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rational solutions of High-Order Algebraic Ordinary Differential Equations

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Journal of Systems Science & Complexity 2020年第3期33卷 821-835页

作者： VO Thieu N. ZHANG Yi Fractional Calculus Optimization and Algebra Research GroupFaculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam Johann Radon Institute for Computational and Applied Mathematics(RICAM) Austrian Academy of SciencesAustria Department of Mathematical Sciences The University of Texas at Dallas(UTD)USA

This paper considers algebraic ordinary differential equations(AODEs)and study their polynomial and rational *** authors first prove a sufficient condition for the existence of a bound on the degree of the possible polynomial solutions to an *** AODE satisfying this condition is called *** the authors prove that some common classes of low-order AODEs are *** rational solutions,the authors determine a class of AODEs,which are called maximally comparable,such that the possible poles of any rational solutions are recognizable from their *** generalizes the well-known fact that any pole of rational solutions to a linear ODE is contained in the set of zeros of its leading ***,the authors develop an algorithm to compute all rational solutions of certain maximally comparable AODEs,which is applicable to 78.54%of the AODEs in Kamke's collection of standard differential equations.

关键词： Algebraic ordinary differential equations algorithms polynomial solutions rational solutions

rational solutions of Painlevé-Ⅱequation as Gram determinant

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Chinese Physics B 2023年第12期32卷 200-211页

作者：张晓恩陆冰滢 College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590China SISSA Via Bonomea 265TriesteItaly

Under the Flaschka-Newell Lax pair,the Darboux transformation for the Painlevé-Ⅱequation is constructed by the limiting *** the aid of the Darboux transformation,the rational solutions are represented by the Gram determinant,and then we give the large y asymptotics of the determinant and the rational ***,the solution of the corresponding Riemann-Hilbert problem is obtained from the Darboux matrices.

关键词： Painlevé-Ⅱequation Darboux transformation rational solutions

rational solutions with Non-zero Asymptotics of the Modified Korteweg-de Vries Equation

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Communications in Theoretical Physics 2012年第6期57卷 923-929页

作者：张莹莹张大军 Department of Mathematics Shanghai University

Using the relation between the mKdV equation and the KdV-mKdV equation,we derive non-singular rational solutions for the mKdV *** solutions are given in terms of *** of some solutions is investigated by means of asymp... 详细信息

关键词： the modified KdV equation bilinear B^icklund transformation Wronskian rational solutions

Generalized Darboux Transformation and rational solutions for the Nonlocal Nonlinear Schrodinger Equation with the Self-Induced Parity-Time Symmetric Potential

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Journal of Applied Mathematics and Physics 2015年第5期3卷 530-536页

作者： Jian Chen College of Science University of Shanghai for Science and TechnologyShanghaiChina

In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.

关键词： Generalized Darboux Transformation rational solutions Nonlocal Nonlinear Schrodinger Equation

The Riccati Equation, Differential Transform, rational solutions and Applications

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Applied Mathematics 2022年第9期13卷 774-792页

作者： Malick Ndiaye Department of Mathematics and Computer Sciences Marist College Poughkeepsie NY USA

In this article, the Riccati Equation is considered. Various techniques of finding analytical solutions are explored. Those techniques consist mainly of making a change of variable or the use of Differential Transform. It is shown that the nonconstant rational functions whose numerator and denominator are of degree 1, cannot be solutions to the Riccati equation. Two applications of the Riccati equation are discussed. The first one deals with Quantum Mechanics and the second one deal with Physics.

关键词： Riccati Equation Differential Transform rational solutions

rational and Periodic Wave solutions of Two-Dimensional Boussinesq Equation

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Communications in Theoretical Physics 2008年第4期49卷 815-824页

作者： ZHANG yi YE Ling-Ya Department of Mathematics Zhejiang Normal University Jinhua 321004 China

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.

关键词： soliton Hirota bilinear method Riemann theta function periodic wave solutions rational solutions two-dimensional Boussinesq equation

New Families of Nontravelling Wave solutions to Two （3＋1）-Dimensional Equations

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Communications in Theoretical Physics 2005年第3期43卷 417-422页

作者： BAICheng-Lin LIUXi-Qiang ZHAOHong SchoolofPhysicsScienceandInformationEngineering LiaochengUniversityLiaocheng252059China SchoolofMathematicsScience LiaochengUniversityLiaocheng252059China

In this paper, two (3+1)-dimensional equations are ***-Baecklund transformation is obtained, which is used with some ansatze to seek new types ofexact solutions including some arbitrary functions. When these arbitrary functions are taken assome special functions, these solutions possess abundant structures. These solutions containsoliton-like solutions and rational solutions.

关键词： (3+1)-dimensional equations soliton-like solutions rational solutions

Generalized Wronskian solutions to Differential-Difference KP Equation

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Communications in Theoretical Physics 2007年第5期47卷 769-772页

作者： JI Jie YAO Yu-Qin LIU Yu-Qing CHEN Deng-Yuan Department of Mathematics Shanghai University Shanghai 200444 China Department of Information Science Jiangsu Polytechnic University Changzhou 213016 China

A new method for constructing the Wronskian entries is proposed and applied to the differential-difference Kadomtsev-Petviashvilli （DΔKP） equation. The generalized Wronskian solutions to it are obtained, including ... 详细信息

关键词： Wronskian technique DΔKP equation rational solutions Matveev solutions

Novel Wronskian solutions to Boussinesq Equation

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Communications in Theoretical Physics 2007年第4X期48卷 577-583页

作者： YAO Yu-Qin LIU Yu-Qing JI Jie CHEN Deng-Yuan Department of Mathematics Shanghai University Shanghai 200444 China

The new method for constructing the Wronskian entries is applied to the Boussinesq equation. The novel Wronskian solutions to it are obtained, including solitons, rational solutions, Matveev solutions, and complexitons.

关键词： Boussinesq equation Wronskiail technique rational solutions Matveev solutions complexitons