Compact Discrete Gradient Schemes for Nonlinear Schrödinger Equations

AbstractThis paper proposes two schemes for a nonlinear Schrödinger equation with four-order spacial derivative by using compact scheme and discrete gradient methods. They are of fourth-order accuracy in space. We analyze two discrete invariants of the schemes. The numerical experiments are implemented to investigate the efficiency of the schemes.

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Numerical Methods for the Derivative Nonlinear Schrödinger Equation

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2016-0184 ◽

2018 ◽

Vol 19 (3-4) ◽

pp. 239-249 ◽

Cited By ~ 2

Author(s):

Shu-Cun Li ◽

Xiang-Gui Li ◽

Fang-Yuan Shi

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

Second Order ◽

Compact Scheme ◽

Order Accuracy ◽

Nonlinear Schrödinger ◽

Accuracy Order ◽

Derivative Nonlinear Schrödinger Equation

AbstractIn this work, a second-order accuracy in both space and time Crank–Nicolson (C-N)-type scheme, a fourth-order accuracy in space and second-order accuracy in time compact scheme and a sixth-order accuracy in space and second-order accuracy in time compact scheme are proposed for the derivative nonlinear Schrödinger equation. The C-N-type scheme is tested to satisfy the conservation of discrete mass. For the two compact schemes, the iterative algorithm and the Thomas algorithm in block matrix form are adopted to enhance the computational efficiency. Numerical experiment is given to test the mass conservation for the C-N-type scheme as well as the accuracy order of the three schemes. In addition, the numerical simulation of binary collision and the influence on the solitary solution by adding a small random perturbation to the initial condition are also discussed.

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A Compact Scheme for Coupled Stochastic Nonlinear Schrödinger Equations

Communications in Computational Physics ◽

10.4208/cicp.300815.180416a ◽

2016 ◽

Vol 21 (1) ◽

pp. 93-125 ◽

Cited By ~ 6

Author(s):

Chuchu Chen ◽

Jialin Hong ◽

Lihai Ji ◽

Linghua Kong

Keyword(s):

Conservation Law ◽

Numerical Experiments ◽

Nonlinear Schrödinger Equations ◽

Compact Scheme ◽

Inelastic Interaction ◽

Schrodinger Equations ◽

Schrödinger Equations ◽

Discrete Energy ◽

Nonlinear Schrödinger ◽

Nonlinear Schrodinger Equations

AbstractIn this paper, we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrödinger equations. We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law, discrete charge conservation law and discrete energy evolution law almost surely. Numerical experiments confirm well the theoretical analysis results. Furthermore, we present a detailed numerical investigation of the optical phenomena based on the compact scheme. By numerical experiments for various amplitudes of noise, we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time. In particular, if the noise is relatively strong, the soliton will be totally destroyed. Meanwhile, we observe that the phase shift is sensibly modified by the noise. Moreover, the numerical results present inelastic interaction which is different from the deterministic case.

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On the inhomogeneous fourth-order nonlinear Schrödinger equation

Journal of Mathematical Physics ◽

10.1063/1.4929657 ◽

2015 ◽

Vol 56 (9) ◽

pp. 093502 ◽

Cited By ~ 11

Author(s):

Nakao Hayashi ◽

Pavel I. Naumkin

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Fourth Order ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger

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ASYMPTOTIC STABILITY OF NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL

Reviews in Mathematical Physics ◽

10.1142/s0129055x05002522 ◽

2005 ◽

Vol 17 (10) ◽

pp. 1143-1207 ◽

Cited By ~ 21

Author(s):

ZHOU GANG ◽

I. M. SIGAL

Keyword(s):

Asymptotic Stability ◽

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Initial Conditions ◽

Nonlinear Schrodinger Equation ◽

Schrödinger Equations ◽

Nonlinear Schrödinger ◽

Nonlinear Schrodinger Equations ◽

Generalized Nonlinear Schrödinger Equation

We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.

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