Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

Abstract

We study scattering problems for the one-dimensional nonlinear Dirac equation (∂t + α∂x + iβ)Φ = λ|Φ|p−1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(−t) and t∂x + x∂t − α/2, where {D(t)}t∈ℝ is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(−t) and t∂x + x∂t − α/2.