Qualitative theory of $L^{2}$-normalized solitary waves of the Maxwell-Dirac system: uniqueness, multiplicity, regularity, and stability

The Maxwell-Dirac equations couple the Dirac equation for a charged spin-1/2 fermion to the electromagnetic field generated by the fermion's own charge and current. Nolasco (2021) proved the existence of an $L^{2}$-normalized solitary wave of this system in (3+1)-dimensional Minkowski space: a time-periodic, spatially localized solution with prescribed charge $\|\psi\|_{L^{2}}^{2}=1$, constructed as a critical point of the strongly indefinite Maxwell-Dirac energy at its lowest positive critical level, with the spinor a priori only in $H^{1/2}(\mathbb{R}^{3};\mathbb{C}^{4})$. Beyond this single existence statement, essentially nothing is known about these objects. The open problem is to develop the qualitative theory of the normalized solitary waves: whether the lowest-level normalized solution is unique up to phase and the symmetries of the problem and how all normalized solutions can be classified; whether normalized solutions exist at higher positive critical levels (excited states), with which symmetries, and how the solution set branches as the frequency $\omega$ varies in $(0,m)$; whether every $H^{1/2}$ normalized solution is more regular and decays at infinity; how the solutions depend quantitatively on the coupling $e^{2}$ (the fine-structure constant) and the mass $m$, and whether a small-coupling limit connects them to Choquard-type ground states; and whether the normalized solitary waves are orbitally (or asymptotically) stable under the full Maxwell-Dirac time evolution. The known follow-up literature on normalized solutions of nonlinear Dirac equations concerns different models (Klein-Gordon-Dirac, Maxwell-Klein-Gordon-Dirac, or local power-type nonlinearities), so the entire qualitative program for the Maxwell-Dirac system remains open.

Background

The Maxwell-Dirac (MD) system is the classical field-theoretic model of a single charged, relativistic spin-1/2 particle, such as an electron, interacting with its own electromagnetic field; it can be read as the classical (pre-second-quantization) limit of quantum electrodynamics. Writing $\Psi:\mathbb{R}^{1+3}\to\mathbb{C}^{4}$ for the Dirac spinor (the wave function of the fermion), $A^{\mu}$ for the electromagnetic four-potential, $\gamma^{\mu}$ for the $4\times 4$ Dirac matrices, $m>0$ for the particle mass and $e$ for its charge, the Euler-Lagrange equations in the Lorenz gauge (the gauge condition $\partial_{\mu}A^{\mu}=0$) are

$$(i\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}A_{\mu})\Psi-m\Psi=0,\qquad \partial_{\nu}\partial^{\nu}A^{\mu}=4\pi e\,\bar{\Psi}\gamma^{\mu}\Psi,$$

where $\bar{\Psi}=\Psi^{\dagger}\gamma^{0}$ and $e\bar{\Psi}\gamma^{\mu}\Psi$ is the conserved charge-current density. A solitary wave is a solution that is periodic in time and localized in space, of the special form $\Psi(t,x)=e^{i\omega t}\psi(x)$, $A^{\mu}(t,x)=A^{\mu}(x)$; such non-dispersing, particle-like lumps are the classical analogues of extended particles. Solving the Maxwell equations for $A^{\mu}$ in terms of the spinor via the Coulomb kernel $1/|x|$ reduces the search to a nonlinear eigenvalue problem for $\psi$, in which the frequency $\omega$ plays the role of a Lagrange multiplier.

The existence theory for MD solitary waves has a long history. Numerical evidence goes back to Lisi (1995). The first rigorous construction is due to Esteban, Georgiev and Séré (1996), who used a variational linking argument to produce, for every prescribed frequency $\omega\in(0,m)$, a stationary solution with smooth, exponentially decaying spinor; Abenda (1998) extended existence to all $\omega\in(-m,m)$ using an axial-symmetry ansatz and also proved multiplicity results for stationary solutions of the Coulomb-Dirac model (the simplified system in which the spinor couples only to the electrostatic potential $A^{0}$ through an attractive Coulomb self-interaction). Comech and Stuart (2018) constructed small-amplitude solitary waves bifurcating from the ground state of the Choquard equation (the nonlocal Hartree-type equation with nonlinearity $|\psi|^{2}\ast|x|^{-1}$). A common feature of all these results is that the $L^{2}$ norm of the spinor, which is proportional to the physical charge and is a conserved quantity of the time-dependent system, is left free: none of these constructions prescribes it.

Nolasco (2021) obtained the first $L^{2}$-normalized MD solitary wave: a pair $(\psi,\omega)$ with $\|\psi\|_{L^{2}}^{2}=1$ and $\omega\in(0,m)$ solving the stationary system, constructed as a critical point of the MD energy functional restricted to the unit sphere of $L^{2}(\mathbb{R}^{3};\mathbb{C}^{4})$, at the lowest positive critical level $E\in(0,m)$. The construction faces two structural obstacles that explain why so little more is known. First, the functional is strongly indefinite: its quadratic part is the free Dirac operator $D=i\boldsymbol{\alpha}\cdot\nabla-m\beta$ (with $\boldsymbol{\alpha}$, $\beta$ the standard Dirac matrices), whose spectrum is $(-\infty,-m]\cup[m,+\infty)$, so the energy is unbounded above and below even on the $L^{2}$ sphere, and the direct method of the calculus of variations is unavailable; the proof instead adapts the min-max characterization used for the first eigenvalue of Dirac operators with Coulomb-type potentials. Second, the natural energy space is the fractional Sobolev space $H^{1/2}(\mathbb{R}^{3};\mathbb{C}^{4})$, the domain of the half-order operator $(-\Delta+m^{2})^{1/4}$, so the constructed spinor is a priori only weakly differentiable; no uniqueness, higher regularity, decay rate, or multiplicity statement is obtained. As a byproduct, the Coulomb-Dirac model is shown to possess a normalized energy minimizer.

