High-order numerical method for the derivative nonlinear Schrödinger equation

Li, Shu-Cun; Li, Xiang-Gui; Cao, Jun-Jie; Li, Wenbo

doi:10.1142/s1793962317500179

Cited by 11 publications

(8 citation statements)

References 16 publications

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“…Example Random perturbation. By adding a small random perturbation to the initial data , that is

u_{1}^{0}

and

u_{2}^{0}

in Example 1, we solve NLDE using the TSCN scheme.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Time‐splitting methods with charge conservation for the nonlinear Dirac equation

Li¹,

Li²,

Shi

2017

Numerical Methods Partial

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In this work, four numerical time‐splitting methods are proposed for the (1 + 1)‐dimensional nonlinear Dirac equation. All of these methods (or schemes) are proved to satisfy the charge conservation in the discrete level. To enhance the computation efficiency, the block Thomas algorithm is adopted. Numerical experiments are given to test the accuracy order for these schemes, to simulate numerically the binary collision including two standing waves and two moving solitons, meanwhile, the dynamic properties for the nonlinear Dirac equation are discussed. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1582–1602, 2017

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“…Example Random perturbation. By adding a small random perturbation to the initial data , that is

u_{1}^{0}

and

u_{2}^{0}

in Example 1, we solve NLDE using the TSCN scheme.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Random perturbation. By adding a small random perturbation to the initial data [37], that is u 0 1 and u 0 2 in Example 1, we solve NLDE using the TSCN scheme. Figure 3 shows the exact data of ρ Q (x, t), and the right one demonstrates the numerical data of ρ Q (x, t) after adding a 0-0.5% random perturbation.…”

Section: B Dynamic Propertiesmentioning

confidence: 99%

Time‐splitting methods with charge conservation for the nonlinear Dirac equation

Li¹,

Li²,

Shi

2017

Numerical Methods Partial

Self Cite

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“…with w being the differentiable functions about x and t. Next, we will construct the numerical one-soliton solutions for Eq. ( 16) via a finite difference method [47].…”

Section: Numerical Simulations For System (1)mentioning

confidence: 99%

Singular soliton, shock-wave, breather-stripe soliton, hybrid solutions and numerical simulations for a (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada system in fluid mechanics

Liu

Tian

2022

Nonlinear Dyn

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In this paper, a (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada system is investigated in fluid mechanics via the symbolic computation. With the help of the Hirota method, we derive some singular soliton, shock-wave, breather-stripe soliton and hybrid solutions. Based on the finite difference method, we get some numerical one-soliton solutions. We graphically show the singular and shock-wave solutions, and observe that the singular one-soliton solutions are explosive and unstable, but the shock-wave solutions are nonsingular and stable. We observe that the breatherstripe soliton moves along the negative direction of the y axis, where y is a variable, and the amplitude and shape of the breather-stripe soliton remain invariant during the propagation. We graphically demonstrate the interaction among a rogue wave, a periodic wave and a pair of the stripe solitons: the rogue wave arises from the one stripe soliton; the rogue wave interacts with the periodic wave, the rogue wave splits into two waves and then the two waves merge into a wave; the rogue wave fuses with the other stripe soliton. We graphically present the numerical one-soliton solutions which agree with the analytic one-soliton solutions.

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“…In addition, the iteration termination condition for the above numerical techniques can be set as [33] 儨…”

Section: The Temporal Discretizationmentioning

confidence: 99%

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

Zhang

Shao

2017

Discrete Dynamics in Nature and Society

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A high-order accuracy numerical method is proposed to solve the (1 + 1)-dimensional nonlinear Dirac equation in this work. We construct the compact finite difference scheme for the spatial discretization and obtain a nonlinear ordinary differential system. For the temporal discretization, the implicit integration factor method is applied to deal with the nonlinear system. We therefore develop two implicit integration factor numerical schemes with full discretization, one of which can achieve fourth-order accuracy in both space and time. Numerical results are given to validate the accuracy of these schemes and to study the interaction dynamics of the nonlinear Dirac solitary waves.

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High-order numerical method for the derivative nonlinear Schrödinger equation

Cited by 11 publications

References 16 publications

Time‐splitting methods with charge conservation for the nonlinear Dirac equation

Time‐splitting methods with charge conservation for the nonlinear Dirac equation

Singular soliton, shock-wave, breather-stripe soliton, hybrid solutions and numerical simulations for a (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada system in fluid mechanics

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

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