Asymptotic behaviour for schrödinger equations with a quadratic nonlinearity in one-space dimension

We consider the Cauchy problem for the Schrödinger equation with a quadratic nonlinearity in one space dimensio iu_t + 1/2u_xx = t^-a\u_x|², u(0,x) = U₀(X), where a G (0, 1). From the heuristic point of view, solutions to this problem should have a quasilinear character when o. G (1/2, 1). We show in this paper that the solutions do not have a quasilinear character for all a G (0, 1) due to the special structure of the nonlinear term. We also prove that for a G [1/2, 1) if the initial data UQ G H ' H H ' are small, then the solution has a slow time decay such as t-^a/2 . For o. G (0, 1/2), if we assume that the initial data U₀ are analytic and small, then the same time decay occurs.