Symmetry reduction and new interaction solutions for the negative-order potential KdV equation

2020 ◽  
pp. 2150108
Author(s):  
Hengchun Hu ◽  
Zhenya Zhang
Keyword(s):  
Kdv Equation ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Local Symmetry ◽  
Similarity Solutions ◽  
Seed Solution ◽  
Interaction Solutions ◽  
New Exact Solutions ◽  
Negative Order ◽  
Non Local

New soliton–cnoidal interaction solutions for the negative-order potential KdV equation are studied with the help of the consistent tanh expansion method. The non-local symmetry for the negative-order potential KdV equation is derived from the truncated Painlevé expansion method. The non-local symmetry is transformed into the standard Lie point symmetry by introducing new dependent variables and the finite symmetry transformation is also presented to construct new exact solutions with non-zero seed solution. The similarity solutions of the enlarged negative-order potential KdV system are obtained through different constant selections.

Nonlocal Symmetries, Conservation Laws and Interaction Solutions of the Generalised Dispersive Modified Benjamin–Bona–Mahony Equation

2018 ◽  
Vol 73 (5) ◽  
pp. 399-405 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Xiu-Bin Wang ◽  
Tian-Tian Zhang
Keyword(s):  
Direct Method ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Local Symmetry ◽  
Dispersive Media ◽  
Jacobi Elliptic Function ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Non Local ◽  
Truncated Painlevé Expansion

AbstractWe consider the generalised dispersive modified Benjamin–Bona–Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painlevé expansion method, we derive its non-local symmetry and Bäcklund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves.

Residual Symmetries and Interaction Solutions for the Classical Korteweg-de Vries Equation

2017 ◽  
Vol 72 (3) ◽  
pp. 217-222 ◽  
Author(s):  
Jin-Xi Fei ◽  
Wei-Ping Cao ◽  
Zheng-Yi Ma
Keyword(s):  
Vries Equation ◽  
Expansion Method ◽  
Point Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Dependent Variables ◽  
Non Linear ◽  
De Vries ◽  
Finite Transformation ◽  
Non Local

AbstractThe non-local residual symmetry for the classical Korteweg-de Vries equation is derived by the truncated Painlevé analysis. This symmetry is first localised to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we then obtain the finite transformation for the localised residual symmetry. Based on the consistent tanh expansion method, some exact interaction solutions among different non-linear excitations are explicitly presented finally. Some special interaction solutions are investigated both in analytical and graphical ways at the same time.

Non-local symmetries and the nth finite symmetry transformations for the (2+1)-dimensional Korteweg–de Vries equation

2020 ◽  
pp. 2050432
Author(s):  
Xiazhi Hao ◽  
Xiaoyan Li
Keyword(s):  
Spectral Function ◽  
Soliton Solution ◽  
Vries Equation ◽  
Kdv Equation ◽  
Symmetry Transformation ◽  
De Vries ◽  
Symmetry Transformations ◽  
Non Local ◽  
Local Symmetries

Non-local symmetries in forms of square spectral function and residue over the (2+1)-dimensional Korteweg–de Vries (KdV) equation are studied in some detail. Then, we present [Formula: see text]-soliton solution to this equation with the help of symmetry transformation.

New Exact Solutions of the CDGSK Equation Related to a Non-local Symmetry

1994 ◽  
Vol 11 (10) ◽  
pp. 593-596 ◽  
Author(s):  
Senyue Lou ◽  
Hangyu Ruan ◽  
Weizhong Chen ◽  
Zhenli Wang ◽  
Lili Chen
Keyword(s):  
Exact Solutions ◽  
Local Symmetry ◽  
New Exact Solutions ◽  
Non Local

scholarly journals F-Expansion Method and New Exact Solutions of the Schrödinger-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ali Filiz ◽  
Mehmet Ekici ◽  
Abdullah Sonmezoglu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function ◽  
Weierstrass Elliptic Function ◽  
New Exact Solutions

F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the Schrödinger-KdV equation with a source to illustrate the validity and advantages of the proposed method. A number of Jacobi-elliptic function solutions are obtained including the Weierstrass-elliptic function solutions. When the modulusmof Jacobi-elliptic function approaches to 1 and 0, soliton-like solutions and trigonometric-function solutions are also obtained, respectively. The proposed method is a straightforward, short, promising, and powerful method for the nonlinear evolution equations in mathematical physics.

scholarly journals Some New Exact Solutions of (1+2)-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wei-Xiong Chen ◽  
Ji Lin
Keyword(s):  
Gordon Equation ◽  
Direct Method ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Sine Gordon Equation ◽  
Interaction Solutions ◽  
New Exact Solutions ◽  
Propagation Properties ◽  
A Direct Method

We use a generalized tanh function expansion method and a direct method to study the analytical solutions of the (1+2)-dimensional sine Gordon (2DsG) equation. We obtain some new interaction solutions among solitary waves and periodic waves, such as the kink-periodic wave interaction solution, two-periodic solitoff solution, and two-toothed-solitoff solution. We also investigate the propagation properties of these solutions.

scholarly journals Nonlocal symmetry and interaction solutions for the new (3+1)-dimensional integrable Boussinesq equation

2022 ◽  
Author(s):  
Hengchun Hu ◽  
Xiaodan Li
Keyword(s):  
Soliton Solution ◽  
Boussinesq Equation ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Point Symmetry ◽  
Periodic Waves ◽  
Nonlocal Symmetry ◽  
Interaction Solutions ◽  
Nonlinear Excitations ◽  
Dependent Variables

The nonlocal symmetry of the new (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlev\'{e} method. The nonlocal symmetry can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables. The finite symmetry transformation related to the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is studied. Meanwhile, the new (3+1)-dimensional Boussinesq equation is proved by the consistent tanh expansion method and many interaction solutions among solitons and other types of nonlinear excitations such as cnoidal periodic waves and resonant soliton solution are given.

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