Two neighboring results delineate what a complete qualitative theory should aim for. In the bosonic analogue, the pseudo-relativistic Hartree model describing boson stars, Frohlich, Jonsson and Lenzmann (2007) proved both the existence and the orbital stability of $L^{2}$-normalized solitary waves; orbital stability means that initial data close to the solitary wave remain close to its orbit under the natural symmetry group (phase rotations, translations, Lorentz boosts) for all later times. In the nonrelativistic Choquard equation, the ground state is unique (Lieb, 1977). Whether analogous uniqueness and stability statements hold for the fermionic Maxwell-Dirac system is entirely open. On the dynamical side, the required Cauchy theory has only recently progressed for small data: Cho and Lee (2024), and Herr, Ifrim and Spitz (2025), established global well-posedness and modified scattering for the massive Maxwell-Dirac system in three space dimensions for small, decaying initial data in the Lorenz gauge; nothing is known about the evolution of data at the size of the large-amplitude normalized solitary waves.

Problem Statement

Consider the stationary Maxwell-Dirac system in $(3+1)$-dimensional Minkowski space, written for the spinor $\psi:\mathbb{R}^{3}\to\mathbb{C}^{4}$ as the nonlinear eigenvalue problem

$$\begin{cases} (i\boldsymbol{\alpha}\cdot\nabla-m\beta)\psi-eA_{0}\psi+e\,\boldsymbol{\alpha}\cdot\boldsymbol{A}\,\psi=\omega\,\psi,\\ A_{0}(x)=e\,|\psi|^{2}\ast\dfrac{1}{|x|},\qquad \boldsymbol{A}(x)=e\,(\psi,\boldsymbol{\alpha}\psi)_{\mathbb{C}^{4}}\ast\dfrac{1}{|x|},\\ \|\psi\|_{L^{2}}^{2}=1, \end{cases}$$

where $\boldsymbol{\alpha}$ and $\beta$ are the standard Dirac matrices, $e$ is the charge (in units $\hbar=c=1$ the coupling $e^{2}$ is the fine-structure constant), $m>0$ is the mass, the asterisk denotes convolution in $\mathbb{R}^{3}$, and both the spinor $\psi$ and the frequency $\omega$ are unknown. Any solution $(\psi,\omega)$ with $\omega\in(0,m)$ yields a solitary wave $\Psi(t,x)=e^{i\omega t}\psi(x)$ of the time-dependent Maxwell-Dirac equations. The currently known fact is a single existence statement: a solution exists at the lowest positive critical level $E\in(0,m)$ of the Maxwell-Dirac energy functional on the $L^{2}$ unit sphere, with $\psi\in H^{1/2}(\mathbb{R}^{3};\mathbb{C}^{4})$.

The problem is to characterize these normalized solitary waves beyond this existence statement, with the following objectives, which are independent sub-questions rather than jointly required parts (a rigorous resolution of any one of them, or of any single sub-claim stated within one of them, stands on its own, without requiring the others): (1) uniqueness and classification — determine whether the normalized solution at the lowest positive critical level is unique up to phase and the symmetries of the problem, and classify all solutions of the normalized eigenvalue problem; (2) multiplicity — determine whether normalized solutions exist at positive critical levels above the lowest one (excited states), whether there are finitely or infinitely many, what symmetries (for example axial symmetry) they can possess, and how the set of normalized solutions branches as $\omega$ varies in $(0,m)$; (3) qualitative properties — determine whether every $H^{1/2}$ normalized solution enjoys higher regularity (for example membership in $H^{1}$ or smoothness) and spatial decay (for example algebraic or exponential decay of $\psi$ and of the induced potentials); (4) parameter dependence — determine how the normalized solutions depend on the coupling $e^{2}$ and the mass $m$, and whether in a small-coupling or semiclassical regime they converge to Choquard-type ground states, making rigorous the perturbative connection suggested by the known small-amplitude bifurcation results; and (5) dynamical stability — determine whether the normalized solitary waves are orbitally stable (or unstable) under the full Maxwell-Dirac time evolution, that is, whether initial data close to a normalized solitary wave remain, for all times, close to its orbit under phase rotations, translations, and Lorentz boosts. These objectives share one scientific aim: to establish whether the first known normalized Maxwell-Dirac solitary wave is a robust, classifiable object — the ground state of a well-posed constrained variational structure — rather than an isolated existence artifact. Results obtained after dropping the magnetic vector-potential coupling $\boldsymbol{A}$, replacing the kernel by a purely attractive Choquard nonlinearity, prescribing $\omega$ instead of the $L^{2}$ norm, or otherwise modifying the stated system address neighboring problems, not this one.

References

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