New Solitary Wave Solutions to the KdV-Burgers Equation

Based on the analysis on the features of the Burgers equation and KdV equation as well as KdV-Burgers equation, a superposition method is proposed to construct the solitary wave solutions of the KdV-Burgers equation from those of the Burgers equation and KdV equation, and then by using it we obtain many solitary wave solutions to the KdV-Burgers equation, some of which are new ones.PACS: 02.30.Jr; 03.65.Ge     

Demonstration of Soliton Solutions of DNLS Equation by Liouville Theorem

In the inverse scattering transform (IST), the reflectionless Jost solutions are combined by their analytic properties in the complex spectrum parameter plane, and then can be shown to satisfy the two Lax equations indeed by Liouville theorem. So the corresponding soliton solutions certainly satisfy the nonlinear equation by compatibility condition. Especially the multi-soliton solutions of DNLS equation can be demonstrated in this way. PACS Numbers: 05.45.Yv, 02.30.-f, 11.10.Ef     

New Explicit Exact Solutions of the Born–Infeld Equation

By introducing an auxiliary ordinary differential equation and solving it by means of the method of separation of variables, many new explicit exact solutions to the Born–Infeld equation are found in a concise manner. PACS: 02.30.Jr; 03.40.Kf     

Soliton Solutions of DNLS Equation Found by IST Anew and its Verification in Marchenko Formalism

A general procedure is proposed to derive the multi-soliton solutions of DNLS equation with vanishing boundary value, and the two-soliton solutions of it is given as an example. Furthermore, the verification of multi-soliton solutions is done through Marchenko formalism. PACS numbers: 05.45.Yv; 02.30.-f; 11.10.Ef     

The Application of Extended Tanh-Function Approach in Toda Lattice Equations

In this paper, we generalize the extended tanh-function approach, which used to find new exact travelling wave solutions of nonlinear partial differential equations (NPDES) or coupled nonlinear partial differential equations, to nonlinear differential-difference equations (NDDES). As illustration, we discuss some Toda lattice equations, and solitary wave and periodic wave solutions of these Toda lattice equations are obtained by means of the extended tanh-function approach. PACS numbers: 05.45.Yv, 02.30.Jr, 02.30.Ik.     

Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.     

New Types of Localized Coherent Structures in the Bogoyavlenskii-Schiff Equation

Based on the singular structure analysis, we derive some new types of localized coherent structures for the Bogoyavlenskii-Schiff equation by suitably utilizing the arbitrary function present in the singular manifold equations. PACS 05.45. Yv, 02.30.Ik, 02.30.Jr.     

Nonpropagating Solitary Waves in (2+1)-Dimensional Generalized Dispersive Long Wave Systems

An extended mapping approach is used to obtain a new type of variable separation excitation, with three arbitrary functions, of the 2+1-dimensional generalized dispersive long wave equation (DLWE). By selecting appropriate functions, the richness of nonpropagating solitons, such as nonpropagating dromion, nonpropagating ring, nonpropagating lump, and nonpropagating foldon, etc., is displayed for the ($2+1$)-dimensional generalized dispersive long wave equation (DLWE) in this paper. Meanwhile, we conclude that the solution v₁ and v₂ are essentially equivalent to the “universal” formula. PACS numbers: 02.30.Jr, 02.30.Ik, 05.45.Yv     

Soliton Perturbation Theory for the Compound KdV Equation

The soliton perturbation theory is used to study the solitons that are governed by the compound Korteweg de-Vries equation in presence of perturbation terms. The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained. AMS Codes: 35Q51; 35Q53; 37K10. PACS Codes: 02.30.Jr; 02.30.Ik.     

Construction of New Variable Separation Excitations via Extended Projective Ricatti Equation Expansion Method in (2+ 1)- Dimensional Dispersive Long Wave Systems

Utilizing the extended projective Ricatti equation expansion method, abundant variable separation solutions of the (2+1)-dimensional dispersive long wave systems are obtained. From the special variable separation solution (38) and by selecting appropriate functions, new types of interaction between the multi-valued and the single-valued solitons, such as semi-foldon and dromion, semi-foldon and peakon, semi-foldon and compacton are found. Meanwhile, we conclude that the solution v is essentially equivalent to the ’universal” formula (1). PACS numbers 05.45.Yv, 02.30.Jr, 03.65.Ge     

The Construction of Dirac Wave Packets for a Fermionic Particle Non-Minimally Coupling with an External Magnetic Field

We shall proceed with the construction of normalizable Dirac wave packets for fermionic particles (neutrinos) with dynamics governed by a “modified” Dirac equation with a non-minimal coupling with an external magnetic field. We are not only interested on the analytic solutions of the “modified” Dirac wave equation but also on the construction of Dirac wave packets which can be used for describing the dynamics of some observable physical quantities which are relevant in the context of the quantum oscillation phenomena. To conclude, we discuss qualitatively the applicability of this formal construction in the treatment of chiral (and flavor) oscillations in the theoretical context of neutrino physics. PACS numbers: 02.30.Cj, 03.65.Pm     

(n+1)维双Sine-Gordon方程的新精确解

给出包含第一种椭圆方程的三角函数型辅助方程及其解的叠加公式.在一般函数变换下,借助符号计算系统Mathematica,构造了(n+1)维双sine-Gordon方程新的Jacobi椭圆函数精确解.这些解包括了行波变换下的Jacobi椭圆函数精确解、精确孤立波解和三角函数解.     

