Existence of 3D-Vortices in Abelian Gauge Theories

We consider the Abelian gauge theory related to the interaction between the solitary waves of the nonlinear Klein-Gordon equation and the electromagnetic field. We analyze the existence of electromagnetostatic solutions and, in particular, of vortices. Dedicated to the memory of Professor Aldo Cossu     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

New compacton-like and solitary patterns-like solutions to nonlinear wave equations with linear dispersion terms

It has been shown that many fully nonlinear wave equations with nonlinear dispersion terms possess compacton solutions and solitary patterns solutions. In this paper, with the aid of Maple, the mKdV equation, the equation with a source term, the five order KdV-like equation and the KdV–mKdV equation are investigated using some new, generalized transformations. As a consequence, it is shown that these equations with linear dispersion terms admit new compacton-like solutions and solitary patterns-like solutions. These transformations can be also extended to other nonlinear wave equations with nonlinear dispersion terms to seek new compacton-like solutions and solitary patterns-like solutions.     

Gauge Equivalence among Quantum Nonlinear Many Body Systems

Transformations performing on the dependent and/or the independent variables are an useful method used to classify PDE in class of equivalence. In this paper we consider a large class of U(1)-invariant nonlinear Schrödinger equations containing complex nonlinearities. The U(1) symmetry implies the existence of a continuity equation for the particle density ρ≡|ψ|² where the current j _ψ has, in general, a nonlinear structure. We introduce a nonlinear gauge transformation on the dependent variables ρ and j _ψ which changes the evolution equation in another one containing only a real nonlinearity and transforms the particle current j _ψ in the standard bilinear form. We extend the method to U(1)-invariant coupled nonlinear Schrödinger equations where the most general nonlinearity is taken into account through the sum of an Hermitian matrix and an anti-Hermitian matrix. By means of the nonlinear gauge transformation we change the nonlinear system in another one containing only a purely Hermitian nonlinearity. Finally, we consider nonlinear Schrödinger equations minimally coupled with an Abelian gauge field whose dynamics is governed, in the most general fashion, through the Maxwell-Chern-Simons equation. It is shown that the nonlinear transformation we are introducing can be applied, in this case, separately to the gauge field or to the matter field with the same final result. In conclusion, some relevant examples are presented to show the applicability of the method.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

A Method for Constructing Traveling Wave Solutions to Nonlinear Evolution Equations

In this paper, an analytical method is proposed to construct explicitly exact and approximate solutions for nonlinear evolution equations. By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation are obtained explicitly. These solutions include solitary wave solutions, singular traveling wave solutions and periodical wave solutions. These results indicate that in some cases our analytical approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations. It can also be applied to some related nonlinear dynamical systems.     

Explicit and exact travelling wave solutions for the generalized derivative Schrödinger equation

In this paper, a new auxiliary equation expansion method and its algorithm is proposed by studying a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. Being concise and straightforward, the method is applied to the generalized derivative Schrödinger equation. As a result, some new exact travelling wave solutions are obtained which include bright and dark solitary wave solutions, triangular periodic wave solutions and singular solutions. This algorithm can also be applied to other nonlinear wave equations in mathematical physics.     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

Construction of a series of travelling wave solutions to nonlinear equations

In this paper, based on new auxiliary ordinary differential equation with a sixth-degree nonlinear term, we study the (1 + 1)-dimensional combined KdV–MKdV equation, Hirota equation and (2 + 1)-dimensional Davey–Stewartson equation. Then, a series of new types of travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.     

高维弱扰动破裂孤子波方程行波解

研究了一类高维弱扰动破裂孤子波方程.首先讨论了对应的典型破裂孤子波方程, 利用待定系数投射方法得到了孤子波精确解.再利用泛函分析和摄动理论得到了原弱扰动破裂孤子波方程的孤子行波渐近解.最后, 举出例子说明了用该方法得到的弱扰动破裂孤子波方程的行波渐近解具有简捷、有效和较高精度的优点.     

