Weakly localized states for nonlinear Dirac equations

Abstract

We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization as least action critical points of a suitable action functional. We also indicate how the content of the present paper allows to extend our previous results for the massive case [5] to more general nonlinearities.

Bifurcating standing waves for effective equations in gapped honeycomb structures

“…Existence and qualitative properties of solutions to the massless version of (1) have been studied in [8,9]. The massive case in (1) has been addressed in [10,11] for a special choice of parameters in (1).…”

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

“…where H := {w ∈ H 1 (G) : (15) holds} with the norm induced by H 1 (G) (this can be easily proved arguing as in [1,Proposition 3.3]). It is also worth mentioning that a Palais-Smale sequence for J is a sequence (w n ) ⊂ H such that dJ(w n ) → 0 in H * , namely sup ϕ H 1 dJ(w n )|ϕ → 0, as n → ∞.…”

“…Precisely, [50] suggests the study again of the stationary solutions, that is χ(t, x) = e −iωt ψ(x), with ω ∈ R, that solve Dψ − |ψ| p−2 ψ = ωψ . (5) recall that the existence of stationary solutions for cubic and Hartree-type Dirac equations for honeycomb structures and graphene samples has been investigated in [14,13,15]; whereas, for an overview on global existence results for one dimensional NLDE we refer to [18,44].…”

“…The usual definition of the space H 1 (G) consists also of a global continuity requirement, which forces all the components of a function that are incident to a vertex to assume the same value at that vertex. However, for the aims of this paper it is worth keeping this global continuity notion separate and introduce it when it is actually required (see (15)).…”

Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks

“…In that case one may expect to recover the metric graph model in an appropriate scaling regime. We also mention that a rigorous mathematical study of the dynamics and the existence of Dirac solitons on lattices has been recently treated in [42][43][44][45][46].…”