Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation {i∂tu+14∂x4u=iλ∂x(|u|3u),t>0,x∈R,u(0,x)=u0(x),x∈R with critical nonlinearity, where λ∈R. We assume that the initial data are such that u0∈H1,1, with sufficiently small norm |u0|H1,1. We prove that there exists a unique global solution e−it4∂x4u∈C([0,∞);H1,1) of the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation such that |u(t)|L∞≤C(1+t)−14. Moreover we show that if the total mass 12π∫Ru0(x)dx≠0, then the large time asymptotics is determined by the self-similar solution.