Solving coupled nonlinear Schrödinger equation using finite difference method and hybrid cubic B-spline collocation method

2019 ◽  
Author(s):  
Hanis Safirah Saiful Anuar ◽  
Amirah Azmi ◽  
Ahmad Izani Md. Ismail ◽  
Nur Nadiah Abd Hamid
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Collocation Method ◽  
Difference Method ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
Coupled Nonlinear Schrödinger Equation

scholarly journals The Hybrid Finite Difference and Moving Boundary Methods for Solving a Linear Damped Nonlinear Schrödinger Equation to Model Rogue Waves and Breathers in Plasma Physics

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Alvaro H. Salas ◽  
S. A. El-Tantawy ◽  
Jairo E. Castillo H.
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Schrödinger Equation ◽  
Plasma Physics ◽  
Difference Method ◽  
Schrodinger Equation ◽  
Moving Boundary ◽  
Rogue Waves ◽  
Nonlinear Schrodinger Equation ◽  
Time Interval

In this paper, the dissipative and nondissipative modulated pulses such as rogue waves (RWs) and breathers (Kuznetsov-Ma breathers and Akhmediev breathers) that can exist and propagate in several fields of sciences, for example, plasma physics, have been analyzed numerically. For this purpose, the fluid dusty plasma equations with taking the kinematic dust viscosity into account are reduced to the linear damped nonlinear Schrödinger equation using a reductive perturbation technique. It is known that this equation is not integrable and, accordingly, does not have analytical solution. Thus, for modelling both dissipative RWs and breathers, the improved finite difference method is introduced for this purpose. It is found that FDM is a good numerical technique for small time interval but for large time interval it becomes sometimes unacceptable. Therefore, to describe these waves accurately, the new improved numerical method is considered, which is called the hybrid finite difference method and moving boundary method (FDM-MBM). This last and updated method gives an accurate and excellent description to many physical results, as it was applied to the dust plasma results and the results were good.

scholarly journals A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method

2019 ◽  
Vol 9 (2) ◽  
pp. 223-235 ◽  
Author(s):  
Ali Başhan
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Schrödinger Equation ◽  
Difference Method ◽  
Schrodinger Equation ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
B Spline ◽  
Wide Range

The present manuscript include, finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) to obtain the numerical solutions for the nonlinear Schr¨odinger (NLS) equation. For this purpose, firstly Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using special type of classical finite difference method namely, Crank-Nicolson scheme. After that, Rubin and Graves linearization techniques have been utilized and differential quadrature method has been applied. So, partial differential equation turn into algebraic equation system. Next, in order to be able to test the accuracy of the newly hybrid method, the error norms L2 and L? as well as the two lowest invariants I1 and I2 have been calculated. Besides those, the relative changes in those invariants have been given. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison has clearly indicated that the currently utilized method, namely FDM-DQM, is an effective and efficient numerical schemeand allowed us to propose to solve a wide range of nonlinear equations.

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