Adiabatic Limit for Some Nonlinear Equations of Gauge Field Theory

We consider the adiabatic limit for nonlinear dynamic equations of gauge field theory. Our main example of such equations is given by the Abelian (2+1)-dimensional Higgs model. We show next that the Taubes correspondence, which assigns pseudoholomorphic curves to solutions of Seiberg--Witten equations on symplectic 4-manifolds, may be interpreted as a complex analogue of the adiabatic limit construction in the (2+1)-dimensional case.     

求解非线性方程的双函数法

基于齐次平衡法和李志斌的tanh函数法,得到简单有效的求解非线性发展方程的双函数法,这种方法利用非线性发展方程孤立波的局部性特点,把非线性方程的孤波解表示为函数f和g的多项式,并用这种方法求出了非线性波理论中的基本模型KdV方程的多组孤波解。     

一类高阶非线性波方程的子方程与精确行波解

结合子方程和动力系统分析的方法研究了一类五阶非线性波方程的精确行波解.得到了这类方程所蕴含的子方程, 并利用子方程在不同参数条件下的精确解, 给出了研究这类高阶非线性波方程行波解的方法, 并以Sawada Kotera方程为例, 给出了该方程的两组精确谷状孤波解和两组光滑周期波解.该研究方法适用于形如对应行波系统可以约化为只含有偶数阶导数、一阶导数平方和未知函数的多项式形式的高阶非线性波方程行波解的研究.     

非线性耦合标量场方程显式解析解的研究

利用两种不同的变换,获得了一类非线性耦合标量场方程的若干类型的精确解析解,其中包括孤子解、奇性孤波解和三角函数解,从而丰富了方程解的内容。这些结论可以应用于其它的非线性方程。此外还纠正了一些文献的部分结论。     

有限变形弹性杆中三种非线性弥散波

在一维弹性细杆拉压、扭转和弯曲波的经典线性理论基础上,分别计入有限变形和弥散效应,借助Hamilton变分原理,由统一的方法导出了3种非线性弥散波的演化方程．对3种演化方程进行了定性分析．结果表明,这些方程在相平面上存在同宿轨道或异宿轨道,分别相应于孤波解或冲击波解．根据齐次平衡原理,用Jacobi椭圆函数展开对这些演化方程进行了求解,在一定的条件下它们均可能存在孤立波解或冲击波解,这与方程的定性分析完全一致．     

The Riccati Equation Method Combined with the Generalized Extended $(G'/G)$-Expansion Method for Solving the Nonlinear KPP Equation

An analytic study of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation is presented in this paper. The Riccati equation method combined with the generalized extended $(G''/G)$-expansion method is an interesting approach to find more general exact solutions of the nonlinear evolution equations in mathematical physics. We obtain the traveling wave solutions involving parameters, which are expressed by the hyperbolic and trigonometric function solutions. When the parameters are taken as special values, the solitary and periodic wave solutions are given. Comparison of our new results in this paper with the well-known results are given.     

Coupling Bäcklund trasnsformation of Riccati equation and division theorem method for traveling wave solutions of some class of NLPDEs

In this paper, a effective method for searching infinite sequence periodic and solitary wave solutions to nonlinear partial differential equations (NLPDEs) is proposed. A simple transformation technique and division theorem are used to reduce some class of NLPDEs to the Riccati equation, and then the infinite sequence periodic and solitary wave solutions of some class of NLPDEs are constructed by using Bäcklund transformation of Riccati equation and nonlinear superposition principle. As illustrative examples, we obtain the infinite sequence travelling-wave solutions of the three special equations, respectively.     

Modified extended tanh-function method for solving nonlinear partial differential equations

Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers–Fisher equation and Drinfeld–Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.     

基于吴方法的孤波自动求解软件包及其应用

基于非线性代数方程组的吴特征列方法,在计算机代数系统Maple上实现了非线性微分方程孤波解的自动求解,编制了一个小型实用的软件包。作为应用,考虑了一个一般的五阶模型方程,利用该软件包获得了此方程新的孤波解以及孤子解存在的条件。