This study aimed to investigate the traveling solutions of the Camassa-Holm and Degasperis-Procesi (CH-DP) equation. By using the qualitative theory of dynamical systems and integrating along special phase orbits, we present exact explicit expressions of compacton and generalized kink wave solutions to the CH-DP equation. Our results will enrich the previous literature and help to understand the propagation of nonlinear waves.

In this paper, we consider the symmetry properties of the positive solutions of a $p$-Laplacian problem of the form

$$\begin{array}{r}\{\begin{array}{l}-{\mathrm{\Delta }}_{p}u=f(x,u),\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=g(x),\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$$

where $\mathrm{\Omega }$ is an open smooth bounded domain in $R^N,N\ge 2$, and symmetric w.r.t. the hyperplane $T_0^{\nu }(\nu$ is a direction vector in $R^N,|\nu |=1$ $)$, $f$: $\mathrm{\Omega }\times R^{+}\to R^{+}$ is a continuous function of class $C^1$ w.r.t. the second variable, $g\ge 0$ is continuous, and both $f$ and $g$ are symmetric w.r.t. $T_0^{\nu }$, respectively. Introducing some assumptions on nonlinearities, we get that the positive solutions of the problem above are symmetric w.r.t. the direction $\nu$ by a new simple idea even if $\mathrm{\Omega }$ is not convex in the direction $\nu$.

In this paper, we are interested in the existence of multiple nontrivial $T$-periodic solutions of the nonlinear second ordinary differential equation $\ddot{x}+V_x(t,x)=0$ in $N(\ge 1)$ dimensions. Using homological linking and morse theory, we get at least two critical points of the functional corresponding to our problem. And, we also prove that two critical points are different by critical groups. Then, we obtain there are at least two nontrivial $T$-periodic solutions of the problem.