Periodic and quasi-periodic solutions for the complex Swift-Hohenberg equation

In this paper, we consider the complex Swift-Hohenberg(CSH) equation $\frac{\partial u}{\partial t}=\lambda u-(\alpha+\mathrm{i}\beta)\l(1+\frac{\partial^2}{\partial x^2}\r)^2u-(\sigma+\mathrm{i}\rho)|u|^2u $ subject to periodic boundary conditions. Using an infinite dimensional KAM theorem, we prove that there exist a continuous branch of periodic solutions and a Cantorian branch of quasi-periodic solutions for the above equation.     

Sub-ODE method and soliton solutions for the variable-coefficient mKdV equation

In this work, an auxiliary equation is used for an analytic study on the time-variable coefficient modified Korteweg-de Vries (mKdV) equation. Five sets of new exact soliton-like solutions are obtained. The results show that the pulse parameters are time-dependent variable coefficients. Moreover, the basic conditions for the formation of derived solutions are presented.     

一类时滞Duffing微分方程同宿解的存在性

利用重合度理论和一些分析技巧,得到一类二阶时滞Duffing微分方程的2kT周期解,通过对该微分方程的一系列周期解取极限获得同宿解的存在性.同时,β(t)是可变号的.     

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.     

组合Zakharov-Kuznetsov方程的显式孤波解

借助于Ｍａｔｈｅｍａｔｉｃａ是吴消元法,本文通过用一个新的假设,获得了组合Ｚａ－ｋｈａｒｏｖ－Ｋｕｚｎｅｔｓｏｖ方程的１２种孤波解,其中包括钟状与扭状组合型孤波解和周期型孤波解。这种假设也能用于其他的非线性演化方程（组）。     

Explicit peaked wave solutions to the generalized Camassa-Holm equation

By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.     

Exact solutions and dynamics of the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity

This paper investigate the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity by using the method of dynamical systems. The functions $q(x,t)=\phi(\xi)\exp(i(-kx+\omega t))$ are solutions of the equation (1.1) that governs the propagation of Raman solitons through optical metamaterials, where $\xi=x-vt$ and $\phi(\xi)$ in the solutions satisfy a singular planar dynamical system (1.5) which has two singular straight lines. By using the bifurcation theory method of dynamical systems to the equation of $\phi(\xi)$, bifurcations of phase portraits for this dynamical system are obtained under 28 different parameter conditions. Based on those phase portraits, 62 exact solutions of system (1.5) including periodic solutions, heteroclinic and homoclinic solutions, periodic peakons and peakons as well as compacton solutions are derived.     

Multiple periodic solutions for system of first-order differential equation

Sufficient conditions have been obtained for the existence of at least two non-negative periodic solutions to a system of first-order nonlinear functional differential equations. Applications to some ecological models are given.     

Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity

In this paper, we consider the nonlinear Schr\"{o}dinger equation with an anti-cubic nonlinearity. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system under different parameter conditions. Corresponding to different level curves defined by the Hamiltonian, we derive all exact explicit parametric representations of the bounded solutions (including periodic peakon solutions, periodic solutions, homoclinic solutions, heteroclinic solutions and compacton solutions).     

Explicit solutions of the Bogoyavlensky-Konoplechenko equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the Bogoyavlensky-Konoplechenko equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the BK equation. Then we get the reductions using the symmetry and give some exact solutions of the BK equation.     

Explicit rational solutions of Knizhnik-Zamolodchikov equation

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions generated by elements of the symmetric group _n . We assume that parameter ρ = ±1. In previous paper [5] we proved that the fundamental solution of the corresponding KZ-equation is rational. Now we construct this solution in the explicit form.      

On the -dimensional Gardner equation: Determinant solutions and pfaffianization

In this paper, we first present the Casorati and Grammian determinant solutions to the (2+1)-dimensional Gardner equation. Then, by using the pfaffianization procedure of Hirota and Ohta, a new integrable coupled system is generated from the Gardner equation. Moreover, Gramm-type pfaffian solutions to the pfaffianized system are proposed.     

Planar bifurcation method of dynamical system for investigating different kinds of bounded travelling wave solutions of a generalized Camassa-Holm equation

In this study, by using planar bifurcation method of dynamical system, we study a generalized Camassa-Holm (gCH) equation. As results, under different parameter conditions, many bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given. The dynamic properties of these exact solutions are investigated.     

On the solutions of a max‐type difference equation system

In this paper, we study behavior of the solution of the following max‐type difference equation system: where , the parameter A is positive real number, and the initial values x₀,y₀ are positive real numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Bäcklund transformations and soliton solutions for the KdV6 equation

Using Hirota technique, a Bäcklund transformation in bilinear form is obtained for the KdV6 equation. Furthermore, we present a modified Bäcklund transformation by a dependent variable transformation, it is shown that a new representation of N-soliton solution and some novel solutions to the KdV6 equation are derived by performing an appropriate limiting procedure on the known soliton solutions.     

mKdV-SineGordon方程的多孤子解

该文首先给出了mKdV-SineGordon方程的双线性形式和双线性Backlund变换,然后利用Hirota方法、Backlund变换方法和Wronskian技巧三种不同的方法分别得到mKdV-SineGordon方程的孤子解,最后验证了这三种解的一致性。     

Exact traveling wave solutions and bifurcations for the Dullin-Gottwald-Holm equation

Utilizing the methods of dynamical system theory, the Dullin-Gottwald-Holm equation is studied in this paper. The dynamical behaviors of the traveling wave solutions and their bifurcations are presented in different parameter regions. Furthermore, the exact explicit forms of all possible bounded solutions, such as solitary wave solutions, periodic wave solutions and breaking loop wave solutions are obtained.     

New exact periodic wave solutions for the Dullin-Gottwald-Holm equation

In this paper, the Dullin-Gottwald-Holm equation is studied using semi-inverse method and integral bifurcation method. New periodic waves such as peakon-like periodic wave, compacton-like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion. The exact parametric representations of these waves are also presented.     

On the reflecting function and the qualitative behavior of solutions of some non-autonomous differential equations

In this article, we use the Mironenko''s method to discuss the qualitative behavior of some non-autonomous differential equations. We study the structure of the reflecting functions of the simplest differential equations, and obtain some sufficient conditions under which these equations have the rational reflecting functions. We apply the obtained results to discuss the numbers of periodic solutions of the non-autonomous differential systems and derive some sufficient conditions for a critical point of theirs to be a center.     

Bifurcations and exact travelling wave solutions of M-N-Wang equation

By using the method of dynamical systems to Mikhailov-Novikov-Wang Equation, through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system of the derivative $\phi(\xi)$ of the wave function $\psi(\xi)$. Under different parameter conditions, for $\phi(\xi)$, exact explicit solitary wave solutions, periodic peakon and anti-peakon solutions are obtained. By integrating known $\phi(\xi)$, nine exact explicit traveling wave solutions of $\psi(\xi)$ are given.