应用指数函数展开法求解非线性发展方程

利用指数函数展开法,研究BBM方程与KG方程,在一个特定的变换下,借助Maple软件的符号运算功能,获得BBM方程与KG方程指数函数型新的孤立波解与周期解.这种方法用于求解非线性发展方程是简单而有效的.     

一类广义BBM方程周期行波解的存在性

本文引入和讨论一类新的广义BBM方程,得到了关于这类广义BBM方程周期行波解的一些存在性定理。     

新型广义BBM方程B(m,n)的孤立波模型解

引进了一类带强色散项的新型广义BBM方程B(m,n):ut+ux+a(um)x+b(un)xx t=0,研究了B(m,n)方程的孤立波模型解,分别得到了它的双曲正弦,双曲余弦,双曲正切形式的孤立波模型解.     

具任意次非线性项的Lienard方程的精确解及其应用

该文推导了具任意次非线性项的Liénard方程a″(ξ)+la(ξ)+ma\+q(ξ)+na\+\{2q-1\}(ξ)=0和\{a″(ξ)\}+ra′(ξ)+la(ξ)+ma\+q(ξ)+na\+\{2q-1\}(ξ)=0解的若干性质，通过适当变换，并结合假设待定法求出了它们的钟状和扭状显式精确解.据此，求出了一批具任意次非线性项的发展方程的钟状和扭状显式精确孤波解，其中包括广义BBM型方程、二维广义Klein Gordon方程、广义Pochhammer Chree方程和非线性波方程等.     

The limit behavior of the solutions to a class of nonlinear dispersive wave equations

In this paper, we study the limit behavior of the solutions to a class of nonlinear dispersive wave equations. We also demonstrate that the solutions to Eq. (1.1) converge to the solution to the corresponding BBM equation as the parameter γ converges to zero. And the convergence of solutions to Eq. (1.1) as α²→0 is studied in Hs(R), . Given the discussion of the parameters in the nonlinear dispersive wave equation (1.1), one will obtain some conditions in which compacton and peakon occur.     

EXACT TRAVELLING WAVE SOLUTIONS TO BBM EQUATION

Abundant new travelling wave solutions to the BBM (Benjamin-Bona-Mahoni) equation are obtained by the generalized Jacobian elliptic function method. This method can be applied to other nonlinear evolution equations.     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.     

Shock wave solution of the variable coefficient nonlinear partial differential equations

In this paper, we consider a variable coefficient Calogero–Degasperis equation, a variable coefficient potential Kadomstev–Petviashvili equation and the generalized (3+1)‐dimensional variable coefficient Kadomtsev–Petviashvili equation with time‐dependent coefficients. Shock wave solutions for the considered models are obtained by using ansatz method in the form of tanhp function. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Copyright © 2016 John Wiley & Sons, Ltd.     

An algebraic method with computerized symbolic computation for the one-dimensional generalized BBM equation of any order

In this paper, an extended algebraic method with symbolic computation is applied to construct a series of travelling wave solutions of the one-dimensional generalized BBM equation of any order with positive and negative exponents. As a result, the proposed method gives many explicit exact solutions such as solitary wave solutions, periodic solutions, solitary patterns solutions and compacton 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.     

Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients

A generalized KdV equation with time-dependent coefficients will be studied. The BBM equation with time-dependent coefficients and linear damping term will also be examined. The wave soliton ansatz will be used to obtain soliton solutions for both equations. The conditions of existence of solitons are presented.     

Attractors for generalized Ginzburg-Landau-BBM equations in an unbounded domain

By the interpolation inequality and a priori estimates in the weighted space, the existence of global solutions for generalized Ginzburg-Landau equation coupled with BBM equation in an unbounded domain is considered, and the existence of the maximal attractor is obtained. This research is supported by the National Natural Science Foundation of China (No.19861004).     

Symmetry reduction and some exact solutions of a nonlinear five-dimensional wave equation

By using decomposable subgroups of the generalized Poincaré group P(1,4), we perform a symmetry reduction of a nonlinear five-dimensional wave equation to differential equations with a smaller number of independent variables. On the basis of solutions of the reduced equations, we construct some classes of exact solutions of the equation under consideration.     

A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order

In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations.     

Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.     

The extended F-expansion method and its application for a class of nonlinear evolution equations

By means of a simple transformation technique, we have shown that the higher-order nonlinear Schrödinger equation in nonlinear optical fibers, a new Hamiltonian amplitude equation, generalized Hirota–Satsuma coupled system and generalized ZK-BBM equation can be reduced to the elliptic-like equation. Then, the extended F-expansion method is used to obtain a series of solutions including the single and the combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear evolution equations.     

On the exact solitary wave solutions of a special class of Benjamin-Bona-Mahony equation

The general form of Benjamin-Bona-Mahony equation (BBM) is $u_t + au_x + bu_{xxt} + (g(u))_x = 0,a,b \in \mathbb{R},$ where ab ≠ 0 and g(u) is a C ₂-smooth nonlinear function, has been proposed by Benjamin et al. in [1] and describes approximately the unidirectional propagation of long wave in certain nonlinear dispersive systems. In this payer, we consider a new class of Benjamin-Bona-Mahony equation (BBM) $u_t + au_x + bu_{xxt} + (pe^u + qe^{ - u} )_x = 0,a,b,p,q \in \mathbb{R},$ where ab ≠ 0, and qp ≠ 0, and we obtain new exact solutions for it by using the well-known (G′/G)-expansion method. New periodic and solitary wave solutions for these nonlinear equation are formally derived.     

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer–Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations.     

The Exact Solutions for the Benjamin-Bona-Mahony Equation

The Benjamin-Bona-Mahony (BBM) equation represents the unidirectional propagation of nonlinear dispersive long waves, which has a clear physical background, and is a more suitable mathematical and physical equation than the KdV equation. Therefore, the research on the BBM equation is very important. In this article, we put forward an effective algorithm, the modified hyperbolic function expanding method, to build the solutions of the BBM equation. We, by utilizing the modified hyperbolic function expanding method, obtain the traveling wave solutions of the BBM equation. When the parameters are taken as special values, the solitary waves are also derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The modified hyperbolic function expanding method is direct, concise, elementary and effective, and can be used for many other nonlinear partial differential equations.     

The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation

The asymptotic behaviors of solutions of an initial-boundary value problem for the generalized BBM equation with non-convex flux are discussed in this paper. It is proved that under the conditions of constant boundary data and small perturbation for the initial data, the global solutions exist and converge time-asymptotically to a stationary wave or the superposition of a stationary wave and a rarefaction wave. The proof is given by a technical L ²-weighted energy method.