Solutions of Two-Dimensional Nonlinear Sine-Gordon Equation via Triple Laplace Transform Coupled with Iterative Method

This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

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

Advances in Mathematical Physics ◽

10.1155/2022/6977692 ◽

2022 ◽

Vol 2022 ◽

pp. 1-15

Author(s):

Alemayehu Tamirie Deresse

Keyword(s):

Iterative Method ◽

Gordon Equation ◽

Approximate Analytical Solution ◽

Test Problems ◽

Sine Gordon Equation ◽

One Dimensional ◽

Nonlinear Part ◽

Sumudu Transform ◽

The One ◽

The Given

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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Duality of positive and negative integrable hierarchies via relativistically invariant fields

Journal of High Energy Physics ◽

10.1007/jhep07(2021)058 ◽

2021 ◽

Vol 2021 (7) ◽

Author(s):

S. Y. Lou ◽

X. B. Hu ◽

Q. P. Liu

Keyword(s):

Integrable Hierarchies ◽

Gordon Equation ◽

Formal Series ◽

Two Dimensional ◽

Relativistic Invariance ◽

Sine Gordon Equation ◽

Recursion Operators ◽

Symmetry Approach ◽

Duality Relations ◽

Duality Problem

Abstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

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