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symbolic computation FOR THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS

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Acta Mathematica Scientia 2022年第6期42卷 2478-2504页

作者： Bo HUANG Wei NIU Dongming WANG LMIB-School of Mathematical Sciences Beihang UniversityBeijing 100191China Ecole Centrale de Pekin Beihang UniversityBeijing 100191China Beihang Hangzhou Innovation Institute Yuhang Hangzhou 310051China Centre National de la Recherche Scientifique 75794 Paris Cedea 16France

This paper provides a survey on symbolic computational approaches for the analysis of qualitative behaviors of systems of ordinary differential equations,focusing on symbolic and algebraic analysis for the local stability and bifurcation of limit cycles in the neighborhoods of equilibria and periodic orbits of the systems,with a highlight on applications to computational biology.

关键词： biological systems center-focus limit cycles qualitative analysis symbolic computation

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 WANG Deng-Shan CHEN Lan-Xin 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 R/ccati equations rational expansion method was further extended. Many double soliton-like and other novel combined forms of exact solutions of the （2＋1）-dimensional Breaking soliton equation are derived by using the extended multiple Riccatl equations expansion method.

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

A Bilinear Bcklund Transformation and N-Soliton-Like Solution of Three Coupled Higher-Order Nonlinear Schrdinger Equations with symbolic computation

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Communications in Theoretical Physics 2008年第9期50卷 689-695页

作者： ZHU Hong-Wu TIAN Bo MENG Xiang-Hua LI Juan XU Tao School of Science P.O. Box 122 Beijing University of Posts and Telecommunications Beijing 100876 China Key Laboratory of Optical Communication and Lightwave Technologies Ministry of Education Beijing University of Posts and Telecommunications Beijing 100876 China Hunan Meteorological Observatory Changsha 410007 China

A bilinear Baecklund transformation is presented for the three coupled higher-order nonlinear Schroedinger equations with the inclusion of the group velocity dispersion, third-order dispersion and Kerr-law nonlinearity, which can describe the dynamics of alpha helical proteins in living systems as well as the propagation of ultrashort pulses in wavelength-division multiplexed system. Starting from the Baecklund transformation, the analytical soliton solution is obtained from a trivial solution. Simultaneously, the N-soliton-like solution in double Wronskian form is constructed, and the corresponding proof is also given via the Wronskian technique. The results obtained from this paper might be valuable in studying the transfer of energy in biophysics and the transmission of light pulses in optical communication systems.

关键词： coupled higher-order nonlinear Schroedinger equations Baecklund transformation soliton solution Wronskian technique symbolic computation

Darboux Transformation and Grammian Solutions for Nonisospectral Modified Kadomtsev-Petviashvili Equation with symbolic computation

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Communications in Theoretical Physics 2008年第8期50卷 411-416页

作者： LI Juan FIAN Bo ZHANG Hai-Qiang XU Tao ZHANG Ya-Xing School of Science P.O. Box 122 Beijing University of Posts and Telecommunications Beijing 100876 China State Key Laboratory of Software Development Environment Beijing University of Aeronautics and Astronautics Beijing 100083 China Key Laboratory of Optical Communication and Lightwave Technologies Ministry of Education Beijing University of Posts and Telecommunications Beijing 100876 China

In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painleve branches through the Painleve analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.

关键词： nonisospectral modified Kadomtsev-Petviashvili equation Darboux transformation Grammiansolution symbolic computation

Conservation Laws and Analytic Soliton Solutions for Coupled Integrable Dispersionless Equations with symbolic computation

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Communications in Theoretical Physics 2010年第10期54卷 687-696页

作者：王盼田播刘文军屈启兴江彦 School of Science Beijing University of Posts and Telecommunications State Key Laboratory of Software Development Environment Beijing University of Aeronautics and Astronautics Key Laboratory of Information Photonics and Optical Communications(BUPT) Ministry of EducationBeijing University of Posts and Telecommunications

Under investigation in this paper are two coupled integrable dispersionless (CID) equations modelingthe dynamics of the current-fed string within an external magnetic *** a set of the dependent variabletransformations, the bilinear forms for the CID equations are *** on the Hirota method and symboliccomputation, the analytic N-soliton solutions are *** many conservation laws for the CID equationsare given through the known spectral *** characteristics and interaction behaviors of the solitons areanalyzed graphically.

关键词： coupled integrable dispersionless equations conservation laws soliton solutions hirota method symbolic computation

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

Darboux Transformation and Soliton Solutions for the (2+1)-Dimensional Generalization of Shallow Water Wave Equation with symbolic computation

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Communications in Theoretical Physics 2013年第8期60卷 194-200页

作者：闻小永孟祥花 Department of Mathematics School of Applied ScienceBeijing Information Science and Technology University

In this paper, the （2＋l）-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, we prove that the （2＋ l）-dimensional generalization of shallow water wave equation possesses the Palnlev6 property under a certain condition, and its Lax pair is constructed by applying the singular manifold method. Based on the obtained Lax representation, the Darboux transformation （DT） is constructed. The first iterated solution, second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT. Relevant properties are graphically illustrated, which might be helpful to understanding the propagation processes for ocean waves in shallow water.

关键词： (2+1)-dimensional generalization of shallow water wave equation singular manifold method Laxpair Darboux transformation symbolic computation

Using symbolic computation to Exactly Solve the Integrable Broer-Kaup Equations in （2＋1）-Dimensional Spaces

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Communications in Theoretical Physics 2005年第4期43卷 585-590页

作者： CHENJing XIEFu-Ding LüZhuo-Sheng DepartmentofComputerScience LiaoningNormalUniversityDalian116029China KeyLaboratoryofMathematicsMechanization InstituteofSystemsSciencesAcademyofMathematicsandSystemsSciencestheChineseAcademyofSciencesBeijing100080China

The extended tanh method is further improved by generalizing the Riccati equation and introducing its twenty seven new solutions. As its application, the (2+ 1)-dimensional Broer-Kaup equation is investigated and then its fifty four non-travelling wave solutions have been obtained. The results reported in this paper show that this method is more powerful than those, such as tanh method, extended tanh method, modified extended tanh method and Riccati equation expansion method introduced in previous literatures.

关键词： BK equations symbolic computation non-travelling wave solution

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

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Communications in Theoretical Physics 2006年第2X期46卷 199-202页

作者： LUE Zhuo-Sheng 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 mathematlcal 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