The Boussinesq equation: Lax pair, Bäcklund transformation, symmetry group transformation and consistent Riccati expansion solvability

Abstract Painleve analysis is used to study the complete integrability of the recently proposed Konno-Onno equation, which also leads to a general form of solutions of the system. An independent study, using the prolongation theory, gives the explicit form of the Lax pair which is then used to obtain the Backlund transformation connecting two sets of solutions of the system. The existence of the Lax pair and the positive result of the Painleve test indicate the complete integrability of the system

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Nonlocal symmetries, Bäcklund transformation and interaction solutions for the integrable Boussinesq equation

Modern Physics Letters B ◽

10.1142/s0217984920502887 ◽

2020 ◽

Vol 34 (26) ◽

pp. 2050288

Author(s):

Jun Cai Pu ◽

Yong Chen

Keyword(s):

Boussinesq Equation ◽

Bäcklund Transformation ◽

Superposition Principle ◽

Symmetry Transformation ◽

Backlund Transformation ◽

Nonlocal Symmetry ◽

Linear Superposition ◽

Nonlocal Symmetries ◽

Interaction Solutions ◽

Physical Phenomena

The nonlocal symmetry of the integrable Boussinesq equation is derived by the truncated Painlevé method. The nonlocal symmetry is localized to the Lie point symmetry by introducing auxiliary-dependent variables and the finite symmetry transformation related to the nonlocal symmetry is presented. The multiple nonlocal symmetries are obtained and localized base on the linear superposition principle, then the determinant representation of the [Formula: see text]th Bäcklund transformation is provided. The integrable Boussinesq equation is also proved to be consistent tanh expansion (CTE) form and accurate interaction solutions among solitons and other types of nonlinear waves are given out analytically and graphically by the CTE method. The associated structure may be related to large variety of real physical phenomena.

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Bilinear Bäcklund transformation and Lax pair for a coupled Ramani equation

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2009.04.006 ◽

2009 ◽

Vol 357 (1) ◽

pp. 132-136 ◽

Cited By ~ 5

Author(s):

Yi He ◽

Hon-Wah Tam

Keyword(s):

Bäcklund Transformation ◽

Lax Pair ◽

Backlund Transformation ◽

Bilinear Bäcklund Transformation

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Lax pair, Bäcklund transformation and conservation laws for the (2+1)-dimensional extended shallow water wave equation

Computers & Fluids ◽

10.1016/j.compfluid.2013.10.039 ◽

2014 ◽

Vol 89 ◽

pp. 153-156 ◽

Cited By ~ 6

Author(s):

Na Liu ◽

Feng Ding

Keyword(s):

Wave Equation ◽

Conservation Laws ◽

Shallow Water ◽

Water Wave ◽

Bäcklund Transformation ◽

Lax Pair ◽

Backlund Transformation ◽

Shallow Water Wave

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Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

Modern Physics Letters B ◽

10.1142/s0217984921504212 ◽

2021 ◽

Author(s):

Tian-Yu Zhou ◽

Bo Tian ◽

Su-Su Chen ◽

Cheng-Cheng Wei ◽

Yu-Qi Chen

Keyword(s):

Burgers Equation ◽

Bäcklund Transformation ◽

Lax Pair ◽

Ocean Dynamics ◽

Bäcklund Transformations ◽

Backlund Transformation ◽

Dispersive Waves ◽

Backlund Transformations ◽

Hybrid Solutions ◽

Kink Waves

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

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