Scattering Problem for the Supercritical Nonlinear Schrödinger Equation in 1d

Abstract

We consider the one dimensional nonlinear Schrödinger equation iu_t + u_xx/2 = f(u), x ∈ R, t > 0, u(0,x) = u₀(x), x ∈ R, with a super critical nonlinearity f(u) = ∑_j≠0 f_j(u), and f_j(u) are such that f_j(u) = λ_j|u|^σ_j−ju^j, where λ_j ∈ C, σ_j > 3. We prove the existence of the scattering operator in the weighted Sobolev spaces.