Related papers: The heteroclinic connection problem for general do…
We are concerned with conservative systems $\ddot{q}=\nabla V(q), \; q\in\mathbb{R}^N$ for a general class of potentials $V\in C^1(\mathbb{R}^N)$. Assuming that a given sublevel set $\{V\leq c\}$ splits in the disjoint union of two closed…
We study the existence of solutions $u:\R^{3}\to\R^{2}$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:\R^{2}\to\R$ is a double well symmetric potential. We…
The existence of hyperbolic orbits is proved for a class of singular Hamiltonian systems $\ddot{u}(t)+\nabla V(u(t))=0$ by taking limit for a sequence of periodic solutions which are the variational minimizers of Lagrangian actions.
We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the…
For a class of second-order discrete Hamiltonian systems $\Delta^2x(t-1)-L(t)x(t)+V'_x(t,x(t))=0$, we investigate the existence of homoclinic orbits by applying variational method, where $L$ and $V(\cdot,x)$ are periodic functions. Further,…
We consider the minimal action problem min \int\_R 1/2 |$\gamma$'|^2 + W($\gamma$) dt among curves lying in a non-locally-compact metric space and connecting two given zeros of W $\ge$ 0. For this problem, the optimal curves are usually…
In this paper, the dynamical heteroclinic orbit and attractor have been employed to make the late-time behaviors of the model insensitive to the initial condition and thus alleviates the fine tuning problem in cosmological dynamical system…
We study the existence of traveling waves for the parabolic system \begin{equation} \partial_t w - \partial_{x}^2 w = -\nabla_{\mathbb{u}} W(w) \mbox{ in } [0,+\infty) \times \mathbb{R} \end{equation} where $W$ is a potential bounded below…
We interpret the close link between the critical points of Mather's barrier functions and minimal homoclinic orbits with respect to the Aubry sets on $\mathbb{T}^n$. We also prove a critical point theorem for barrier functions, and the…
This paper introduces techniques of symplectic topology to the study of homoclinic orbits in Hamiltonian systems. The main result is a strong generalization of homoclinic existence results due to Sere and to Coti-Zelati, Ekeland and Sere,…
In the scalar case, the nondegeneracy of heteroclinic orbits is a well-known property, commonly used in problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. On the other hand, Schatzman proved that in the vector case this…
We consider the system {\Delta}u - W_u (u) = 0, for u: R^2 -> R^2, W: R^2 -> R, where W_u (u) is a smooth potential, symmetric with respect to the u_1, u_2 axes, possessing two global minima a^\pm := (\pma,0) and two connections e^\pm(x_1)…
In this paper, we consider the scalar reaction-diffusion equations $\partial_t u=\Delta u + f(x,u,\nabla u)$ on a bounded domain $\Omega\subset\mathbb{R}^d$ of class $C^2$. We show that the heteroclinic and homoclinic orbits connecting…
We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions…
About twenty years ago, Rabinowitz showed firstly that there exist heteroclinic orbits of autonomous Hamiltonian system joining two equilibria. A special case of autonomous Hamiltonian system is the classical pendulum equation. The phase…
We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but…
n this paper we show the existence of traveling waves $w: [0,+\infty) \times \mathbb{R}^2 \to \mathbb{R}^k$ ($k \geq 2$) for the parabolic Allen-Cahn system \begin{equation} \partial_t w - \Delta w = -\nabla_u V(w) \mbox{ in } [0,+\infty)…
The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type $$ -div\left(\frac{\nabla u}{\sqrt{1+|\nabla…
In this paper, we prove the existence of infinitely many homoclinic orbits for the first order Hamiltonian systems $J\dot{x}-M(t)x+ R'(t,x)=0$, by the minimax methods in critical point theory, when $R(t,y)$ satisfies the superquadratic…
We study the relationship between a homological capacity $c_{\mathrm{SH}^+}(W)$ for Liouville domains $W$ defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on $W$: If the positive…