Remarks on nonlinear Schrödinger equations in one space dimension

Abstract

We consider the initial value problem for nonlinear Schödinger equations, where ∂ = ∂x = ∂/∂x and F: C4 → C is a polynomial having neither constant nor linear terms. Without a smallness condition on the data u0, it is shown that (+) has a unique local solution in time if u0 is in H3, 0 ∩ H2, 1, where Hm, s = {f ∈ S’ ∥f∥m, s = ∥(1 + x2)s/2 (1-Δ)f∥