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In this paper, we study the influences of dually environmental noises on the traveling wave which develops from the deterministic KPP equation. We prove that if the strengths of noises satisfy some condition, the solution of the stochastic KPP equation with Heaviside initial condition develops a random traveling wave, whose wave speed is deterministic and depends on the strengths of noises. If the strengths of noises satisfy some other conditions, the solution tends to zero as time tends to infinity. Therefore, there exist bifurcations of asymptotic behaviors of solution induced by the strengths of dual noises. [ABSTRACT FROM AUTHOR]

We use the bifurcation method of dynamical systems to study the (2+1)-dimensional Broer-Kau-Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow-up wave solutions, periodic smooth wave solutions, periodic blow-up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow-up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the ZK–BBM equation. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions, unbounded solutions and blow-up wave solutions. [Copyright &y& Elsevier]

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey-Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears. [ABSTRACT FROM AUTHOR]

In this paper, we study the explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation $$ u_t+2ku_x-u_{xxt}+3u^2u_x=2u_xu_{xx}+uu_{xxx}. $$ Not only are the precise expressions of the explicit nonlinear wave solutions obtained, but some interesting bifurcation phenomena are revealed. Firstly, it is verified that k = 3/8 is a bifurcation parametric value for several types of explicit nonlinear wave solutions. When k < 3/8, there are five types of explicit nonlinear wave solutions, which are (i) hyperbolic peakon wave solution, (ii) fractional peakon wave solution, (iii) fractional singular wave solution, (iv) hyperbolic singular wave solution, (v) hyperbolic smooth solitary wave solution. When k = 3/8, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution. When k > 3/8, there is not any type of explicit nonlinear wave solutions. Secondly, it is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately. Thirdly, it is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the generalization of the Camassa–Holm equation where is a positive constant and . The bifurcation and some explicit expressions of peakons and periodic cusp wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. Further, in the process of obtaining the bifurcation of phase portraits, we show that is the peakon bifurcation parameter value for the equation. From the bifurcation theory, in general, the peakons can be obtained by taking the limit of the corresponding periodic cusp waves. However, we find that in the cases of , when tends to the corresponding bifurcation parameter value, the periodic cusp waves will no longer converge to the peakons, instead, they will still be the periodic cusp waves. To the best of our knowledge, up until now, this phenomenon has not appeared in any other literature. By further studying the cause of this phenomenon, we show that this planar system has some different characters from the previous Camassa–Holm systems. What is more, we obtain some periodic cusp wave solutions in the form of polynomial functions, which are different from those in the form of exponential functions. Some previous results are extended. [Copyright &y& Elsevier]

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation ut + a(l + bu)uux + uxxx = 0, and PC-like equation vtt-vttxx - (aiv + a2v² + a3v³) xx = 0, respectively. Via some special phase orbits, we obtain some new explicit periodic wave solutions which are called trigonometric function periodic wave solutions because they are expressed in terms of trigonometric functions. We also show that the trigonometric function periodic wave solutions can be obtained from the limits of elliptic function periodic wave solutions. It is very interesting that the two equations have similar periodic wave solutions. Our work extend previous some results. [ABSTRACT FROM AUTHOR]

In this paper, through qualitative analysis and integration, we study the explicit periodic wave solutions and their bifurcations for the generalized Camassa-Holm equation \begin{eqnarray*} u_t+2ku_x-u_{xxt}+3u^2u_x=2u_xu_{xx}+uu_{xxx}. \end{eqnarray*} When the parameter k satisfies k < 3/8 and the constant wave speed c satisfies $({1}/{2})(3-\sqrt{3(3-8k)})\, < c \, < ({1}/{2})(3+\sqrt{3(3-8k)})$, we obtain two types of explicit periodic wave solutions, elliptic smooth periodic wave solution and elliptic periodic blow-up solutions. These solutions include a bifurcation parameter α which has four bifurcation values αi (i = 1, 2, 3, 4). When α tends to the bifurcation values, the elliptic periodic wave solutions become three types of other solutions, the hyperbolic smooth solitary wave solution, the hyperbolic blow-up solution and the trigonometric periodic blow-up solution. Especially, a new bifurcation phenomenon is found, that is, the periodic blow-up solution can become a smooth solitary wave solution when α varies. When k > 3/8, we guess that there is no other explicit solution except the explicit periodic blow-up solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE)where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended. [Copyright &y& Elsevier]

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the higher-order KdV equation ut+aunux+uxxx=0, where n&ges;1 and a∈R. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, explicit solitary wave solutions are obtained. Specially, some new solitary wave solutions are found for KdV or MKdV equation. [Copyright &y& Elsevier]

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined. [ABSTRACT FROM AUTHOR]

From the mathematical point of view, we use the bifurcation method of phase plane to study the CH equation given by Camassa and Holm [Phys. Rev. Lett. 71(11) (1993) 1661]. In case of k=0, they showed that it has solitary waves of the form ce−|x−ct|, which were called “peakons”. When k≠0, we obtain the peakons of the form (3k/2)exp(−x−(kt/2))−k in three different ways, which are also applicable to the case of k=0. Note that our approaches can also be used to study the traveling waves in various nonlinear integrable equations. [Copyright &y& Elsevier]

Bifurcations of solitary waves and domain wall waves for a KdV-like equation with higher order nonlinearity are studied, by using bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown. Numbers of solitary waves and domain wall waves are given. Under some parameter conditions, a lot of explicit formulas of solitary wave solutions and domain wall solutions are obtained. [ABSTRACT FROM AUTHOR]

Camassa and Holm [1993] recently derived a new dispersive shallow water equation known as the Camassa-Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called "peakons". In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa-Holm equation u[sub t] + 2ku[sub x] - u[sub xxt] + au[sup m]u[sub x] ... 2u[sub x]u[sub xx] + uu[sub xxx] with a > 0, k ∈ R, m ∈ N N and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For m = 1, 2, 3, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations. [ABSTRACT FROM AUTHOR]