scholarly journals CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Wenguang Cheng ◽  
Biao Li
Keyword(s):  
Boussinesq Equation ◽  
Wave Interaction ◽  
Original System ◽  
Cnoidal Wave ◽  
Nonlocal Symmetry ◽  
Residual Symmetry ◽  
Symmetry Approach ◽  
Modified Boussinesq Equation ◽  
The Relationship ◽  
Truncated Painlevé Expansion

It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained. The residual symmetry is localized after embedding the original system in an enlarged one. The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach.

scholarly journals Resonant Soliton and Soliton-Cnoidal Wave Solutions for a (3+1)-Dimensional Korteweg-de Vries-Like Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei ◽  
Jun-Chao Chen ◽  
Quan-Yong Zhu
Keyword(s):  
Wave Interaction ◽  
Elliptic Functions ◽  
Cnoidal Wave ◽  
Linear Superposition ◽  
Jacobian Elliptic Functions ◽  
Multiple Wave ◽  
Residual Symmetry ◽  
Wave Solutions ◽  
De Vries ◽  
Truncated Painlevé Expansion

The residual symmetry of a (3+1)-dimensional Korteweg-de Vries (KdV)-like equation is constructed using the truncated Painlevé expansion. Such residual symmetry can be localized and the (3+1)-dimensional KdV-like equation is extended into an enlarged system by introducing some new variables. By using Lie’s first theorem, the finite transformation is obtained for this localized residual symmetry. Further, the linear superposition of multiple residual symmetries is localized and the n-th Bäcklund transformation in the form of the determinants is constructed for this equation. For illustration more detail, the first three multiple wave solutions-the collisions of resonant solitons are depicted. Finally, with the aid of the link between the consistent tanh expansion (CTE) method and the truncated Painlevé expansion, the explicit soliton-cnoidal wave interaction solution containing three kinds of Jacobian elliptic functions for this equation is derived.

New interaction solutions of the similarity reduction for the integrable (2+1)-dimensional Boussinesq equation

2021 ◽  
Author(s):  
Hengchun Hu ◽  
Xiaodan Li
Keyword(s):  
Boussinesq Equation ◽  
Rational Solution ◽  
Symmetry Transformation ◽  
Nonlocal Symmetry ◽  
Similarity Reduction ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Dependent Variables ◽  
New Formula ◽  
Truncated Painlevé Expansion

The nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation is studied by the standard truncated Painlevé expansion. This nonlocal symmetry can be localized to the Lie point symmetry of the prolonged system by introducing two auxiliary dependent variables. The corresponding finite symmetry transformation and similarity reduction related to the nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation are studied. The rational solution, the triangle solution, two solitoff-interaction solution and the soliton–cnoidal interaction solutions for the new [Formula: see text]-dimensional Boussinesq equation are presented analytically and graphically by selecting the proper arbitrary constants.

Nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

2016 ◽  
Vol 30 (23) ◽  
pp. 1650293 ◽  
Author(s):  
Xiazhi Hao ◽  
Yinping Liu ◽  
Xiaoyan Tang ◽  
Zhibin Li
Keyword(s):  
Spectral Function ◽  
Darboux Transformation ◽  
Bäcklund Transformation ◽  
Original System ◽  
Lax Pair ◽  
Nonlocal Symmetries ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Truncated Painlevé Expansion ◽  
Consistent Riccati Expansion

The nonlocal symmetries of the residual symmetry and the spectral function symmetry of Sawada–Kotera (SK) equation can be derived from the truncated Painlevé expansion and the Lax pair, respectively. By localizing the nonlocal symmetries of the original system to local ones of the prolonged system, the Bäcklund transformation and the Darboux transformation for both the original and the prolonged systems are obtained. Moreover, by the truncated Painlevé expansion, we further study the integrability of the SK quation in the sense of having a consistent Riccati expansion (CRE).

Residual Symmetry and Explicit Soliton–Cnoidal Wave Interaction Solutions of the (2+1)-Dimensional KdV–mKdV Equation

2016 ◽  
Vol 71 (4) ◽  
pp. 351-356 ◽  
Author(s):  
Wenguang Cheng ◽  
Biao Li
Keyword(s):  
Elliptic Integral ◽  
Wave Interaction ◽  
Elliptic Functions ◽  
Mkdv Equation ◽  
Jacobi Elliptic Functions ◽  
Cnoidal Wave ◽  
Residual Symmetry ◽  
The Third ◽  
Interaction Solutions ◽  
Consistent Riccati Expansion

