Decay of solutions of nonlinear Dirac equations: the 2D case

TL;DR

This paper investigates the long-time decay of small solutions to 2D nonlinear Dirac equations with various nonlinearities, establishing decay conditions, ruling out localized structures, and introducing new virial identities for analysis.

Contribution

It provides new decay results for small solutions of 2D nonlinear Dirac equations, extending previous understanding and introducing virial identities specific to Dirac models.

Findings

Small solutions decay to zero locally in L^2 as time tends to infinity.

No small, localized structures like breathers or solitary waves exist in the studied regimes.

Decay results depend on the power of the nonlinearity and boundedness conditions.

Abstract

We study the long-time behavior of small solutions for a broad class of 2D Dirac-type equations with suitable nonlinearities. First, we prove that for nonlinearities with power $p\geq 5$ (massless case) and $p\geq7$ (massive case), any small globally bounded radial solution with vorticity $S\ne -1,0$ decays to zero locally in $L^2_{loc}$, as time tends to infinity. For solutions uniformly bounded in time in a weighted $H^1$ space, this decay result extends to lower powers $p\geq 3$ (massless) and $p\geq5$ (massive). Our main results apply to several physical models of current interest, such as the 2D Dirac equation with a honeycomb potential described by Fefferman and Weinstein. Finally, we rule out the existence of small, localized structures such as standing breathers or solitary waves in the 2D regimes considered. To prove these results, we introduce new virial identities with a particular algebra that are applied directly to the Dirac model, and without resorting to the nonlinear Klein-Gordon equation.