We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result [
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Figure 1. (Left) The numerically computed solution $ f_1 $ in the system (16) with $ p = 5 $ and three choices of $ \omega $. The asymptotically flat solution corresponds to $ \omega_1\approx1.49171 $, at which we find an even resonance. The other two solutions correspond to values of $ \omega $ slightly above/below this value. (Right) The analogous situation when searching for an odd resonance, which we find at $ \omega_2\approx 19.5722 $
Figure 2. (Left) A closer look at the shape of the $ f_1 $ component of the even resonance near $ x = 0 $. We obtain the full solution by taking an even reflection across $ x = 0 $. (Right) The shape of the $ f_1 $ and $ f_2 $ components of the odd resonance for $ x>0 $. We obtain the full solution by taking an odd reflection across $ x = 0 $. These solutions both correspond to the case $ p = 5 $, with the even resonance occurring at $ \omega_1\approx 1.49171 $ and the odd resonance occurring at $ \omega_2 \approx 19.5722 $
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(Left) The numerically computed solution
(Left) A closer look at the shape of the
(Left) A plot the curves
A plot of