The 1-D Dirac equation with concentrated nonlinearity

2016/07/03 by Claudio Cacciapuoti, Cacciapuoti, Claudio, Raffaele Carlone +5
Mathematics · Physics and Astronomy · #35A01 #35B25 #35Q41 #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1607.00665

openalex publication_date 2016/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define and study the Cauchy problem for a 1-D nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy-Šeba models and the concentrated versions of the Bragg Resonance, Gross-Neveu, and Soler type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable (the "charge").

Citations

• Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory

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