This talk is based on the joint work with S.Cuccagna (arXiv:2004.01366). In this talk, I explain a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrödinger equations (NLS). We consider a NLS with a Schrödinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the "refined profile", a quasi-periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in the previous work, giving us also a better understanding of the Fermi Golden Rule.

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