On the asymptotic stability of ground states of the pure power NLS on the line at 3rd and 4th order Fermi Golden Rule

TL;DR

This paper proves the asymptotic stability of certain ground states in the nonlinear Schrödinger equation with pure power nonlinearity on the line, focusing on third and fourth order Fermi Golden Rule interactions, under hypotheses supported by numerical results.

Contribution

It establishes asymptotic stability for ground states at specific power exponents where FGR occurs, extending previous results to higher order interactions.

Findings

Asymptotic stability proven for generic p in 3rd order FGR case.

Stability shown for specific p satisfying FGR and non-resonance in 4th order case.

Results depend on hypotheses supported by numerical evidence from prior work.

Abstract

Assuming as hypotheses the results proved numerically by Chang et al. \cite{Chang} for the exponent $p\in (3,5)$, we prove that some of the ground states of the nonlinear Schr\"odinger equation (NLS) with pure power nonlinearity of exponent $p$ in the line are asymptotically stable for a certain set of values of the exponent $p$ where the FGR occurs by means of a discrete mode 3rd or 4th order power interaction with the continuous mode. For the 3rd the result is true for generic $p$ while for the 4th order case we assume that there are $p$'s satisfying Fermi Golden rule and the non-resonance condition of the threshold of the continuous spectrum of the linearization. The argument is similar to our recent result valid for $p$ near 3 contained in \cite{CM24D1}.