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.     

mKdV方程和mKP方程组的新的精确孤立波解

用三角函数假设法和一种新辅助方程的解构造mK dV方程和mKP方程组的精确孤立波解.这种方法也可用于寻找其它非线性发展方程的新的孤立波解.     

Jacobian elliptic function method for nonlinear differential-difference equations

An algorithm is devised to derive exact travelling wave solutions of differential-difference equations by means of Jacobian elliptic function. For illustration, we apply this method to solve the discrete nonlinear Schrödinger equation, the discretized mKdV lattice equation and the Hybrid lattice equation. Some explicit and exact travelling wave solutions such as Jacobian doubly periodic solutions, kink-type solitary wave solutions are constructed.     

Sub-manifold and traveling wave solutions of Ito's 5th-order mKdV equation

In this paper, we study Ito''s 5th-order mKdV equation with the aid of symbolic computation system and by qualitative analysis of planar dynamical systems. We show that the corresponding higher-order ordinary differential equation of Ito''s 5th-order mKdV equation, for some particular values of the parameter, possesses some sub-manifolds defined by planar dynamical systems. Some solitary wave solutions, kink and periodic wave solutions of the Ito''s 5th-order mKdV equation for these particular values of the parameter are obtained by studying the bifurcation and solutions of the corresponding planar dynamical systems.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

Auxiliary equation method for the mKdV equation with variable coefficients

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients. As a result, new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation.     

广义组合KdV-mKdV方程的显式精确解

Abstract. With the aid of Mathematica and Wu-elimination method,via using a new generalizedansatz and well-known Riccati equation,thirty-two families of explicit and exact solutions forthe generalized combined KdV and mKdV equation are obtained,which contain new solitarywave solutions and periodic wave solutions. This approach can also be applied to other nonlinearevolution equations.     

Solitary wave solutions to the generalized coupled mKdV equation with multi-component

Using the Darboux matrix method, the multi-solitary wave solutions of the generalized coupled mKdV equation with multi-component are obtained. The obtained solution formulas provide us with a comprehensive approach to construct exact solutions for the generalized coupled mKdV equation by some basic solutions of the Boiti and Tu spectral problem.     

Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics

We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a new class of liner connections which are ‘N-adapted’, metric compatible and uniquely defined by the metric structure. We prove that for such a linear connection, one yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in solitonic hierarchies described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrödinger map equation. All geometric constructions can be re-defined for the Levi-Civita connection but with “noholonomic mixing” of solitonic interactions. Finally, we speculate why certain methods and results from the geometry of nonholonmic manifolds and solitonic equations have general importance in various directions of modern mathematics, geometric mechanics, fundamental theories in physics and applications, and briefly analyze possible nonlinear wave configurations for modeling gravitational interactions by effective continuous media effects.     

有限变形弹性圆杆中的孤波

在同时引入横向惯性和横向剪切应变的情况下,导出了有限变形弹性圆杆的非线性纵向波动方程,方程中包含了二次和三次的非线性项以及由横向剪切与横向惯性导致的两种几何弥散效应．借助Mathematica软件,利用双曲正割函数的有限展开法,对该方程和对应的截断的非线性方程进行求解,得到了非线性波动方程的孤波解,同时给出了这些解存在的必要条件．     

NEW PERIODIC SOLUTIONS OF ITO＇S 5th-ORDER mKdV EQUATION AND ITO＇S 7th-ORDER mKdV EQUATION

Based on the modified Jocobi elliptic function expansion method and the modified extended tanh-function method, a new algebraic method is presented to obtain multiple travelling wave solutions for nonlinear wave equations. By using the method ,Ito‘s 5th-order and 7th-order mKdV equations are studied in detail and more new exact Jocobi elliptic function periodic solutions are found. With modulus m→1 or m→0, these solutions degenerate into corresponding solitary wave solutions, shock wave solutions and trigonometric function solutions.     

All meromorphic solutions of an ordinary differential equation and its applications

Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions w_r,2(z) and simply periodic solutions w_s,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Constructing exact solutions to discrete systems with the trial function method

Based on the homogenous balance method and the trial function method, several trial function methods composed of exponential functions are proposed and applied to nonlinear discrete systems. With the.help of symbolic computation system, the new exact solitary wave solutions to discrete nonlinear mKdV lattice equation, discrete nonlinear （2 ＋ 1） dimensional Toda lattice equation, Ablowitz-Ladik-lattice system are constructed.The method is of significance to seek exact solitary wave solutions to other nonlinear discrete systems.     

Symmetry group analysis and similarity solutions of variant nonlinear long-wave equations

Symmetry group properties and similarity solutions of the variant nonlinear long-wave equations in the form of system of nonlinear partial differential equations are analyzed. Lie symmetry group analysis of the variant nonlinear long-wave equations presents that the system has only two-parameter point symmetry group that corresponds to only traveling wave solutions. The symmetry groups yield the general reduced similarity form of the system, which is in the system of nonlinear ordinary differential equations. By using the improved tanh method the similarity solutions are obtained from the reduced system of equations. In addition, some graphical representations of the solitary and periodic solutions are presented.     

新的双曲函数有理展开法和符号计算构造差分mkdv的孤波解

本文通过利用一个广泛的题设提出一种推广的双曲函数展开法, 并利用此方法求解了 离散的 mKdV 方程, 获得了丰富的显式精确解. 此方法可以用于求其他非线性系统的精确解.     

The multi-component second modified Korteweg–de Vries equation and its two distinct integrable coupling types

A new isospectral problem is designed and the multi-component second mKdV equation is worked out from it. It follows that two distinct types of integrable couplings of the multi-component second mKdV equation are obtained by constructing two types of new loop algebras. As its reduction, two distinct types of integrable couplings of the multi-component KdV equation, the multi-component mKdV equation and the multi-component KdV–mKdV equation are presented.     

On Lie symmetry analysis,conservation laws and solitary waves to a longitudinal wave motion equation

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.     

mKdV方程的对称与群不变解

主要考虑mKdV方程的一些简单对称及其构成的李代数,并利用对称约化的方法将mKdV方程化为常微分方程,从而得到该方程的群不变解,这是对该方程群不变解的进一步扩展.     

New sets of solitary wave solutions to the KdV,mKdV, and the generalized KdV equations

In this work we introduce new schemes, each combines two hyperbolic functions, to study the KdV, mKdV, and the generalized KdV equations. It is shown that this class of equations gives conventional solitons and periodic solutions. We also show that the proposed schemes develop sets of entirely new solitary wave solutions in addition to the traditional solutions. The analysis can be used to a wide class of nonlinear evolutions equations.     

A fifth order semidiscrete mKdV equation

In this paper,aiming to get more insight on the relation between the higher order semidiscrete mKdV equations and higher order mKdV equations,we construct a fifth order semidiscrete mKdV equation from the three known semidiscrete mKdV fluxes.We not only give its Lax pairs,Darboux transformation,explicit solutions and infinitely many conservation laws,but also show that their continuous limits yield the corresponding results for the fifth order mKdV equation.We thus conclude that the fifth order discrete mKdV equation is extremely an useful discrete scheme for the fifth order mKdV equation.