Travelling wave solutions of the Fornberg-Whitham equation

Zhou and Tian [J.B. Zhou, L.X. Tian, A type of bounded travelling wave solutions for the Fornberg-Whitham equation, J. Math. Anal. Appl. 346 (2008) 255-261] successfully found a type of bounded travelling wave solutions of the Fornberg-Whitham equation. In this paper, we improve the previous result by using the phase portrait analytical technology. Moreover, some smooth periodic wave, smooth solitary wave, periodic cusp wave and loop-soliton solutions are given, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.     

广义特殊Tzitzeica-Dodd-Bullough类型方程的行波解

本文研究了广义特殊Tzitzeica-Dodd-Bullough类型方程,利用动力系统分支理论方法,证明该方程存在周期行波解,无界行波解和破切波解,并求出了一些用参数表示的显示精确行波解.     

Travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation

The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation ut-uxxt + (1 + b)umux = buxuxx + uuxxx are considered where b > 1 and m are positive integers. The qualitative analysis methods of planar autonomous systems yield its phase portraits. Its soliton wave solutions, kink or antikink wave solutions, peakon wave solutions, compacton wave solutions, periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also give...     

Explicit and implicit solutions of a generalized Camassa-Holm Kadomtsev-Petviashvili equation

In this paper, a generalized Camassa-Holm Kadomtsev-Petviashvili (gCH-KP) equation is studied. As a result, under different parameter conditions, the bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given, and the dynamic characters of these solutions are investigated.     

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.     

Exact loop solutions,cusp solutions,solitary wave solutions and periodic wave solutions for the special CH–DP equation

By using the method of dynamical systems, the travelling wave solutions of a special CH–DP equation are studied. Exact explicit parametric representations of smooth solitary waves, solitary cusp waves, breaking waves and uncountably infinitely many smooth periodic wave solutions are given. In different regions of the parametric plane, different phase portraits are determined. The so called loop soliton solution is discussed.     

Periodic and solitary travelling-wave solutions of a CH–DP equation

By using the dynamical system theory and the integral bifurcation method, a modified Camassa–Holm and Degasperis–Procession (CH–DP) equation are studied. The bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given, and the dynamic characters of these solutions are investigated.     

The Bäcklund transformations and abundant explicit exact solutions for the AKNS–SWW equation

The Bäcklund transformations and abundant explicit exact solutions to the AKNS shallow water wave equation are obtained by combining the extended homogeneous balance method with the extended hyperbolic function method. The solutions obtained admit of multiple arbitrary parameters. These solutions include (a) a compound of the rational fractional function and a linear function, (b) a compound of solitary wave solution and a linear function, (c) a compound of the singular travelling wave solutions and a linear function, and (d) a compound of the periodic wave solutions of triangle function and a linear function. In special cases, we can obtain a series of soliton solutions, singular travelling wave solutions, periodic travelling wave solutions, and rational fractional function solution. In addition to re-deriving some known solutions in a systematic way, some brand-new exact solutions are also established.     

小展弦比波的Green-Naghdi渐进模型的行波解

本文研究了小展弦比波的Green-Naghdi渐进模型. 利用平面自治系统的稳定性分析方法, 在不同的参数条件下, 讨论了它的行波系统的分岔并且给出了对应的相图, 得到了光滑周期波解, 广义扭波解, 广义反扭波解, 广义紧波解, 周期尖波解, 孤波解和孤立尖波解的精确表达式. 进一步, 通过数学软件Maple模拟了这些解.     

Bifurcations of travelling wave solutions for the generalized KP-BBM equation

By using the bifurcation theory of dynamical systems to the generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation, the existence of solitary wave solutions, compactons solution, non-smooth periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Bifurcations of travelling wave solutions for the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation

By using the bifurcation theory of dynamical systems, we study the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation, the existence of solitary wave solutions, compacton solutions, periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

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.     

The bifurcation and exact peakons,solitary and periodic wave solutions for the Kudryashov–Sinelshchikov equation

In this paper, the Kudryashov–Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, the existence of peakon, solitary wave, smooth and non-smooth periodic waves is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some new exact travelling wave solutions are presented through some special phase orbits.     

Periodic travelling waves in an artificial neural network

In a study (Szekely, Acta Physiol. Hung. 27 (1965), pp. 285–289) of the locomotion of salamanders, it is observed that a ‘doubly periodic travelling wave solution’ of a logical neural network can be used to explain a dynamic pattern of movements. We show here that a relatively simple (nonlogical) artificial neural network can also be built and necessary and sufficient conditions for the existence of doubly periodic travelling wave solutions can be found. It is hoped that our investigation will set some foundation in the future design of other artificial neural networks that also allow periodic travelling wave solutions.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.     

Bifurcation of travelling wave solutions for （2＋1）-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.     

Bifurcations of exact travelling wave solutions for the generalized R-K-L equation

Using the method of dynamical systems for the the generalized Radhakrishnan, Kundu, Lakshmanan equation, the existence of soliton solutions, uncountably infinite many periodic wave solutions and unbounded wave solution are obtained. Exact explicit parametric representations of the above travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS TO A COUPLED NONLINEAR WAVE SYSTEM

Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations, the existence of solitary wave solutions, kink wave solutions, anti-kink wave solutions and periodic wave solutions is obtained. Under different parametric conditions, various suffcient conditions to guarantee the existence of the above so-lutions are given. Some exact explicit parametric representations of travelling wave solutions are derived.     

Smooth and non-smooth traveling wave solutions of the Fornberg-Whitham equation with linear dispersion term

In this paper, the Fornberg-Whitham equation with linear dispersion term is investigated by employing the bifurcation method of dynamical systems. As a result, the existence of smooth and non-smooth traveling wave solutions is obtained. And the analytic expressions of solitary wave solutions, periodic cusp wave solutions and peakons are given under some parameter conditions.     

Peakon, pseudo-peakon, loop, and periodic cusp wave solutions of a three-dimensional 3DKP(2, 2) equation with nonlinear dispersion

In this paper, we study the three-dimensional Kadomtsev-Petviashvili equation (3DKP(m, n)) with nonlinear dispersion for m=n=2. By using the bifurcation theory of dynamical systems, we study the dynamical behavior and obtain peakon, pseudo-peakon, loop and periodic cusp wave solutions of the three-dimensional 3DKP(2, 2) equation. The parameter expressions of peakon, pseudo-peakon, loop and periodic cusp wave solutions are obtained and numerical graph are provided for those peakon, pseudo-peakon, loop and periodic cusp wave solutions.