Decay of solutions of nonlinear Dirac equations
Sebastian Herr, Christopher Maulén, Claudio Muñoz
TL;DR
The paper advances the understanding of long-time behavior for nonlinear Dirac equations across dimensions by developing Dirac-adapted virial identities that yield decay and rule out breathers in key regimes. It proves massless 1D decay inside an expanding window, extends decay to 1D massive data under harmonic Cauchy–Riemann conditions, and establishes local $L^2$ decay for small radial 3D solutions without parity constraints; it also demonstrates $L^2$ decay in exterior light-cone regions for general $n$ with real nonlinearities. The core methodology hinges on carefully constructed weighted virial functionals that emphasize massless dynamics in the Dirac framework, avoiding energy-sign issues. Together, these results illuminate the nonexistence of nontrivial localized oscillations (breathers/solitons) in broad nonlinear Dirac models and lay groundwork for future asymptotic stability analyses of Dirac solitary waves.
Abstract
We study the long-time behavior of small and large solutions to a broad class of nonlinear Dirac-type equations. Our results are classified in 1D massless and massive cases, 3D general and $n$ dimensional in generality. In the 1D massless case we prove that any globally defined solution converges to zero as time tends to infinity, within a spatial region expanding at a rate proportional to $ t \log^{-2} t$. This result holds without assumptions on the smallness of initial data or specific power of nonlinearity, ruling out the existence of standing breather-like or solitary wave structures in this regime. In the 1D massive case, solitary waves are known to exist. Introducing new virial identities adapted to the Dirac's distinctive algebra, we prove that there are ``holomorphic'' odd nonlinearities under which globally defined small odd solutions decay to zero on spatial compact sets as time tends to infinity. This result is extended to the 3D case under boundedness of the $H^1$ norm but without requiring the parity condition on the data, giving decay proofs for an important class of nonlinear Dirac models, and opening the door to the future use of virial identities to prove asymptotic stability of well-chosen Dirac solitary waves. Finally, in higher dimensions $ n \geq 1$, we prove the $L^2$ decay for global solutions of nonlinear Dirac equations in the ``exterior light-cone'' region. This confirms the non-existence of breathers and other solutions propagating faster than the speed of light. Our proofs rely on carefully constructed weighted virial identities.