Asymptotic stability of solitary waves for the 1D focusing cubic Schr\"odinger equation under even perturbations

We establish the full asymptotic stability of solitary waves for the focusing cubic Schr\"odinger equation on the line under small even perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the symmetries of the problem, namely the invariance under scaling and phase shifts. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix Schr\"odinger operator around the solitary waves. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.