Coordinates at Small Energy and Refined Profiles for the Nonlinear Schrödinger Equation

• DOI: 10.1007/s40818-021-00105-2
• Publication Date: 2021-07-20T16:03:26Z
• Authors: Scipio Cuccagna; Masaya Maeda
• Keywords: FOS: Physical sciences; Stability; standing waves; Mathematical Physics (math-ph); 01 natural sciences; standing waves; 35Q55; Mathematics - Analysis of PDEs; FOS: Mathematics; 0101 mathematics; Stability; Mathematical Physics; Analysis of PDEs (math.AP)

In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schr��dinger equations (NLS) that we gave in \cite{CM15APDE}. We consider a NLS with a Schr��dinger 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 \cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.<br/>27 pages<br/>