Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity

In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity λ|u|²u, where λ is a complex constant satisfying Im λ < 0, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)^-1/2 as t → ∞.