Exact solutions to the generalized lienard equation and its applications

Some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to solve nonlinear wave equations with nonlinear terms of any order directly. The generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz (A) equation and the generalized Gerdjikov-Ivanov (GI) equation are investigated and abundant new exact travelling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions.      

New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.     

修正的F展开法和推广的KdV方程新的孤波解和精确解

对F展开法进行了修正,附加含任意常数的负指数项,解决了精确解的奇点问题.用此方法讨论推广的KdV方程的周期波解.在特别情形下,获得了新的Jacobi椭圆函数精确解、孤波解和三角函数解. 关键词： 齐次平衡原则 F展开法 推广的KdV方程 孤波解     

Exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation

The extended mapping method with symbolic computation is developed to obtain exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation. Limit cases are studied and new solitary wave solutions and triangular periodic wave solutions are obtained. The method is applicable to a large variety of non-linear partial differential equations, as long as odd-and even-order derivative terms do not coexist in the equation under consideration.     

Instability of oscillatory shock profile solutions of the generalized Burgers-KdV equation

Oscillatory shock profile solutions of the Burgers-KdV equation are studied in the limit as the viscosity → 0. A rigorous proof of their instability is obtained by showing that the linearized operator about such a solution has many unstable eigenvalues for sufficiently small . The result is obtained by applying a topological method introduced by Alexander, Gardner and Jones to extract spectral information about the perturbed wave with slightly positive viscosity from particular “pieces” of the underlying wave which are approximated by a solitary wave solution of the reduced equation with no viscous dissipation.     

两组新的广义的Ito方程组的多种行波解

利用推广的形变映射法，得到两组广义的Ito方程组的丰富的行波解，包括孤子解，三角函数解，椭圆函数解，幂函数解. 关键词： 广义的Ito方程 非线性Klein-Gordon方程 行波解     

Interaction phenomena between lump and solitary wave of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

In this paper, a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation is studied in liquid with gas bubbles. Based on the Hirota's bilinear form and symbolic computation, lump and interaction solutions between lump and solitary wave are obtained,which include a periodic-shape lump solution, a parabolic-shape lump solution, a cubic-shape lump solution, interaction solutions between lump and one solitary wave, and between lump and two solitary waves. The spatial structures called the bright lump wave and the bright-dark lump wave are discussed. Interaction behaviors of two bright-dark lump waves and a periodic-shape bright lump wave are also presented. Their interactions are shown in some 3D plots.     

广义Boussinesq方程的同伦映射近似解

研究了一个广义非线性Boussinesq方程. 利用同伦映射方法,首先构造了相应的同伦变换;其次选取了适当的初始近似;然后用同伦映射方法得到了孤立子波近似解;最后得到了微扰渐近表示式. 关键词： Boussinesq方程 非线性 孤立子 近似方法     

Novel solitary and shock wave solutions for the generalized variable-coefficients (2+1)-dimensional KP-Burger equation arising in dusty plasma

The 2-D generalized variable-coefficient Kadomtsev-Petviashvili-Burgers equation representing many types of acoustic waves in cosmic and/or laboratory dusty plasmas is reduced by the modified classical direct similarity reduction method to nonlinear ordinary differential equation of fourth-order. Using the extended Riccati equation mapping method for solving the reduced equation, many new shock wave, solitary wave and periodic wave solutions are obtained with some constraints between the variable coefficients. Finally, some physical interpretations for the obtained solutions as, bright and dark solitons, periodic solitary wave, and shock wave in dust plasma and quantum plasma are achieved.     

用推广的李群约化法求解非线性薛定谔方程

用推广的经典李群约化法，得到了色散系数、非线性系数、补偿（或损失）系数为时、空变量函数时的非线性薛定谔方程的精确解.深入研究了非线性薛定谔模型的一般孤波解与线性调频孤波解在光纤通讯与光纤放大器中的潜在应用. 关键词： 李群约化 非线性薛定谔方程 光纤通讯     

一类广义非线性扰动色散方程孤立波的近似解

采用了一个简单而有效的技巧,研究了一类非线性扰动色散方程.首先引入求解相应典型方程的孤立波解.然后利用同伦映射方法得到了原非线性扰动色散方程奇异孤立波的近似解.     

Solitary wave solution to a singularly perturbed generalized Gardner equation with nonlinear terms of any order

This paper is concerned with the existence of travelling wave solutions to a singularly perturbed generalized Gardner equation with nonlinear terms of any order. By using geometric singular perturbation theory and based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of solitary wave solutions of this equation is proved when the perturbation parameter is sufficiently small. The numerical simulations verify our theoretical analysis.     

广义(3+1)维非线性Burgers系统孤波级数解

研究了一类广义(3+1)维非线性Burgers系统.首先,利用同伦映射方法构造了相应的映射关系式.其次,利用迭代方法得到了扰动系统的一个孤波非行波的级数解.     

Homotopic mapping solutions for generalized method of solitary wave complex Burgers equation

A class of generalized complex Burgers equation is considered. First, a set of equations of the complex value functions are solved by using the homotopic mapping method. The approximate solution for the original generalized complex Burgers equation is obtained. This method can find the approximation of arbitrary order of precision simply and reliably.     

The solitary waves,quasi-periodic waves and integrability of a generalized fifth-order Korteweg-de Vries equation

Under investigation in this paper is a fifth-order Korteweg-de Vries (fKdV) equation, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the binary Bell polynomials, a lucid and systematic approach is proposed to systematically study its bilinear representation, bilinear Bäcklund transformations and Lax pairs with explicit formulas, respectively. These results can be reduced to the ones of several integrable equations such as Sawada-Kotera equation, Caudrey-Dodd-Gibbon equation, Lax equation, Kaup-Kuperschmidt equation and Ito equation, etc. Furthermore, the N-solitary wave solutions formula and quasi-periodic wave solutions are obtained by using bilinear form of the fKdV equation. Finally, the relation between the periodic wave solution and solitary wave solution is rigorously established.     

Bessel solitary waves in strongly nonlocal nonlinear media with distributed parameters

Bessel solitary wave solutions to a two-dimensional strongly nonlocal nonlinear Schrödinger equation with distributed coefficients are obtained. Bessel solitary wave solutions have unique characteristics compared with Gaussian solitary wave solutions, Laguerre-Gaussian solitary wave solutions, and Hermite-Gaussian solitary wave solutions. The generalized two-dimensional nonlocal nonlinear Schrödinger equation with distributed coefficients is investigated for the first time to our knowledge.     

Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method

In this Letter, the Fan sub-equation method is used to construct exact solutions of a generalized Hirota-Satsuma coupled KdV equation. Many exact traveling wave solutions are successfully obtained, which contain more general solitary wave solutions and Jacobian elliptic function solutions with double periods. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Travelling solitary wave solutions for the generalized Burgers--Huxley equation with nonlinear terms of any order

In this paper, the travelling wave solutions for the generalized Burgers--Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.     

Exact solitary wave solutions to a class of generalized odd-order KdV equations

Using a proper ansatz, we have obtained a series of exact solitary wave solutions to a class of generalized odd-order KdV equations.     

1-soliton solution of the generalized Radhakrishnan, Kundu, Lakshmanan equation

This Letter carries out the integration of the generalized Radhakrishnan, Kundu, Lakshmanan equation to obtain the 1-soliton solution. The solitary wave ansatz is used to carry out the integration, to obtain an exact solution of this equation.