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Homoclinic solutions of an infinite-dimensional Hamiltonian system. (English) Zbl 1008.37040

Summary: We consider the system \[ \left\{ \begin{aligned} \partial_t u-\Delta_x u+V(x)u &= H_v(t,x,u,v) \\ -\partial_t v-\Delta_x v+V(x)v &= H_u(t,x,u,v) \end{aligned}\right. \qquad\text{for }(t,x)\in\mathbb{R}\times\mathbb{R}^N \] which is an unbounded Hamiltonian system in \(L^2(\mathbb{R}^N,\mathbb{R}^{2M})\). We assume that the constant function \((u_0,v_0) \equiv (0,0) \in \mathbb{R}^{2M}\) is a stationary solution, and that \(H\) and \(V\) are periodic in the \(t\) and \(x\) variables. We present a variational formulation in order to obtain homoclinic solutions \(z=(u,v)\) satisfying \(z(t,x)\to 0\) as \(|t|+|x|\to\infty\). It is allowed that \(V\) changes sign and that \(-\Delta+V\) has essential spectrum below (and above) 0. We also treat the case of a bounded domain \(\Omega\) instead of \(\mathbb{R}^N\) with Dirichlet boundary conditions.

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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