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

TL;DR

This paper proves that classical self-similar blowup solutions do not exist for nonlinear Dirac equations in (1+1) dimensions, indicating such equations likely do not develop self-similar singularities in finite time.

Contribution

It establishes the nonexistence of classical self-similar blowup solutions for a broad class of nonlinear Dirac equations in one spatial dimension.

Findings

No classical self-similar blowup solutions in bounded function space.

Characterization of unbounded self-similar solutions for cubic Dirac equations.

Suggests smooth solutions do not form self-similar singularities in finite time.

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.