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

TL;DR

The paper investigates local decay for small radial solutions to broad 2D nonlinear Dirac equations with polynomial nonlinearities, focusing on vorticity values $S\in\mathbb{Z}\setminus\{-1,0\}$. It introduces a novel 2D radial virial framework with four functionals $\mathcal{J}_1$, $\widetilde{\mathcal{J}}_1$, $\mathcal{J}_2$, $\widetilde{\mathcal{J}}_2$ and a carefully chosen weight $\varphi$ to obtain coercive estimates directly for Dirac dynamics, avoiding Klein–Gordon reductions. The main results establish local decay to zero for small data when the nonlinearity power satisfies $p\ge 5$ (massless) or $p\ge 7$ (massive), with improved thresholds $p\ge 3$ and $p\ge 5$ under additional weighted $H^1$ or $L^\infty$ bounds, and exclude the existence of small breathers or solitary waves in the regimes studied. The results apply to physically relevant models, including the 2D Dirac equation with a honeycomb potential, and provide a rigorous nonexistence theory for localized stationary structures in 2D Dirac flows. Overall, the work advances the understanding of long-time dynamics for low-dimensional Dirac-type systems via a direct Dirac-virial approach.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: HMM
• Claim 2.1
• Lemma 2.2: Strauss
• Remark 2.1
• Remark 2.2: On the absence of volume terms
• Proposition 2.3
• Remark 2.3: On the sum \ref{['sum']}
• Corollary 2.4