Nonlinear superposition rule for Abel's equation

A nonlinear superposition rule for Abel's cubic differential equation involving four particular solutions can be replaced by a superposition rule involving only two particular solutions. Chiellini's restrictions on the particular solutions are found to be satisfied.     

sine-Gordon型方程的无穷序列新精确解

为了获得sine-Gordon型方程的无穷序列精确解,给出三角函数型辅助方程和双曲函数型辅助方程及其Bäcklund变换和解的非线性叠加公式,借助符号计算系统Mathematica,构造了sine-Gordon方程、mKdV-sine-Gordon方程、(n+1)维双sine-Gordon方程和sinh-Gordon方程的无穷序列新精确解.其中包括无穷序列三角函数解、无穷序列双曲函数解、无穷序列Jacobi椭圆函数解和无穷序列复合型解. 关键词： sine-Gordon型方程 解的非线性叠加公式 辅助方程 无穷序列精确解     

Degasperis-Procesi 方程的无穷序列尖峰孤立波解

本文为了构造非线性发展方程的无穷序列尖峰精确解,给出了Riccati方程的Bäcklund 变换和解的非线性叠加公式,并借助符号计算系统Mathematica,用Degasperis-Procesi方程为应用实例,构造了无穷序列尖峰孤立波解和无穷序列尖峰周期解. 关键词： Riccati方程 解的非线性叠加公式 尖峰孤立波解 Degasperis-Procesi 方程     

(2+1)维广义Calogero-Bogoyavlenskii-Schiff方程的无穷序列类孤子解

为了构造高维非线性发展方程的无穷序列类孤子新解, 研究了二阶常系数齐次线性常微分方程, 获得了新结论. 步骤一, 给出一种函数变换把二阶常系数齐次线性常微分方程的求解问题转化为一元二次方 程和Riccati方程的求解问题. 在此基础上, 利用Riccati方程解的非线性叠加公式, 获得了二阶常系数齐次线性常微分方程的无穷序列新解. 步骤二, 利用以上得到的结论与符号计算系统Mathematica, 构造了(2+1)维广义Calogero-Bogoyavlenskii-Schiff (GCBS)方程的无穷序列类孤子新解. 关键词： 常微分方程 非线性叠加公式 高维非线性发展方程 无穷序列类孤子新解     

构造非线性发展方程的无穷序列复合型类孤子新解

为了构造非线性发展方程的无穷序列复合型类孤子新解, 进一步研究了G'(ξ)/G(ξ) 展开法. 首先, 给出一种函数变换, 把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题. 然后, 利用Riccati方程解的非线性叠加公式, 获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解. 在此基础上, 借助符号计算系统Mathematica, 构造了改进的(2+1)维色散水波系统和(2+1)维色散长波方程的无穷序列复合型类孤子新精确解. 关键词： G'(ξ)/G(ξ)展开法')" href="#">G'(ξ)/G(ξ)展开法 非线性叠加公式 非线性发展方程 复合型类孤子新解     

New application to Riccati equation

<正>To seek new infinite sequence of exact solutions to nonlinear evolution equations,this paper gives the formula of nonlinear superposition of the solutions and Backlund transformation of Riccati equation.Based on the tanhfunction expansion method and homogenous balance method,new infinite sequence of exact solutions to Zakharov-Kuznetsov equation,Karamoto-Sivashinsky equation and the set of(2+l)-dimensional asymmetric Nizhnik-Novikov-Veselov equations are obtained with the aid of symbolic computation system Mathematica.The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.     

构造非线性发展方程无穷序列复合型精确解的一种方法

为了获得非线性发展方程新的无穷序列复合型精确解,给出了Riccati方程的Bäcklund变换和解的非线性叠加公式,符号计算系统Mathematica的帮助下,以广义Boussinesq方程为应用实例,获得了无穷序列复合型精确解.这里包括双曲函数、三角函数与有理函数复合解、双曲函数与三角函数复合解等几种新的无穷序列复合型精确解.该方法在构造非线性发展方程无穷序列复合型精确解方面具有普遍意义. 关键词： 非线性发展方程 非线性叠加公式 Riccati方程 无穷序列精确解     

A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions

Abstract

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a B¨acklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.