Open Access

ARTICLE

Numerical Solution of System of N–Coupled Nonlinear Schrödinger Equations via Two Variants of the Meshless Local Petrov–Galerkin (MLPG) Method

by M. Dehghan¹, M. Abbaszadeh², A. Mohebbi³

1 Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave.,15914, Tehran, Iran, Corresponding author. Email: mdehghan@aut.ac.ir , mdehghan.aut@gmail.com, (M. Dehghan)
2 Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran. Email: m.abbaszadeh@aut.ac.ir (M. Abbaszadeh)
3 Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran. Email: a_mohebbi@kashanu.ac.ir (A. Mohebbi).

Computer Modeling in Engineering & Sciences 2014, 100(5), 399-444. https://doi.org/10.3970/cmes.2014.100.399

Abstract

In this paper three numerical techniques are proposed for solving the system of N-coupled nonlinear Schrödinger (CNLS) equations. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via the forward finite difference formula, then for obtaining a full discretization scheme, we use the Kansa’s approach to approximate the spatial derivatives via radial basis functions (RBFs) collocation methodology. We introduce the moving least squares (MLS) approximation and radial point interpolation method (RPIM) with their shape functions, separately. It should be noted that the shape functions of RPIM unlike the shape functions of the MLS approximation have kronecker delta property. Also, we implement the local meshless Petrov-Galerkin (MLPG) and local RPIM (LRPIM) techniques for obtaining two full discretization schemes for the numerical solution of the mentioned equation in the two-dimensional case. In the meshless local weak forms for obtaining an approximate solution for the node i in every sub-domain we use the shape functions of the moving least squares (MLS) and RPIM meshless approximations. The main aim of this paper is to show that the meshless methods based on the global form i.e. radial basis functions collocation method and local weak form i.e. MLPG and LRPIM techniques are also simple in implementation and suitable for the treatment of the system of coupled nonlinear Schrödinger equations. We show that the RBFs collocation scheme provides a simple implementation for computing long-range solitary solutions considered by coupled nonlinear Schrödinger equations and the conserved quantities mass and energy almost are constant. Of course selecting small enough time step, obtains conserved quantities which are exactly fixed. Also several test problems including the two-dimensional case are given and numerical simulations are reported. We compare the obtained numerical results with together. The numerical results confirm the efficiency of the proposed schemes.

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