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The Lump Solutions of the (1 + 1)-Dimensional Ito-Equation  [PDF]
Xiangmin Meng, Hongcai Ma
Open Journal of Applied Sciences (OJAppS) , 2019, DOI: 10.4236/ojapps.2019.93011
Abstract: In this paper, several kinds of lump solutions for the (1 + 1)-dimensional Ito-equation are introduced. The proposed method in this work is based on a Hirota bilinear differential equation. The form of the solutions to the equation is constructed and the solutions are improved through analysis and symbolic computations with Maple. Finally, figure of the solution is made for specific examples for the lump solutions.
Variable separation solutions and new solitary wave structures to the (1+1)-dimensional Ito system
Variable separation solutions and new solitary wave structures to the (l+l)-dimensional Ito system

Xu Chang-Zhi,He Bao-Gang,Zhang Jie-Fang,
徐昌智
,何宝钢,张解放

中国物理 B , 2006,
Abstract: A variable separation approach is proposed and extended to the (1+1)-dimensional physics system. The variable separation solution of (1+1)-dimensional Ito system is obtained. Some special types of solutions such as non-propagating solitary wave solution, propagating solitary wave solution and looped soliton solution are found by selecting the arbitrary function appropriately.
Multiple travelling wave solutions of two newly generalized Ito systems
两组新的广义的Ito方程组的多种行波解

Li Hua-Mie,Lin Ji,Xu You-Sheng,
李画眉
,林 机,许友生

物理学报 , 2004,
Abstract: By use of the extended mapping deformation method,many explicit and exact travelling wave solutions of the two newly generalized Ito systems are obtained which contain solitary wave solutions,trigonometric function solutions,Jacobian elliptic function solutions and rational solutions.
几种广义的函数展开法在构建偏微分方程精确解中的文献综述与应用(G/G2)-展开法、(exp)-展开法构建(2 + 1)维Boiti-Leon-Pempinelli方程精确解
A Literature Review and Application of Sev Eral Generalized Function Expansion Methods in Constructing Exact Solutions of Partial Differential Equations(G/G2)-Expansion Method,(exp)-Expansion Method Construction (2 + 1) Exact Solution of the Dimensional Boiti-Leon-Pempinelli Equation
 [PDF]

吴大山, 孙峪怀, 杜玲禧
Advances in Applied Mathematics (AAM) , 2019, DOI: 10.12677/AAM.2019.810196
Abstract:
首先,系统给出(G/G2)-展开法、F-展开法、(exp)-展开法、改进的Kudryashov方法、直接截断法,构建偏微分方程的精确解的起源与研究现状的文献综述。接下来,采用对比方式给出上述五种广义的函数展开法在构建偏微分方程精确解的步骤。最后,通过上述五种广义的函数展开法中的(G/G2)-展开法、(exp)-展开法构建(2 + 1)维Boiti-Leon-Pempinelli方程的精确解,并使用控制变量法进行数学实验分析了(2 + 1)维Boiti-Leon-Pempinelli方程中三个变量对于精确解的影响。
First, the system gives(G/G2)-expansion method, F-expansion method, (exp)-expansion method, improved Kudryashov method, direct truncation method, to construct the literature review of the origin and research status of the exact solutions of partial differential equations. Next, the steps of constructing the exact solutions of the partial differential equations by the above five generalized function expansion methods are given in comparison. Finally, through the above five generalized (G/G2)-expansion method, (exp)-expansion method in the function expansion method constructs the exact solution of the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation. The control variable method is used to analyze the influence of three variables on the exact solution in the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation.
A Generalized Ito Formula  [PDF]
Kenneth L. Kuttler,Ji Li
Mathematics , 2013,
Abstract: An Ito formula is developed in a context consistent with the development of abstract existence and unique- ness theorems for nonlinear stochastic partial differential equations, which are singular or degenerate. This is a generalization of an earlier Ito formula for Gelfand triples. After this, an existence theorem is presented for some singular and degenerate stochastic equations followed by a few examples.
Solitary wave series solution for generalized (3+1)-dimensional nonlinear Burgers system
广义(3+1)维非线性Burgers系统孤波级数解