AbstractThe truncated Painlevé method is developed to obtain the nonlocal residual symmetry and the Bäcklund transformation for the (2+1)-dimensional KdV–mKdV equation. The residual symmetry is localised after embedding the (2+1)-dimensional KdV–mKdV equation to an enlarged one. The symmetry group transformation of the enlarged system is computed. Furthermore, the (2+1)-dimensional KdV–mKdV equation is proved to be consistent Riccati expansion (CRE) solvable. The soliton–cnoidal wave interaction solution in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral is obtained by using the consistent tanh expansion (CTE) method, which is a special form of CRE.

scholarly journals Residual Symmetries and Bäcklund Transformations of Strongly Coupled Boussinesq–Burgers System

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1365 ◽  
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Bäcklund Transformation ◽  
Point Symmetry ◽  
Backlund Transformation ◽  
Nonlocal Symmetry ◽  
Linear Superposition ◽  
Strongly Coupled ◽  
Residual Symmetry ◽  
Commutative Subalgebra ◽  
Truncated Painlevé Expansion ◽  
Burgers System

In this article, we construct a new strongly coupled Boussinesq–Burgers system taking values in a commutative subalgebra Z 2 . A residual symmetry of the strongly coupled Boussinesq–Burgers system is achieved by a given truncated Painlevé expansion. The residue symmetry with respect to the singularity manifold is a nonlocal symmetry. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is obtained with the help of Lie’s first theorem. Further, the linear superposition of multiple residual symmetries is localized to a Lie point symmetry, and a N-th Bäcklund transformation is also obtained.

Nonlocal Symmetries and Consistent Riccati Expansions of the (2+1)-Dimensional Dispersive Long Wave Equation

2017 ◽  
Vol 72 (5) ◽  
pp. 425-431 ◽  
Author(s):  
Lian-Li Feng ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang
Keyword(s):  
Wave Equation ◽  
Elliptic Integral ◽  
Wave Interaction ◽  
Elliptic Functions ◽  
Nonlocal Symmetry ◽  
Constant Depth ◽  
Nonlocal Symmetries ◽  
Long Wave ◽  
Truncated Painlevé Expansion ◽  
Consistent Riccati Expansion

AbstractIn this article, the (2+1)-dimensional dispersive long wave equation (DLWE) is investigated, which is derived in the context of a water wave propagating in narrow infinitely long channels of finite constant depth. By using of the truncated Painlevé expansion, we construct its nonlocal symmetry and Bäcklund transformation. After implanting the equation into an enlarged one, then the residual symmetry is localised. Meanwhile, the symmetry group transformation can be computed from the prolonged system. Furthermore, the equation is verified to be consistent Riccati expansion (CRE) solvable. Outing from the CRE, the soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral are studied, respectively.

Analytic rogue wave-type solutions for the generalized (2+1)-dimensional Boussinesq Equation

2021 ◽  
pp. 2150380
Author(s):  
Xiu-Rong Guo
Keyword(s):  
Boussinesq Equation ◽  
Rogue Wave ◽  
Wave Interaction ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Wave Type ◽  
Hirota Bilinear Form ◽  
Basic Facts ◽  
2D And 3D ◽  
Solitary Wave Interaction

Based on the Hirota bilinear form of the generalized (2+1)-dimensional Boussinesq equation, which can be expressed as the shallow water wave mechanism appearing in fluid mechanics, we applied the new polynomial functions to construct the rational solutions and rogue wave-type solutions. Next, the system parameters control on the rational solutions and rogue wave-type solutions were also shown. As a result, we found the following basic facts: (i) these parameters may affect the wave shapes, amplitude, and bright/dark for this considered equation, (ii) the solitary wave interaction rogue waves and triplet rogue wave-type solutions can be viewed on [Formula: see text], [Formula: see text], and [Formula: see text] planes, respectively. Their nonlinear dynamic behaviors were presented by numerical simulation of the 2D- and 3D-plots.

scholarly journals Nonlocal Symmetry and Bäcklund Transformation of a Negative-Order Korteweg–de Vries Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Jinxi Fei ◽  
Weiping Cao ◽  
Zhengyi Ma
Keyword(s):  
Vries Equation ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Nonlocal Symmetry ◽  
Linear Superposition ◽  
Residual Symmetry ◽  
Negative Order ◽  
De Vries ◽  
Finite Transformation ◽  
New Variables

The residual symmetry of a negative-order Korteweg–de Vries (nKdV) equation is derived through its Lax pair. Such residual symmetry can be localized, and the original nKdV equation is extended into an enlarged system by introducing four new variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Furthermore, we localize the linear superposition of multiple residual symmetries and construct n-th Bäcklund transformation for this nKdV equation in the form of the determinants.

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