Dynamics and bifurcations of travelling wave solutions of R(<Emphasis Type="Italic">m,n</Emphasis>) equations

By using the bifurcation theory and methods of planar dynamical systems to R(m, n) equations, the dynamical behavior of different physical structures like smooth and non-smooth solitary wave, kink wave, smooth and non-smooth periodic wave, and breaking wave is obtained. The qualitative change in the physical structures of these waves is shown to depend on the systemic parameters. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above waves are given. Moreover, some explicit exact parametric representations of travelling wave solutions are listed.     

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 modified nonlinear dispersive phi-four equation

By using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi-four equation, we analysis all bifurcations and phase portraits in the parametric space, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some explicit exact solution formulas are acquired for some special cases.     

等离子声波方程的行波解的动力学行为

对等离子声波方程, 用平面动力系统理论得到了其光滑、非光滑孤立波解和不可数无穷多光滑、非光滑周期波解的存在性．进一步,在给定的参数条件下,得到了保证上述解存在的充分条件．     

Bifurcations of travelling wave solutions for a two-component camassa-holm equation

By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, smooth and non-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 travelling wave solutions are listed.     

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.     

一类广义四阶非线性Camassa-Holm方程的行波解

用动力系统的分支理论研究了一类广义四阶非线性Camassa-Holm方程的动力学行为和行波解,发现方程存在一些孤立波解,周期波解和一些诸如Compacton类型的非光滑行波解.在不同的参数条件下,给出了这些解存在的条件和一些特殊条件下的精确解.     

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.     

Characterizing defect n-extendable graphs and -critical graphs

A near perfect matching is a matching saturating all but one vertex in a graph. Let G be a connected graph. If any n independent edges in G are contained in a near perfect matching where n is a positive integer and n(|V(G)|-2)/2, then G is said to be defect n-extendable. If deleting any k vertices in G where k|V(G)|-2, the remaining graph has a perfect matching, then G is a k-critical graph. This paper first shows that the connectivity of defect n-extendable graphs can be any integer. Then the characterizations of defect n-extendable graphs and (2k+1)-critical graphs using M-alternating paths are presented.     

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.     

Consecutive customer losses in oscillating GIX/M//n systems with state dependent services rates

We address the probability that k or more Consecutive Customer Losses take place during a busy period of a queue, the so-called k-CCL probability, for oscillating GI ^X /M//n systems with state dependent services rates, also denoted as GI ^X /M(m)−M(m)//n systems, in which the service rates oscillate between two forms according to the evolution of the number of customers in the system. We derive an efficient algorithm to compute k-CCL probabilities in these systems starting with an arbitrary number of customers in the system that involves solving a linear system of equations. The results derived are illustrated for specific sets of parameters.     

Travelling wave solutions for a higher order wave equations of KdV type (I)

Using the method of planar dynamical systems to a higher order wave equations of KdV type, the existence of smooth solitary wave and uncountably infinite many smooth and non-smooth periodic wave solutions is proved. In different regions of the parametric plane, the sufficient conditions to guarantee the existence of the above solutions are given.     

Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation

The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation (B(m,n) equation), u_tt−u_xx−a(uⁿ)_xx+b(u^m)_xxxx=0, is studied by using bifurcation theory of dynamical system. As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given.     

Bifurcations of traveling wave solutions in a coupled non-linear wave equation

By using the theory of planar dynamical systems to a coupled non-linear wave equation, the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.     

Bifurcations of travelling wave solutions for the Gilson–Pickering equation

In this paper, the qualitative behavior and exact travelling wave solutions of the Gilson–Pickering equation are studied by using the qualitative theory of polynomial differential system. The phase portraits of the system are given under different parametric conditions. Some exact travelling wave solutions of the Gilson–Pickering equation are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given.     

Characterization for commutators of n -dimensional fractional Hardy operators

In this paper, it was proved that the commutator generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L ^p1 (ℝ ⁿ ) to L ^p2 (ℝ ⁿ ) if and only if b is a CṀO(ℝ ⁿ ) function, where 1/p ₁ − 1/p ₂ = β/n, 1 < p ₁ < ∞, 0 ⩽ β < n. Furthermore, the characterization of on the homogenous Herz space (ℝ ⁿ ) was obtained. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)     

Bifurcation of travelling wave solutions for the generalized ZK-BBM equations

By using the bifurcation theory of planar dynamical systems to the generalized ZK-BBM equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Bifurcation of travelling wave solutions for the generalized ZK equations

By using the bifurcation theory of planar dynamical systems to the generalized ZK equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Bifurcation of travelling wave solutions for the generalized KP-MEW equations

By using the theory of bifurcations of planar dynamical systems to the generalized KP-MEW equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.     

Bifurcation of travelling wave solutions for the generalized Camassa–Holm–KP equations

By using the bifurcation theory of planar dynamical systems to the generalized Camassa–Holm–KP equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.