WAVE OPERATORS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH A NONLINEARITY OF LOW DEGREE IN ONE OR TWO SPACE DIMENSIONS.

In this paper, we study the asymptotic behavior of solutions to the cubic and the quadratic nonlinear Schrödinger equations in one and two space dimensions, respectively. When the nonlinearity is of a form f = |u|[sup p-1]u, it is known that there exist scattered states if p > 1+2/n and there does not otherwise. Therefore we may consider the nonlinearities treated in the present paper to be of critical order for the existence of scattered states though their forms differ slightly from that given above. We prove, however, that there exist scattered states for these critical nonlinear Schrödinger equations, in other words, that the wave operators exist on a certain set of final states. Our proof is mainly based on the construction of suitable approximate functions that approach to solutions of nonlinear Schrödinger equations at t = ∞. [ABSTRACT FROM AUTHOR]