The dissipative property of a cubic non-linear Schrödinger equation

We study the large-time behaviour of solutions of the Cauchy problem for a non-linear Schrödinger equation. We consider the interaction between the resonance term and other types of non-linearity. We prove that solutions exist globally in time and find a large-time asymptotic representation for them. We show that the decay of solutions in the far region has the same order as in the linear case, while the solutions in the short-range region acquire an additional logarithmic decay, which is slower than in the case when there is no resonance term in the original equation.