Wen Zhao-Hui,Mo Jia-Qi,
温朝晖
,莫嘉琪

物理学报 , 2010,
Abstract: A class of generalized (3+1)-dimensional nonlinear Burgers system is studied. Using the homotopic mapping method, the corresponding mapping expansions are constructed and using the iteration method the series solution of travelling wave for a solitary wave is obtained.
Solitons in a generalized (2+1)-dimensional Ablowitz-Kaup-Newell-Segur system
郑春龙,张解放,吴锋民,盛正卯,陈立群
中国物理 B , 2003,
Abstract: In the previous Letter (Zheng C L and Zhang J F 2002 Chin. Phys. Lett. 19 1399), a localized excitation of the generalized Ablowitz-Kaup-Newell-Segur (GAKNS) system was obtained via the standard Painlevé truncated expansion and a special variable separation approach. In this work, starting from a new variable separation approach, a more general variable separation excitation of this system is derived. The abundance of the localized coherent soliton excitations like dromions, lumps, rings, peakons and oscillating soliton excitations can be constructed by introducing appropriate lower-dimensional soliton patterns. Meanwhile we discuss two kinds of interactions of solitons. One is the interaction between the travelling peakon type soliton excitations, which is not completely elastic. The other is the interaction between the travelling ring type soliton excitations, which is completely elastic.
Integrability of a Generalized Ito System: the Painleve Test  [PDF]
Ayse Karasu,Atalay Karasu,Sergei Sakovich
Physics , 2001, DOI: 10.1143/JPSJ.70.1165
Abstract: It is shown that a generalized Ito system of four coupled nonlinear evolution equations passes the Painleve test for integrability in five distinct cases, of which two were introduced recently by Tam, Hu and Wang. A conjecture is formulated on integrability of a vector generalization of the Ito system.
Gravitation and Cosmology in Generalized (1+1)-dimensional dilaton gravity  [PDF]
J. S. F. Chan,R. B. Mann
Physics , 1994, DOI: 10.1088/0264-9381/12/2/006
Abstract: The actions of the ``$R=T$'' and string-inspired theories of gravity in (1+1) dimensions are generalized into one single action which is characterized by two functions. We discuss differing interpretations of the matter stress-energy tensor, and show how two such different interpretations can yield two different sets of field equations from this action. The weak-field approximation, post-Newtonian expansion, hydrostatic equilibrium state of star and two-dimensional cosmology are studied separately by using the two sets of field equations. Some properties in the ``$R=T$'' and string-inspired theories are shown to be generic in the theory induced by the generalized action.
Exact Solutions of Generalized Boussinesq-Burgers Equations and (2+1)-Dimensional Davey-Stewartson Equations  [PDF]
Isaiah Elvis Mhlanga,Chaudry Masood Khalique
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/389017
Abstract: We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+1)-dimensional Davey-Stewartson equations. 1. Introduction Most nonlinear physical phenomena that appear in many areas of scientific fields such as plasma physics, solid state physics, fluid dynamics, optical fibers, mathematical biology, and chemical kinetics can be modelled by nonlinear partial differential equations (NLPDEs). The investigation of exact travelling wave solutions of these NLPDEs is important for the understanding of most nonlinear physical phenomena and possible applications. To address this issue, various methods for finding travelling wave solutions to NLPDEs have been proposed. Some of the most important methods include homogeneous balance method [1], the ansatz method [2, 3], variable separation approach [4], inverse scattering transform method [5], B?cklund transformation [6], Darboux transformation [7], Hirota’s bilinear method [8], the -expansion method [9], the reduction mKdV equation method [10], the tri-function method [11, 12], the projective Riccati equation method [13], the sine-cosine method [14, 15], the Jacobi elliptic function expansion method [16, 17], the -expansion method [18], the exp-function expansion method [19], and Lie symmetry method [20–24]. In this paper, we study two systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. We employ the Lie symmetry method and the travelling wave variable approach to obtain exact solutions to both these systems. The Lie symmetry approach is one of the most effective methods to determine solutions of nonlinear partial differential equations. Sophus Lie (1842–1899), with the inspiration from Galois’ theory for solving algebraic equations, discovered this method which is known today as Lie group analysis. He showed that many of the known methods of integration of ordinary differential equations could be derived in a systematic manner using his theory of continuous transformation groups. In the past few decades, a considerable amount of development has been made in symmetry methods for differential equations. This is evident by the number of research papers, books, and many new symbolic softwares devoted to the subject
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