Type: Article

Asymptotics of solutions to the fractional nonlinear Schrödinger equation with a>52

Hayashi, Nakao; Naumkin, Pavel I.; Sánchez-Suárez, Isahi;

Abstract:

We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional nonlinear Schrödinger equation {i?tu?1?|?x|?u=?|u|?u,u(0,x)=u0(x),t>0, x?R, x?R, where ?>0, the fractional derivative |?x|?=F?1|?|?F, ?>52. This paper is a sequel to our previous papers [17] [MR4431246] for 2<?<52 and [36] [MR4040521] for ?=52. We show that solutions decay in time at the rate t?1?(logt)?1?, namely that the nonlinearity acts as a dissipative term, when ?>0. This phenomena does not occur for the cubic problem {i?tu?1?|?x|?u=?|u|2u,u(0,x)=u0(x),t>0, x?R, x?R, with 0<??2.''