Asymptotic behavior for a quadratic nonlinear schrödinger equation

We study the initial-value problem for the quadratic nonlinear Schrödinger equation iu_t + 1/2u_xx= δ_xū², x ∈ ℝ, t > 1, u(1, x) = u₁(x), x ∈ ℝ. For small initial data u₁ ∈ H^2,2 we prove that there exists a unique global solution u ∈ C([1, ∞); H^2,2) of this Cauchy problem. Moreover we show that the large time asymptotic behavior of the solution is defined in the region ΙxΙ ≤ C√t by the self-similar solution 1/√tMS(x/ √t) such that the total mass 1/√t ∫_ℝ MS(x/√t)dx = ∫_ℝ u₁(x)dx, and in the far region ΙxΙ > √t the asymptotic behavior of solutions has rapidly oscillating structure similar to that of the cubic nonlinear Schrödinger equations.