Exact Solutions of Coupled Sine-Gordon Equations Using the Simplest Equation Method

Zhao, Yun-Mei

doi:10.1155/2014/534346

Cited by 5 publications

(3 citation statements)

References 20 publications

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“…In paper [8], the authors used a rational exponential ansatz to derive the exact solutions of a coupled sine-Gordon equation. The simplest equation method has been used for finding the exact solutions of coupled sine-Gordon equations by Zhao [9]. In papers [10][11][12], Zhao et al obtained some new solutions of coupled sine-Gordon equations including Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions by the Jacobi elliptic function expansion method, the hyperbolic auxiliary function method, and the symbolic computation method.…”

Section: Introductionmentioning

confidence: 99%

“…The simplest equation method has been used for finding the exact solutions of coupled sine-Gordon equations by Zhao [9]. In papers [10][11][12], Zhao et al obtained some new solutions of coupled sine-Gordon equations including Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions by the Jacobi elliptic function expansion method, the hyperbolic auxiliary function method, and the symbolic computation method. Sadighi et al [13] employed the homotopy perturbation method (HPM) for solving both sine-Gordon and coupled sine-Gordon equations.…”

Section: Introductionmentioning

confidence: 99%

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Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

Deresse

2022

Advances in Mathematical Physics

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In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

Deresse

2022

Advances in Mathematical Physics

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“…There is a wealth of literature that deals with the solutions and the possible integrability of coupled Sine-Gordon or nonlinear Klein-Gordon equations in (1+1) dimensions (see, e.g., Refs. [ 27 – 36 ]). The generic form of such systems is: …”

Section: Introductionmentioning

confidence: 99%

Spatially Extended Relativistic Particles Out of Traveling Front Solutions of Sine-Gordon Equation in (1+2) Dimensions

Zarmi

2016

PLoS ONE

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Slower-than-light multi-front solutions of the Sine-Gordon in (1+2) dimensions, constructed through the Hirota algorithm, are mapped onto spatially localized structures, which emulate free, spatially extended, massive relativistic particles. A localized structure is an image of the junctions at which the fronts intersect. It propagates together with the multi-front solution at the velocity of the latter. The profile of the localized structure obeys the linear wave equation in (1+2) dimensions, to which a term that represents interaction with a slower-than-light, Sine-Gordon-multi-front solution has been added. This result can be also formulated in terms of a (1+2)-dimensional Lagrangian system, in which the Sine-Gordon and wave equations are coupled. Expanding the Euler-Lagrange equations in powers of the coupling constant, the zero-order part of the solution reproduces the (1+2)-dimensional Sine-Gordon fronts. The first-order part is the spatially localized structure. PACS: 02.30. Ik, 03.65.Pm, 05.45.Yv, 02.30.Ik.

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Adiabatic invariants of the extended KdV equation

et al. 2017

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When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present the direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from almost constant values are indeed small.

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Exact Solutions of Coupled Sine-Gordon Equations Using the Simplest Equation Method

Abstract: The simplest equation method has been used for finding the exact solutions of coupled sine-Gordon equations. Such equations have some useful applications in physics and biology, so finding their exact solutions is of great importance.

Cited by 5 publications

References 20 publications

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

Spatially Extended Relativistic Particles Out of Traveling Front Solutions of Sine-Gordon Equation in (1+2) Dimensions

Adiabatic invariants of the extended KdV equation

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