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

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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### References:

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