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

Abstract

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 $\lambda |u|^{2}u$, where $\lambda$ is a complex constant satisfying $\mathrm{Im}\lambda < 0$, 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 $t \to +\infty$ in a suitable function space. Our modified free dynamics decays like $(t\log t)^{-1/2}$ as $t\to \infty$.