Nonlocal symmetries, consistent tanh expansion solvability and interaction solutions for a new fifth-order nonlinear integrable equation

Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.

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The Application of the exp⁡(-Φ(ξ))-Expansion Method for Finding the Exact Solutions of Two Integrable Equations

Mathematical Problems in Engineering ◽

10.1155/2018/5191736 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Naila Sajid ◽

Ghazala Akram

Keyword(s):

Exact Solutions ◽

Rational Functions ◽

Expansion Method ◽

Integrable Equation ◽

Rational Solutions ◽

Integrable Equations ◽

Nonlinear Integrable Equation ◽

Fifth Order

In this article, the exp⁡-Φξ method is connected to search for new hyperbolic, periodic, and rational solutions of (1+1)-dimensional fifth-order nonlinear integrable equation and (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. The obtained solutions consist of trigonometric, hyperbolic, rational functions and W-shaped soliton. Furthermore, 3D and 2D graphs are plotted by choosing the suitable values of the parameters involved.

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Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

Zeitschrift für Naturforschung A ◽

10.1515/zna-2016-0147 ◽

2016 ◽

Vol 71 (8) ◽

pp. 735-740

Author(s):

Zheng-Yi Ma ◽

Jin-Xi Fei

Keyword(s):

Vries Equation ◽

Kdv Equation ◽

Elliptic Functions ◽

Group Method ◽

Physical Interaction ◽

Nonlocal Symmetries ◽

Interaction Solutions ◽

The Kdv Equation ◽

De Vries ◽

Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

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Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations

Symmetry ◽

10.3390/sym13071268 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1268

Author(s):

Yarong Xia ◽

Ruoxia Yao ◽

Xiangpeng Xin ◽

Yan Li

Keyword(s):

Differential Equation ◽

Nonlinear Partial Differential Equation ◽

Three Dimensional ◽

Dynamical Evolution ◽

Symmetry Transformation ◽

Nonlocal Symmetry ◽

Similarity Reduction ◽

Nonlocal Symmetries ◽

Interaction Solutions ◽

Dependent Variables

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

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