An improved finite integration method for nonlocal nonlinear Schrödinger equations

In this paper, we study the nonlocal nonlinear Schrödinger equation (NNLSE), which is an important extension of the nonlinear Schrödinger equation (NLSE). Since the nonlinearity of NNLSE involves a convolution, numerical approximation of its solution is very expensive. To achieve an efficient numeri...

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Bibliographic Details
Published in:Computers & mathematics with applications (1987) 2022-05, Vol.113, p.24-33
Main Authors: Zhao, Wei, Lei, Min, Hon, Yiu-Chung
Format: Article
Language:English
Subjects:
Convolution
Improved finite integration method
Mathematical analysis
Multilayers
Nonlinearity
Nonlocal nonlinear Schrödinger equation
Numerical methods
Operator-splitting method
Operators (mathematics)
Partial differential equations
PDE method
Quadratures
Run time (computers)
Schrodinger equation
Time integration
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Summary:In this paper, we study the nonlocal nonlinear Schrödinger equation (NNLSE), which is an important extension of the nonlinear Schrödinger equation (NLSE). Since the nonlinearity of NNLSE involves a convolution, numerical approximation of its solution is very expensive. To achieve an efficient numerical method for solving the NNLSE, according to the property of convolution, we firstly present a partial differential equation (PDE) method to transfer the NNLSE from an integro-differential equation to an equivalent or approximate system of PDEs, and the numerical solution of the NNLSE can then be obtained by solving these PDEs. Based on the recently developed finite integration method (FIM), we derive an improved method (IFIM) in this paper. The novelty of this IFIM is that the quadrature method is only used once instead of multiple times for multi-layer integral, so it consumes less running time and provides higher accuracy. In addition, we adapt a second-order operator-splitting method (OS2) at each time-step to ensure a convergent solution for long-time integration. Several numerical experiments are given to verify the efficiency and accuracy of the proposed IFIM for solving the NNLSE.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2022.03.004