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backlund transformation and Exact Solutionsfor (2+1) -dimensional Kolmogoroff- Petrovsky-Piscounov Equation

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Communications in Nonlinear Science and Numerical Simulation 1999年第2期4卷 146-151页

作者： Zhenya YAN Hongqing ZHANG(Institute of Mathematics. Dalian University of Technology. Dalian 116024. China) Institute of Mathematics Dalian University of Technology Dalian 116024 China

Via using the idea of improved homogeneous balance method andwith the aid of MATHEMATICA, backlund transformation (BT) of (2+1)-dimensionalKolmogoroff-Petrovsky-Piscounov (KPP) equation is derived. Then according to the BT,several families of exact solutions for KPP equation are obtained, Which contain solitonsolutions. (2+1)-dimensional Chaffee-Infante (CI) equation, as the special case of KPPequation, is also considered.

关键词： backlund transformation KPP equation CI equation exact solution

backlund transformation AND EXACT SOLUTIONS FOR WHITHAM-BROER-KAUP EQUATIONS IN SHALLOW WATER

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Applied Mathematics and Mechanics(English Edition) 1998年第8期19卷 713-716页

作者：范恩贵张鸿庆 Institute of Mathematics Dalian University of Technology Dalian P. R. China

By using a new method and Mathematica, the backlund transformations for Whitham-Broer-Kaup equations (WBK) are derived. The connections between WBK equation, heat equation and Burgers equation are found, which are use... 详细信息

关键词： WBK equation backlund transformation exact solution solitary wave solution

backlund transformation OF THE SOLUTIONS OF DIFFERENT EQUATIONS

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Chinese Science Bulletin 1988年第21期33卷 1749-1754页

作者：田畴 Department of Mathematics University of Science Technology of ChinaHefei

In [1], the author discussed the Bcklund transformation (in Darboux type) of nonlinear evolution equations by using the gauge transformation. This type can be considered as an explicit form of Bcklund transformation. In this note, we shall discuss the Bcklund transformation (in Darboux type) of the solutions of the two different equations.

关键词： backlund transformation Lax pair gauge transformation completely integrable system

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

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Journal of Ocean Engineering and Science 2023年第3期8卷 226-237页

作者： Shoukry El-Ganaini Sachin Kumar Monika Niwas Department of Mathematics Faculty of ScienceDamanhour UniversityDamanhour 22514Egypt Department of Mathematics Faculty of Mathematical SciencesUniversity of DelhiDelhi 110007India Department of Mathematics University of DelhiDelhi 110007India

By taking advantage of three different computational analytical methods:the Lie symmetry analysis,the generalized Riccati equation mapping approach,and the modified Kudryashov approach,we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation(NCDR)with power-law nonlinearity and density-dependent diffusion.Lie symmetry analysis is one of the pow-erful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables.By the Lie group technique,we obtain one-parameter in-variant transformations,determining equations and corresponding vectors for the considered convection-diffusion-reaction equation.By treating the parameters of the governing equation as constants,the ap-plied methods yield a variety of new closed-form solutions,including inverse function solutions,periodic solutions,exponential function solutions,dark solitons,singular solitons,combo bright-singular solitons,and the combine of bright-dark solitons and dark-bright solitons.Moreover,using the Bäcklund transfor-mation of the generalized Riccati equation and modified Kudryashov method,we can construct multiple solitons and other solutions of the considered equation.The obtained new solutions of this work demon-strate that the used approaches are powerful and effective in dealing with nonlinear equations,and that these solutions are required to explain many biological and physical phenomena.Comparing our obtained solutions of this paper with the ones obtained in the literature,we see that our solutions are new and not reported elsewhere.These newly formed soliton solutions will be more beneficial in the various dis-ciplines of ocean engineering,plasma physics,and nonlinear sciences.

