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A Maple Package on Symbolic Computation of hirota bilinear form for Nonlinear Equations

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Communications in Theoretical Physics 2009年第11期52卷 801-807页

作者： YANG Xu-Dong RUAN Hang-Yu Department of Physics Ningbo University Ningbo 315211 China

An improved algorithm for symbolic computation of hirota bilinear forms of KdV-type equations withlogarithmic transformations is *** the algorithm,the general assumption of hirota bilinear form is successfullyreduced based on the property of uniformity in ***,we discard the integral operation in the *** software package HBFTrans is written in Maple and its running effectiveness is tested by a variety solitonequations.

关键词： hirota bilinear form nonlinear equation, symbolic algebra

Grammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation

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Chinese Physics B 2012年第7期21卷 85-91页

作者：唐亚宁马文秀徐伟 Department of Applied Mathematics Northwestern Polytechnical University Department of Mathematics and Statistics University of South Florida

Based on the Grammian and Pfaffian derivative formulae, Grammian and Pfaffian solutions are obtained for a （3＋1）-dimensional generalized shallow water equation in the hirota bilinear form. Moreover, a Pfaffian extension is made for the equation by means of the Pfaffianization procedure, the Wronski-type and Gramm-type Pfaffian solutions of the resulting coupled system are presented.

关键词： hirota bilinear form Grammian and Pfaffian solutions Wronski-type and Gramm-typePfaffian solutions

A Pair of Resonance Stripe Solitons and Lump Solutions to a Reduced(3+1)-Dimensional Nonlinear Evolution Equation

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Communications in Theoretical Physics 2017年第6期67卷 595-600页

作者：陈美丹李咸王瑶李彪 Ningbo Collaborative Innovation Center of Nonlinear Harzard System of Ocean and Atmosphere and Department of Mathematics Ningbo University

With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.

关键词： hirota bilinear form lump solutions stripe solitons interaction solutions symbolic computation

Determinant Solutions to a (3+1)-Dimensional Generalized KP Equation with Variable Coefficients

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Chinese Annals of Mathematics,Series B 2012年第5期33卷 641-650页

作者： Alrazi ABDELJABBAR Ahmet YILDIRIM Department of Engineering Technology and Mathematics Savannah State University Department of Mathematics and Statistics University of South Florida Department of Mathematics Ege University

1 Introduction Although partial differential equations that govern the motion of solitons are nonlinear, many of them can be put into the bilinear form. hirota, in 1971, developed an ingenious method to obtain exact solutions to nonlinear partial differential equations in the soliton theory, such as the KdV equation, the Boussinesq equation and the KP equation （see [1-2]）.

关键词： hirota bilinear form Wronskian solution Grammian solution

bilinear approach to N=2 supersymmetric KdV equations

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Science China Mathematics 2009年第9期52卷 1973-1981页

作者： ZHANG MengXia LIU QingPing SHEN YaLi WU Ke Department of Mathematics China University of Mining and TechnologyBeijing 100083China School of Mathematical Sciences Capital Normal UniversityBeijing 100037China Department of Applied Mathematics Yuncheng UniversityYuncheng 044000China

The N=2 supersymmetric KdV equations are studied within the framework of hirota bilinear method. For two such equations, namely N=2, a=4 and N=2, a=1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bcklund transformation is given for the N=2, a=1 supersymmetric KdV equation.

关键词： supersymmetric KdV equation hirota bilinear form solitons Backlund transformation Lax representation

A Simple Approach to Derive a Novel N-Soliton Solution for a (3+1)-Dimensional Nonlinear Evolution Equation

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Communications in Theoretical Physics 2010年第5期53卷 812-814页

作者：吴建平 Institute of Electronic Technology Information Engineering University

Based on the hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equat... 详细信息

关键词： (3+1)-dimensional nonlinear evolution equation hirota bilinear form N-soliton solution resonant behavior

Bcklund Transformation and Multisoliton Solutions in Terms of Wronskian Determinant for (2+1)-Dimensional Breaking Soliton Equations with Symbolic Computation

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Communications in Theoretical Physics 2010年第12期 1059-1066页

作者：秦渤田播刘立才孟祥花刘文军 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 EducationUniversity of Posts and Telecommunications School of Science P.O.Box 122Beijing University of Posts and Telecommunications

In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the hirota bilineax forms and Bgcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilineax equations. Via the Wronskian technique, it is proved that the Bgcklund transformations obtained are the ones between the （ N - 1）- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonie properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.

关键词： breaking soliton equations hirota bilinear form B/icklund transformation Wronskian determinant symbolic computation