Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. (English) Zbl 0890.35016

Pillet, Claude-Alain; Wayne, C. Eugene

J. Differ. Equations 141, No. 2, 310-326 (1997).

This paper concerns the evolution equation \[ i\partial_tu= (-\Delta+ V(x)+\lambda|u|^{m-1})u,\quad x\in\mathbb{R}^n,\;t\in\mathbb{R},\tag{1} \] for \(n=3\) and \(m=3\) or 4, for example. Here \(\lambda\in\mathbb{R}\) and the potential \(V(x)\) are chosen so that the spectrum of the linear part \(-\Delta+ V(x)\) consists of a simple eigenvalue \(E_0<0\), the absolutely continuous spectrum filling the positive real half-line.
It is shown that there exists an invariant manifold to (1) consisting of periodic orbits of the form \(e^{-iEt}v(x)\), where \(v(x)\) is a positive solution to the nonlinear eigenvalue problem \[ (-\Delta+ V(x)+ \lambda|v|^{m- 1})v= Ev,\quad x\in\mathbb{R}^n \] and \(E\approx E_0\). Moreover, each solution to (1) with small initial value approaches a particular periodic orbit with particular phase on the invariant manifold.

Reviewer: L.Recke (Berlin)

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35Q55	NLS equations (nonlinear Schrödinger equations)
35G25	Initial value problems for nonlinear higher-order PDEs

Keywords:

periodic orbits; nonlinear eigenvalue problem

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