical solution of differential equations This algorithm was obtained by utilizing exponential B-spline functions for the Galerkin finite element method. Gardner et al [7] applied a cubic B-spline finite element method for the numerical solution of the Burger’s equation. Ersoy et al [10] studied the numerical solution of the NLS equation using an exponential B-spline with collocation method In that paper, they used the Crank. We study the use of the Galerkin method with cubic B-spline function as the weight and trial functions over finite elements to solve the NLS equation.
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