ASYMPTOTIC BEHAVIOUR FOR SCHRODINGER EQUATIONS WITH A QUADRATIC NONLINEARITY IN ONE-SPACE DIMENSION

We consider the Cauchy problem for the Schrodinger equation with a quadratic nonlinearity in one space dimension iu(t) + 1/2 u(xx) - t(-alpha) broken vertical bar u(x)vertical bar(2), u(0,x) - u(0)(x), where alpha epsilon ( 0; 1). From the heuristic point of view, solutions to this problem should have a quasilinear character when alpha epsilon (1/2, 1). We show in this paper that the solutions do not have a quasilinear character for all alpha epsilon ( 0, 1) due to the special structure of the nonlinear term. We also prove that for alpha epsilon [1/2, 1) if the initial data u(0) epsilon H-3,H-0 boolean AND H-2,H-2 are small, then the solution has a slow time decay such as t(-alpha/2). For alpha epsilon ( 0, 1/2), if we assume that the initial data u0 are analytic and small, then the same time decay occurs.