Stationary states of the nonlinear Dirac equation: A variational approach

Abstract

In this paper we prove the existence of stationary solutions of some nonlinear Dirac equations. We do it by using a general variational technique. This enables us to consider nonlinearities which are not necessarily compatible with symmetry reductions.

Solutions of a Nonlinear Dirac Equation with External Fields

“…Thus Φ ∈ C 1 (E, R) and a standard argument shows that critical points of Φ are weak solutions of (P). Moreover, by [17], such solutions are in W 1,s (R 3 , C 4 ) for all s 2 (see also [7]). In order to find critical points of Φ we will use the following abstract theorems.…”

“…In [17], M. Esteban and E. Séré treated the above mentioned system of ODEs variationally, obtaining the existence of infinitely many solutions, under the main additional assumption that H (s) s θ H (s) for some θ > 1, and all s ∈ R.…”

The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries

“…Nonlinear Dirac equations have a long history in the literature, particularly in the context of particle and nuclear theory [12,13,14,15], but also in applied mathematics and nonlinear dynamics [16,17,18,19,20]. As nonlinearity is a ubiquitous aspect of Nature, it is natural to ask how nonlinearity might appear in a relativistic setting.…”

Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system

“…The Maxwell-Dirac system, which have been widely considered in literature (see [1], [15], [17], [21], [23], [24], [29] and references therein), is fundamental in the relativistic description of spin 1/2 particles. It represents the timeevolution of fast (relativistic) electrons and positrons within external and self-consistent generated electromagnetic field.…”

“…For the subcritical case, using the same iterative argument of [15] one obtains easily the following…”