Narrow Results

For a class of nonlinear diffusion–convection–reaction equations, the corresponding travelling wave systems are well known Liénard systems. Under Chiellini’s integrability condition, the first integrals of Liénard systems can be obtained. In this paper, we use the method of dynamical systems to study the dynamical behavior of the corresponding travelling wave systems of two classes of nonlinear wave equations. Under given parametric conditions, some exact explicit parametric representations of the monotonic and nonmonotonic kink wave solutions are obtained.

By using the method of dynamical systems and the results in [Li & Zhang, 2011] to the sixth-order Ramani equation and a coupled Ramani equation, the families of exact traveling wave solutions can be obtained.

By using the method of dynamical systems to study the generalized Pochhammer–Chree equations, the dynamics of traveling wave solutions are characterized under different parameter conditions. Some exact parametric representations of the traveling wave solutions are obtained. Thus, many results reported in the literature can be completed.

In this paper, we apply the method of dynamical systems to a generalized two-component Camassa–Holm system. Through analysis, we obtain solitary wave solutions, kink and anti-kink wave solutions, cusp wave solutions, breaking wave solutions, and smooth and nonsmooth periodic wave solutions. To guarantee the existence of these solutions, we give constraint conditions among the parameters associated with the generalized Camassa–Holm system. Choosing some special parameters, we obtain exact parametric representations of the traveling wave solutions.

The traveling wave system of a microstructured solid model belongs to the second class of singular traveling wave equations studied in [Li et al., 2009]. In this paper, by using methods from dynamical systems theory, bifurcations of phase portraits of such a traveling wave system are analyzed in its corresponding parameter space. The existence of kink wave solutions and uncountably infinitely many bounded solutions is proved. Moreover, the exact parametric representations of periodic solutions and homoclinic orbits are obtained.

For a class of nonlinear diffusion–convection–reaction equations, corresponding to two families of heteroclinic orbits connecting two nodes of the traveling wave system, the existence of uncountably infinite many global monotonic and nonmonotonic wavefront solutions is discussed. By using the method of planar dynamical systems, the dynamical behavior of the corresponding traveling wave system is studied. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink wave solutions are given.

In this paper, we study the dynamical behavior and exact parametric representations of all traveling wave solutions for (2 + 1)-dimensional higher order Broer–Kaup equation. By using the method of dynamical systems, under different parametric conditions, for the solution component U, exact monotonic and nonmonotonic kink wave solutions, two-peak wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained. Exact wave profiles of traveling wave solutions for all solution components U, V, W, P are shown.

A model of dynamics of protons in hydrogen-bonded quasi-one-dimensional networks was derived, which is a singular system of the second kind with three parameters and two singular straight lines. In this paper, we use the method of dynamical systems to discuss the bifurcations of phase portraits of the vector fields defined by the singular system. Corresponding to the phase orbits of the system in different parameter conditions, we compute all possible exact parametric representations of solutions. It is shown that in given parameter conditions, there exist solitary wave solutions, kink wave solutions and periodic wave solutions. The mentioned system has no peakon solution.

The main purpose of this paper is to form a complete group classification for the Gilson–Pickering equation. Exact traveling wave solutions and bifurcations of the Gilson–Pickering equation are studied by the method of dynamical systems. The study on the wave solutions of the model derives a planar Hamiltonian system. Based on phase portraits, many exact explicit parametric representations of wave solutions are obtained under different parametric conditions.