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symbolic computation and q-Deformed Function Solutions of (2+1)-Dimensional Breaking Soliton Equation

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Communications in Theoretical Physics 2007年第2期47卷 270-274页

作者： CAO Li-Na School of Mathematics and Computer Science,Central University for Nationalities,Beijing 100081,ChinaWANG Deng-Shan Mathematics Mechanization Key Lab,Academy of Mathematics and System Sciences,Chinese Academy of Sciences,Beijing 100080,ChinaCHEN Lan-Xin Department of Mathematics,Shijiazhuang University,Shijiazhuang 050035,China School of Mathematics and Computer Science Central University for Nationalities Beijing 100081 China Mathematics Mechanization Key Lab Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China Department of Mathematics Shijiazhuang University Shijiazhuang 050035 China

In this paper,by using symbolic and algebra computation,Chen and Wang’s multiple Riccati equationsrational expansion method was further extended.Many double soliton-like and other novel combined forms of exactsolutions of the (2+1)-dimensional Breaking soliton equation are derived by using the extended multiple Riccati equationsexpansion method.

关键词： q-deformed hyperbolic functions symbolic computation Riccati equation soliton-like solution

symbolic computation Study of (2+1)-Dimensional Dispersive Long Wave Equations

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Communications in Theoretical Physics 2006年第8期46卷 199-202页

作者： LU Zhuo-Sheng~(1,2,+) and XIE Fu-Ding~(2,3) ~1School of Science, Beijing University of Post and Telecommunications, Beijing 100876, China ~2Key Laboratory of Mathematics Mechanization, Academy of Mathematics and System Sciences, the Chinese Academy of Sciences, Beijing 100080, China ~3Department of Computer Science, Liaoning Normal University, Dalian 116029, China School of Science Beijing University of Post and Telecommunications Beijing 100876 China Key Laboratory of Mathematics Mechanization Academy of Mathematics and System Sciences the Chinese Academy of Sciences Beijing 100080 China Department of Computer Science Liaoning Normal University Dalian 116029 China

Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematical physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact solutions of the (2+1)-dimensional dispersive long wave equations are obtained, among which there are soliton-like solutions, mult-soliton-like solutions and formal periodic solutions, etc. Certain special solutions are considered and some interesting localized structures are revealed.

关键词： dispersive long wave equations symbolic computation exact solution localized structure

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

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POINCARE-LIGHTHILL-KUO METHOD AND symbolic computation

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Applied Mathematics and Mechanics(English Edition) 2001年第3期22卷 261-269页

作者： DAI Shi-qiang(戴世强) Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 P R China

This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi-inverse algorithm was proposed by the author, with which the lengthy ports of intermediate expressions are first frozen in the form of symbols till the Fnal stage of seeking perturbation solutions. Tn discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer-extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head-on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher-order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro-commputers. Finally, it is concluded that,vith the aid of symbolic computation, the vitality of the PLK method is greatly. Strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.

关键词： PLK method perturbation methods symbolic computation intermediate expression swell semi-inverse algorithm

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Adaptive H∞control of polynomial Hamiltonian systems via symbolic computation:controller parameterisation

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Journal of Control and Decision 2020年第2期7卷 160-177页

作者： Zhong Cao Xiaorong Hou Wenjing Zhao School of Physics and Electronic Engineering Guangzhou UniversityGuangzhou510006People’s Republic of China Mathematical Educational Software of Engineering Technology Research Center of Guangdong Province Guangzhou510006People’s Republic of China School of Automation Engineering University of Electronic Science and Technology of ChinaChengduPeople’s Republic of China Lab Center Guangzhou UniversityGuangzhouPeople’s Republic of China

This paper deals with controller parameterisation method of adaptive H∞control for polynomial Hamiltonian systems(PHSs)with disturbances and unknown parameters.We design a simplified controller with a set of tuning parameters which can guarantee that the systems are adaptive H∞stable by using Hamiltonian function method.Then,a method for solving the set of tuning parameters of the controller with symbolic computation is presented.The proposed parameterisation method avoids solving Hamilton–Jacobi–Issacs(HJI)equations and the obtained controller is easier as compared to some existing ones.Simulation example shows that the controller is effective as it can optimise adaptive H∞control by adjusting tuning parameters.All these results are expected to be of use in the study of adaptive H∞control for nonlinear systems with disturbances and unknown parameters.

