## 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.)

### Keywords:

stationary solution; variational formulation
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