Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension

Abstract

In this paper, we prove the global existence of a small solution to the Cauchy problem for the nonlinear Schrödinger equation with a class of cubic nonlinearities in one space dimension. Moreover, we also consider the asymptotic behavior in large time of the solution. Our results says that two cubic nonlinearities given in this paper can be considered as nonlinearities of higher degree (more precisely, of degree 5).