Residual Symmetry Reduction and Consistent Riccati Expansion of the Generalized Kaup-Kupershmidt Equation

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

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New interaction solutions from residual symmetry reduction and consistent Riccati expansion of the (2 $$\varvec{+}$$ + 1)-dimensional Boussinesq equation

Nonlinear Dynamics ◽

10.1007/s11071-018-4139-8 ◽

2018 ◽

Vol 92 (4) ◽

pp. 1469-1479 ◽

Cited By ~ 8

Author(s):

Xi-zhong Liu ◽

Jun Yu ◽

Zhi-Mei Lou

Keyword(s):

Boussinesq Equation ◽

Symmetry Reduction ◽

Residual Symmetry ◽

Interaction Solutions ◽

Consistent Riccati Expansion

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Nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

Modern Physics Letters B ◽

10.1142/s0217984916502936 ◽

2016 ◽

Vol 30 (23) ◽

pp. 1650293 ◽

Cited By ~ 2

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).

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

Zeitschrift für Naturforschung A ◽

10.1515/zna-2015-0504 ◽

2016 ◽

Vol 71 (4) ◽

pp. 351-356 ◽

Cited By ~ 4

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.

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