Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation{i∂tu−14∂x4u=λ|u|4u, (t,x)∈R+×R,u(0,x)=u0(x), x∈R. We prove the large time asymptotic behavior of solutions with a logarithmic correction under the conditions ∫u0(x)dx≠0∫u0(x)dx≠0, λ>0λ>0.