关键词： symbolic computation polynomial Hamiltonian systems(PHSs) tuning parameters adaptive H∞control

Lump solutions to a generalized Hietarinta-type equation via symbolic computation

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Frontiers of Mathematics in China 2020年第3期15卷 435-450页

作者： Sumayah BATWA Wen-Xiu MA Department of Mathematics King Abdulaziz UniversityJeddahSaudi Arabia Department of Mathematics Zhejiang Normal UniversityJinhua 321004China Department of Mathematics and Statistics University of South FloridaTampaFL 33620USA School of Mathematics South China University of TechnologyGuangzhou 510640China

Lump solutions are one of important solutions to partial differential equations,both linear and nonlinear.This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms.The key is a Hirota bilinear form.Lump solutions are constructed via symbolic computations with Maple,and specific reductions of the resulting lump solutions are made.Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented,together with three-dimensional plots and density plots of the lump solutions.

关键词： Soliton equation lump solution symbolic computation Hirota derivative dispersion relation

Using symbolic computation to construct travelling wave solutions to nonlinear partial differential equations

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Chinese Physics B 2004年第10期13卷 1639-1643页

作者：李伟谢福鼎 DepartmentofBasicScience LiaoningTechnicalUniversityFuxin123000China MMKeyLaboratory AcademyofMathematicsandSystemSciencesChineseAcademyofScienceBeijing100080China

Based upon the symbolic computation and the coupled projective Riccati equation, the tanh function method is further improved. As its applications, Wu-Zhang equation (which describes a (2+1)-dimensional dispersive long wave)and the (1+1)-dimensional dispersive long wave equation obtained from Wu-Zhang equation by scaling transformation and symmetry reduction are chosen to illustrate the validity of the proposed approach.

关键词： Wu-Zhang equation coupled projective Riccati equations soliton wave solution periodic wave solution symbolic computation

New conservation laws and exact solutions of the special case of the fifth-order KdV equation

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Journal of Ocean Engineering and Science 2022年第4期7卷 377-382页

作者： Arzu Akbulut Melike Kaplan Mohammed K.A.Kaabar Department of Mathematics-Computer Art-Science FacultyEskisehir Osmangazi UniversityEskisehirTurkey Department of Mathematics Art-Science FacultyKastamonu UniversityKastamonuTurkey Institute of Mathematical Sciences Faculty of ScienceUniversity of MalayaKuala Lumpur 50603Malaysia

The current study deals with the Kaup-Kupershmidt(KK)equation to construct formal Lagrangian,con-servation laws,and exact solutions.KK is basically a special case of the 5th-order KdV equation.The conservation laws obtained by using the conservation theorem are trivial conservation laws.In addition,exact solutions are found via the modified simple equation(MSE)method.For a suitable value of solu-tions,the 3D surfaces have been plotted using MAPLE.These plots giving novel exact solutions are made to reveal important wave characteristics.Our obtained results in this work concerning our investigated equation are essential to explain many physical and oceanographic applications involving ocean gravity waves and many other related phenomena.

关键词： Conservation laws Exact solutions symbolic computation Modified simple equation method

New exact solutions of the Mikhailov-Novikov-Wang equation via three novel techniques

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Journal of Ocean Engineering and Science 2023年第1期8卷 103-110页

作者： Arzu Akbulut Melike Kaplan Mohammed K.A.Kaabar Mathematics-Computer Department Art-Science FacultyEski¸s ehir Osmangazi UniversityEski¸s ehirTurkey Department of Computer Engineering Engineering and Architecture FacultyKastamonu UniversityKastamonuTurkey Institute of Mathematical Sciences Faculty of ScienceUniversity of MalayaKuala Lumpur 50603Malaysia

The current work aims to present abundant families of the exact solutions of Mikhailov-Novikov-Wang equation via three different techniques.The adopted methods are generalized Kudryashov method(GKM),exponential rational function method(ERFM),and modified extended tanh-function method(METFM).Some plots of some presented new solutions are represented to exhibit wave characteristics.All results in this work are essential to understand the physical meaning and behavior of the investigated equation that sheds light on the importance of investigating various nonlinear wave phenomena in ocean engineering and physics.This equation provides new insights to understand the relationship between the integrability and water waves’phenomena.

关键词： Exact solutions Generalized Kudryashov method Modified extended tanh-function method symbolic computation PDEs