Solitary Wave Solutions for the Nonlinear Dirac Equations
Abstract
Description
In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation \[ i \sum_{j=0}^{3} \ga^j \pd_j ψ- mψ+ F(\barψψ)ψ=0 \] where $ m >0$ and $ F(s) = |s|^θ$ for $ 1\leq θ< 2.$ More precisely we show that there exists $\e_0 > 0$ such that for $ω\in(m - \e_0, m), $ there exists a solution $ ψ(t,x) = e^{-iωt}ϕ_ω(x), x_0 = t, x = (x_1, x_2, x_3),$ and the mapping from $ ω$ to $ ϕ_ω $ is continuous. We prove this result by relating the stationary solutions to the ground states of nonlinear Schrödinger equations.
18 pages
18 pages