H. Huh and D.E. Pelinovsky

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

Applied Mathematics Letters 92, 176-183 (2019)

Abstract:
We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical 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 smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.

Keywords:
nonlinear Dirac equations, global existence, finite-time blowup, self-similar solutions