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We consider perturbations of the one-dimensional cubic Schr\"odingerequation, under the form $i \, \partial_t \psi + \partial_x^2 \psi + |\psi|^2\psi - g( |\psi|^2 ) \psi = 0$. Under hypotheses on the function g that can beeasily verified in some cases, we show that the linearized problem around asolitary wave does not have internal mode (nor resonance) and we prove theasymptotic stability of these solitary waves, for small frequencies.

Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schr\"odinger equation in the absence of internal modes