Causal solutions of nonlinear wave and spinor equations obtained by Gel’fand–Shilov regularization

Abstract

By extending the Gel’fand–Shilov regularization method to products of locally integrable functions it can be shown that the nonlinear Klein–Gordon equations ∂2u/∂t2−∂2u/∂x21− ⋅⋅⋅ −∂2/∂x2n +ku2p+1=0, where k=const≳0, n⩾3, p=integer⩾1, have causal solutions which have no δ singularities. It is further shown that this also can be expected for the nonlinear Dirac equation γλ∂ψ/∂xλ+l2ψ (⩾ψ) =0.