Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory

Nikan, O.; Avazzadeh, Z.; Rasoulizadeh, Mohammad Navaz

doi:10.1007/s11071-021-06822-4

Cited by 35 publications

(7 citation statements)

References 66 publications

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“…By that reason, the Tzitzéica equation is sometimes also referred to as the Tzitzéica-Bullough-Dodd model [39]. Another well-known example is the sine-Gordon equation [40][41][42][43], which is given by equation (2.1) with f = a sin φ. In this case, truncated expansion (2.2) includes only the odd powers of φ.…”

Section: Nonlinear Klein-gordon Modelmentioning

confidence: 99%

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Fifth-order nonlinear Schrödinger equation as Routhian reduction of the nonlinear Klein–Gordon model

Sedletsky,

Gandzha

2023

Proc. R. Soc. A.

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We consider the nonlinear Klein–Gordon model with polynomial nonlinearity involving odd and even powers. We elaborate a hybrid Hamiltonian and Lagrangian approach, which is generally referred to as Routh’s procedure, to describe the nonlinear modulation of wave packets assuming the wave field envelope to be a slow function of time and coordinate. An extended fifth-order nonlinear Schrödinger equation is obtained from the Routhian equation for a complex canonical variable that couples the wave field and its momentum. This canonical variable is expressed in terms of the amplitude of fundamental harmonic and its derivatives with respect to coordinate. The resulting fifth-order nonlinear Schrödinger equation is demonstrated to be a Hamiltonian system under a certain constraint imposed on its model parameters. When compared with other techniques used to derive high-order nonlinear Schrödinger models, our approach preserves the Hamiltonian structure of the governing equations for the fundamental harmonic, while the input of higher harmonics is taken into account by means of variational principle applied to the averaged Routhian. In this way, it represents the most general tool for the reduction of nonlinear wave equations to high-order envelope models for slowly modulated wave packets.

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Section: Nonlinear Klein-gordon Modelmentioning

confidence: 99%

“…Another well-known example is the sine-Gordon equation [40–43], which is given by equation (2.1) with f=asin⁡ϕ. In this case, truncated expansion (2.2) includes only the odd powers of ϕ.…”

Section: Nonlinear Klein–gordon Modelmentioning

confidence: 99%

Fifth-order nonlinear Schrödinger equation as Routhian reduction of the nonlinear Klein–Gordon model

Sedletsky,

Gandzha

2023

Proc. R. Soc. A.

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“…The meshfree RBF method is used in both local and global forms. Some authors have tried localized-RBF-based strategies, such as the localized-RBF-generated FD (RBF-FD) [ 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ] and the RBF partition of unity (RBF-PU) [ 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 ].…”

Section: Properties Of Rbfsmentioning

confidence: 99%

Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method

Golbabai

Safaei

Molavi-Arabshahi

2022

Entropy

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This paper introduces a direct method derived from the global radial basis function (RBF) interpolation over arbitrary collocation nodes occurring in variational problems involving functionals that depend on functions of a number of independent variables. This technique parameterizes solutions with an arbitrary RBF and transforms the two-dimensional variational problem (2DVP) into a constrained optimization problem via arbitrary collocation nodes. The advantage of this method lies in its flexibility in selecting between different RBFs for the interpolation and parameterizing a wide range of arbitrary nodal points. Arbitrary collocation points for the center of the RBFs are applied in order to reduce the constrained variation problem into one of a constrained optimization. The Lagrange multiplier technique is used to transform the optimization problem into an algebraic equation system. Three numerical examples indicate the high efficiency and accuracy of the proposed technique.

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“…The information provided in [29][30][31][32][33][34] informs the reader thoroughly about the PDEs discussed and piques their interest. The soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory are studied in [35]. A localized hybrid kernel meshless technique is implemented for solving the fractional Rayleigh-Stokes problem for an edge in a viscoelastic fluid in Ref.…”

Section: Introductionmentioning

confidence: 99%

On the solutions of some nonlinear fractional partial differential equations using an innovative and direct procedure

Rab,

Khan,

Tchier

et al. 2023

Phys. Scr.

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In this article, a highly effective technique is implemented to obtain the approximate solutions of strongly nonlinear fractional order partial differential equations (NFPDEs). The findings of this study show the successful behavior of the fractional novel analytical method (FNAM), which can be used successfully for the solutions of common, severe NFPDEs. In the proposed method, the nonlinearity in each mathematical model is directly handled by using fractional Taylor series, which reduces the calculation effort. In this work, the method’s strength is primarily demonstrated on NFPDEs, and the obtained results are displayed via graphs and tables. Fromthe numerical simulations, it is evident that the suggested technique has greater accuracy despite smaller calculations. It is the most straightforward method for determining the formulaic solution to any type of NFPDE and is considered to be the unique numerical methodology.

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Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory

Cited by 35 publications

References 66 publications

Fifth-order nonlinear Schrödinger equation as Routhian reduction of the nonlinear Klein–Gordon model

Fifth-order nonlinear Schrödinger equation as Routhian reduction of the nonlinear Klein–Gordon model

Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method

On the solutions of some nonlinear fractional partial differential equations using an innovative and direct procedure

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