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The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states. (English) Zbl 1222.35183

Summary: We prove that ground states of the NLS which satisfy the sufficient conditions for orbital stability of M. Weinstein, are also asymptotically stable, for seemingly generic equations. The key issue is to prove that a certain coefficient is non-negative because is a square power. We assume that the NLS has a smooth short range nonlinearity. We assume also the presence of a very short range and smooth linear potential, to avoid translation invariance. The basic idea is to perform a Birkhoff normal form argument on the Hamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0 solution for NLKG. But in our case, the natural coordinates arising from the linearization are not canonical. So we need also to apply the Darboux Theorem. With some care though, in order not to destroy some nice features of the initial hamiltonian.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
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