We present analytic exact solutions to the nonlinear Dirac equation: they display a ring singularity for the Nambu–Jona-Lasinio nonlinearity and a shell singularity for the Soler nonlinearity. For both cases, the size of the singular region is of the order of the Compton length. General comments about square-integrability will be addressed in terms of a solution to the linearized system.
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Fabbri et al. (2026) studied this question.
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