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Darboux Transformation and explicit solutions for Discretized Modified Korteweg-de Vries Lattice Equation

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Communications in Theoretical Physics 2010年第5期53卷 825-830页

作者：闻小永高以天 Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics Beijing University of Aeronautics and Astronautics Department of Mathematics College of SciencesBeijing Information Science and Technology University State Key Laboratory of Software Development Environment Beijing University of Aeronautics and Astronautics

The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.

关键词： Darboux transformation discretized modified Korteweg-de Vries lattice equation explicit solutions symbolic computation

Darboux Transformation and explicit solutions for Drinfel'd-Sokolov-Wilson Equation

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Communications in Theoretical Physics 2010年第6期53卷 1090-1096页

作者：耿献国吴丽华 Department of Mathematics Zhengzhou University

A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.

关键词： Drinfel'd-Sokolov-Wilson equation Darboux transformation explicit solutions

Darboux Transformation of a Differential-Difference Equation and Its explicit solutions

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Chinese Physics Letters 2006年第6期23卷 1361-1364页

作者：李文敏耿献国 Department of Mathematics Zhengzhou University Zhengzhou 450052 College of Life Sciences the Northwest Sci-Tech University of Agriculture and Forestry Yangling 712100

The Darboux transformation of a differential-difference equation associated with a 3 × 3 matrix spectral problem is derived. As an application, explicit soliton solutions of the differential-difference equation are p... 详细信息

关键词： Darboux Transformation Differential-Difference Equation explicit solutions

Sign-invariant and explicit solutions of Nonlinear Reaction-Diffusion Systems

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Communications in Theoretical Physics 2008年第6期49卷 1361-1364页

作者： ZHU Xiao-Ning ZHANG Shun-Li Center for Nonlinear Studies Department of Mathematics Northwest University Xi＇an 710069 China Center of Nonlinear Science Ningbo University Ningbo 315211 China

Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.

关键词： sign-invariant theory nonlinear reaction-diffusion system explicit solutions

Group Analysis, Fractional explicit solutions and Conservation Laws of Time Fractional Generalized Burgers Equation

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Communications in Theoretical Physics 2018年第1期69卷 5-8页

作者： Gang-Wei Wang A. H. Kara School of Mathematics and Statistics Hebei University of Economics and Business Shijiazhuang 050061 China School of Mathematics University of the Witwatersrand Private Bag 3 Wits 2050 Johannesburg South Afric

The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular, an exact solution ＆ provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.

关键词： time fractional generalized Burgers equation Lie symmetry method explicit solutions conservation laws

An extension of the modified Sawada-Kotera equation and conservation laws

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Chinese Physics B 2012年第7期21卷 37-42页

作者：何国亮耿献国 Department of Mathematics Zhengzhou University Department of Mathematics and Information Science Zhengzhou University of Light Industry

Based on the modified Sawad^Kotera equation, we introduce a 3 ~ 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawad-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawad-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawad-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.

关键词： spectral problem explicit solutions conservation laws

Consistent Burgers equation expansion method and its applications to high-dimensional Burgers-type equations

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Communications in Theoretical Physics 2022年第8期74卷 35-40页

作者： Gangwei Wang Li Li A H Kara School of Mathematics and Statistics Hebei University of Economics and BusinessShijiazhuang 050061China Library Hebei University of Economics and BusinessShijiazhuang 050061China School of Mathematics University of the WitwatersrandPrivate Bag 3Wits 2050JohannesburgSouth Africa

In this paper,a novel method,named the consistent Burgers equation expansion(CBEE)method,is proposed to solve nonlinear evolution equations(NLEEs)by the celebrated Burgers *** are said to be CBEE solvable if they are satisfied by the CBEE *** order to verify the effectiveness of the CBEE method,we take(2+1)-dimensional Burgers equation as an *** the(1+1)-dimensional Burgers equation,many new explicit solutions of the(2+1)-dimensional Burgers equation are *** obtained results illustrate that this method can be effectively extended to other NLEEs.

关键词： Burgers-type equations consistent Burgers equation expansion(CBEE)method CBEE solvability explicit solutions

Darboux Transformation for a Four-Component KdV Equation

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Communications in Theoretical Physics 2016年第10期65卷 374-378页

作者： Nian-Hua Li Li-nua Wu School of Mathematics Huaqiao University Quanzhou 362021 China

With the aid of a gauge transformation,we propose a Darboux transformation for a four-component KdV *** an application,we obtain some explicit solutions for the four-component KdV equation.

关键词： a four-component KdV equation Darboux transformation explicit solutions

Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation

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Progress in Natural Science:Materials International 2008年第3期18卷 259-266页

作者： Zhengrong Liu a, Boling Guob a School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China b Institute of Applied Physics and Computational Mathematics, Beijing 100088, China School of Mathematical Sciences South China University of Technology Guangzhou 510640 China Institute of Applied Physics and Computational Mathematics Beijing 100088 China

In this paper, we consider the generalized Camassa-Holm equation ut+2kux-uxxt+ au~2ux=2uxuxx+uuxxxUnder substitution ξ=x-ct, some new explicit periodic wave solutions and their limit forms are presented through some special phase orbits. These periodic wave solutions tend to infinity on ξ-u plane periodically. Thus we call them periodic blow-up solutions. To our knowledge, such periodic blow-up solutions have not been found in any other equations.

关键词： Generalized Camassa–Holm equation explicit solutions Periodic blow-up