Nonexistence of self-similar blowup for the nonlinear Dirac equations in (1+1) dimensions

Abstract

We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that self-similar singularities do not develop in the nonlinear Dirac equations in (1+1) dimensions in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.