Large time asymptotics for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation in the super critical case i∂tu+14∂x4u=λ∂x(|u|ρ−1u), where ρ>4ρ>4, λ∈Cλ∈C. We prove the global existence and the large time asymptotics of solutions for small initial data u0∈H1∩H12,1, when the order of the nonlinearity ρ>4ρ>4.