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

摘要

We study scattering problems for the one-dimensional nonlinear Dirac equation ((t)+(x)+i)phi=|phi|(p-1)phi. 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)-/2, where {D(t)}(t) 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)-alpha/2.