Bifurcations of traveling wave solutions for the magma equation

The traveling wave solutions of the magma equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. Under different regions of parametric space, various sufficient conditions to guarantee the existence of solitary wave, periodic wave and breaking wave solutions are given. Moreover, the reason for appearance of breaking waves is explained.     

Approximate solutions for magmon propagation from a reservoir

^** Email: tim_marchant{at}uow.edu.au^*** Email: n.smyth{at}ed.ac.uk A 1D partial differential equation (pde) describing the flowof magma in the Earth's mantle is considered, this equationallowing for compaction and distension of the surrounding matrixdue to the magma. The equation has periodic travelling wavesolutions, one limit of which is a solitary wave, called a magmon.Modulation equations for the magma equation are derived andare found to be either hyperbolic or of mixed hyperbolic/elliptictype, depending on the specific values of the wave number, meanheight and amplitude of the underlying modulated wave. The periodicwave train is stable in the hyperbolic case and unstable inthe mixed case. Solutions of the modulation equations are foundfor an initial-boundary value problem on the semi-infinite line,these solutions representing the influx of magma from a largereservoir. The modulation solutions are found to consist ofa full or partial undular bore. Excellent agreement with numericalsolutions of the governing pde is obtained, except in the limitwhere the wave train becomes a train of magmons. An alternativeapproximation based on the assumption that the wave train isa series of uniform magmons is also derived and is found tobe superior to modulation theory in this limit.     

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.     

Burgers型方程初值问题解的稳定性及孤波解

本文证明Burgers型方程初值问题的解在L₂意义上的稳定性、唯一性;给出该方程存在扭状孤波解的条件及求扭状孤波解的方法。并指出Burgers型方程不存在钟状孤波解。     

Burgers-BBM方程新的精确解

借助两个推广形式的Riccati方程组和Mathematica软件，求出了Burgers-BBM方程，BBM方程，KDV—Burgers方程的大量新的精确解，包括各种形式的孤立波解和三角函数周期解．     

KK方程和改进的Boussinesq方程的新精确解

应用改进的Fan's代数方法,得到了KK方程和改进的Boussinesq方程的一系列新精确解,包括孤立波解、类孤立波解、纽结波解、奇异纽结波解和三角函数周期解.     

二维RLW方程和二维SRLW方程的显式精确解

本文讨论了二维RLW方程和二维SRLW方程孤立波解的性态，通过直接积分的方法求出了这两个方程的显式精确孤立波解，并通过选取初始条件的方法求出了二维RLW方程和二维SRLW方程的另一类精确行波解．     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

几类具5次强非线性项的发展方程的显式精确孤波解

本文道德求出了具５次强非线性项的Ｌｉｅｎａｒｄ方程的二类显式精确解。     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

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.     

Single peak solitary wave solutions for the Fornberg–Whitham equation

The qualitative theory of differential equations is applied to the Fornberg–Whitham equation. Smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation under inhomogeneous boundary condition are obtained. The conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation. The results presented in this article extend and improve the previous results.     

Exact solitary wave solutions for a generalized KdV–Burgers equation

Using the special truncated expansion method, the solitary wave solutions are constructed for the compound Korteweg–de Vries–Burgers (KdVB) equation. Exact and explicit solitary wave solutions for a generalized KdVB equation are obtained by introducing a suitable ansatz equation. The generalized two-dimensional KdVB equation is discussed. Some particular cases of the generalized KdVB equation are solved by using these methods.     

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.     

Evolution of Higher-Order Gray Hirota Solitary Waves

The defocusing Hirota equation has dark and gray soliton solutions which are stable on a background of periodic waves of constant amplitude. In this paper, gray solitary wave evolution for a higher-order defocusing Hirota equation is examined. A direct analysis is used to identify families of higher-order gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to determine the detailed behavior of an evolving higher-order gray Hirota solitary wave. An integral expression for the first-order correction to the wave is found and analytical expressions for the steady-state and transient components of the solitary wave tail are derived. A subtle and complex picture of the development of solitary wave tails emerges. It is found that solitary wave tails develop for two reasons, one is decay of the solitary wave caused by resonance, the second is corrections at first-order to the background wave. Strong agreement is found between the theoretical predictions of the perturbation theory and numerical solutions of the governing equations.     

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.     

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.     

Single peak solitary wave solutions for the generalized Camassa–Holm equation

This Letter presents all possible smooth, peaked and cusped solitary wave solutions for the generalized Camassa–Holm equation under the inhomogeneous boundary condition.The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the generalized Camassa–Holm equation.     

Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation

The bounded traveling wave solutions of a generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.     

NEW EXACT TRAVELLING WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS USING THE DYNAMICAL SYSTEM METHOD

By the method of dynamical system,we construct the exact travelling wave solutions of a new Hamiltonian amplitude equation and the Ostrovsky equation.Based on this method,the new exact travelling wave solutions of the new Hamiltonian amplitude equation and the Ostrovsky equation,such as solitary wave solutions,kink and anti-kink wave solutions and periodic travelling wave solutions,are obtained,respectively.