Subcritical quadratic nonlinear schrödinger equation

We study the initial value problem for the quadratic nonlinear Schrödinger equation iut+1/2uxx=-4iπtγ-1/2u ², x R, t > u(1,x) = u ₁ (x), x ε R, where γ > 0. Suppose that the Fourier transform û ₁ of the initial data u ₁ satisfies estimates û1 L∞ ≤ ε, d/d (ei/2ε ²û1(ε))L∞,1 ≤ ε> 0 is sufficiently small. Also suppose that Re(eu/2ε ² û1(ε))≥Cε5/4 for |ξ| ≤ 1. Assume that γ > 0 is small: γ = O(5/4). Then we prove that there exists a unique solution u ∈ C([1, ∞);L ²) of the Cauchy problem (*). Moreover, the solution u approaches for large time t → +∞ a self-similar solution of the quadratic nonlinear Schrödinger equation (*).