Assuming as hypotheses the results proved numerically by Chang et al. (SIAM J Math Anal 39:1070–1111, 2007/08) for the exponent \(p\in (3,5)\) , we prove that some of the ground states of the nonlinear Schrödinger 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 Cuccagna and Maeda (J Funct Anal 288(11):110861, 2025).
