Solitary wave solutions and kink wave solutions for a generalized Zakharov-Kuznetsov equation

Bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions in the generalized Zakharov-Kuznetsov equation. Under some parameter conditions, their explicit expressions are obtained.     

Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate traveling wave solutions in the (2 + 1)-dimensional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. Under some parameter conditions, exact solitary wave solutions and kink wave solutions are obtained.     

Exact solitary wave solutions of the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the nonlinear dispersive Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

带色散项的Degasperis-Procesi方程的孤立尖波解

用动力系统的定性分析理论研究了带有色散项的Degasperis-Procesi方程的孤立尖波解.在一定的参数条件下,利用Degasperis-Procesi方程对应行波系统的相图分支从两种不同方式给出了孤立尖波解的表达式.     

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.     

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.     

Exact solitary wave solutions of the generalized (2 + 1) dimensional Boussinesq equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the generalized (2 + 1) dimensional Boussinesq equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

Peaked wave solutions of CH-r equation

In this paper the qualitative analysis methods of planar autonomous systems and numerica simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wave solutions and peaked periodic wave solutions are obtained,and some of their relationships arerevealed.Why peaked points are generated is discussed.     

Bifurcation studies on travelling wave solutions for an integrable nonlinear wave equation

By using the method of planar dynamical systems to an integrable nonlinear wave equation, the existence of periodic travelling wave, solitary wave and kink wave solutions is proved in the different parametric conditions. The phase portraits of the travelling wave system are given. It can be shown that the existence of singular curves in the travelling wave system is the reason why the travelling wave solutions lose their smoothness. Moreover, the so-called W/M-shaped solitary wave solutions are obtained.     

New solitons and kink solutions for the Gardner equation

The Gardner equation, also called combined KdV–mKdV equation, is studied. New hyperbolic ansatze are proposed to derive solitons solutions. The tanh method is used as well to obtain kink solutions.     

Compacton and generalized kink wave solutions of the CH-DP equation

The CH-DP equation is investigated by using the bifurcation method of planar systems and simulation method of differential equations. The bifurcation phase portraits are drawn in different regions of parameter plane. The planar graphs of compactons and generalized kink waves are simulated by using mathematical software Maple. Exact explicit parameter expressions of compactons and implicit expressions of generalized kink wave solutions are given, and the dynamic characters of these solutions are investigated.     

Zhiber-Shabat方程的孤立波解与周期波解

结合齐次平衡法原理并利用F展开法,再次研究了Zhiber-Shabat方程的各种椭圆函数周期解.当椭圆函数的模m分别趋于1或0时,利用这些椭圆函数周期解,得到了Zhiber-Shabat方程的各种孤子解和三角函数周期解,从而丰富了相关文献中关于Zhiber-Shabat波方程的解的类型.     

New exact solutions for a generalized variable-coefficient KdV equation

New exact solutions for a generalized variable-coefficient KdV equation were obtained using the generalized expansion method [R. Sabry, M.A. Zahran, E.G. Fan, Phys. Lett. A 326 (2004) 93]. The obtained solutions include solitary wave solutions besides Jacobi and Weierstrass doubly periodic wave solutions.     

Bifurcations of traveling wave solutions for a generalized Camassa-Holm equation

In this paper, the traveling wave solutions for a generalized Camassa-Holm equation $u_t-u_{xxt}=\frac{1}{2}(p+1)(p+2)u^pu_x-\frac{1}{2}p(p-1)u^{p-2}u_x^3-2pu^{p-1}u_xu_{xx}-u^pu_{xxx}$ are investigated. By using the bifurcation method of dynamical systems, three major results for this equation are highlighted. First, there are one or two singular straight lines in the two-dimensional system under some different conditions. Second, all the bifurcations of the generalized Camassa-Holm equation are given for $p$ either positive or negative integer. Third, we prove that the corresponding traveling wave system of this equation possesses peakon, smooth solitary wave solution, kink and anti-kink wave solution, and periodic wave solutions.     

Exact traveling wave solutions for a modified Camassa-Holm equation

In this paper, the bifurcation method of planar dynamical systems is utilized to investigate a modified Camassa-Holm equation. After dividing the parametric space, some explicit parametric conditions are derived for the existence of traveling wave solutions. Several exact traveling solutions are also obtained.     

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.     

The bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation

By using methods from dynamical systems theory, this paper researches the bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation. Implicit exact parametric representations of all travelling wave solutions as well as some explicit analytic solutions are given. Specially, breaking wave solutions are obtained, which KdV equation does not include.     

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）。     

Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation

By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. We notice that some results in [Wazwaz AM. Analytic study on nonlinear variants of the RLW and the PHI-four equations. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2005.03.001] are incorrect.