关键词： Lie symmetry analysis Generalized riccati equation mapping Modified kudryashov approach Nonlinear convection-diffusion-reaction equation Solitary wave solutions Closed form solutions backlund transformation Exact solution Dynamical wave structures Bäcklund transformation

backlund transformations, consistent Riccati expansion solvability, and soliton-cnoidal interaction wave solutions of Kadomtsev-Petviashvili equation

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Chinese Physics B 2020年第2期29卷 91-99页

作者：刘萍程杰任博杨建荣 School of Electronic and Information Engineering University of Electronic Science and Technology of China Zhongshan InstituteZhongshan 528402China School of Physics University of Electronic Science and Technology of ChinaChengdu 610054China Institute of Nonlinear Science Shaoxing UniversityShaoxing 312000China School of Physics and Electronic Information Shangrao Normal UniversityShangrao 334001China

The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevéexpansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE)solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.

关键词： Kadomtsev-Petviashvili(KP)equation consistent Riccati expansion symmetry backlund transformation interaction solution

BCKLUND transformation ON SURFACESWITH CONSTANT MEAN CURVATURE IN R^(2.1)

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Acta Mathematica Scientia 2003年第3期23卷 369-376页

作者：田畴周扣华田涌波 Department of Mathematics University of Science and Technology of ChinaHefei 230026China Department of Mathematics University of Science and Technology of ChinaHefei 230026China Department of MathematicsYangzhouJiangsu 225002China

In this paper, the authors obtain the backlund transformation on time-like surfaces with constant mean curvature in R2.1. Using this transformation, families of surfaces with constant mean curvature from known ones ca... 详细信息

关键词： backlund transformation principal curvature mean curvature

Bcklund transformation, non-local symmetry and exact solutions for (2+1)-dimensional variable coefficient generalized KP equations

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Communications in Nonlinear Science and Numerical Simulation 2000年第1期5卷 31-35页

作者： Zhenya YAN Institute of Mathematical Science, Dalian University of Technology, Dalian 116024, China e-mail: zhanghq@dlut. *** Institute of Mathematical Science Dalian University of Technology Dalian 116024 China

With the aid of MATHEMATICA, the idea of improved homogeneous balance method is extended to (2+1)-dimensional variable coefficients KP equation. As a result, backlund transformation and non-local symmetry are found. Then via using the backlund transformation, six families of exact solutions for (2+1)-dimensional variable coefficients KP equation are obtained, which contain new soliton-like solutions.

关键词： variable coefficient generalized KP equation backlund transformation nonlocal symmetry exact solution soliton

Bcklund transformation and Conservation Laws for the Variable-coefficient N-Coupled Nonlinear Schrdinger Equations with Symbolic Computation

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Acta Mathematica Sinica,English Series 2012年第5期28卷 969-974页

作者： Xiang Hua MENG Bo TIAN Tao XU Hai Qiang ZHANG School of Science Beijing University of Posts and Telecommunications Department of Mathematics School of Applied ScienceBeijing Information Science and Technology University

Considering the integrable properties for the coupled equations, the variable-coefficient N- coupled nonlinear Schrodinger equations are under investigation analytically in this paper. Based on the Lax pair with the nonisospectral parameter, a backlund transformation for such a coupled system denoting in the F functions is constructed with the one-solitonic solution given as the application sample. Furthermore, an infinite number of conservation laws are obtained using symbolic computation.

关键词： Variable-coefficient N-coupled nonlinear SchrSdinger equations backlund transformation conservation laws solitonic solution symbolic computation

Exact solutions and residual symmetries of the Ablowitz–Kaup–Newell–Segur system

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Chinese Physics B 2015年第1期24卷 125-131页

作者：刘萍曾葆青杨建荣任博 College of Electron and Information Engineering University of Electronic Science and Technology of China Zhongshan Institute School of Physical Electronics University of Electronic Science and Technology of China Department of Physics and Electronics Shangrao Normal University Institute of Nonlinear Science Shaoxing University

The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the backlund transformations between the AKNS equations and the Schwarzian AKNS equation.

关键词： residual symmetries Ablowitz-Kaup-Newell-Segur equation exact solution backlund transformation