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.
We consider a Dirac operator with short range potential and with eigenvalues. We add a nonlinear term and we show that the small standing waves of the corresponding nonlinear Dirac equation (NLD) are attractors for small solutions of the NLD. This extends to the NLD results already known for the Nonlinear Schrödinger Equation (NLS)
The aim of this paper is to show the local well-posedness of 2 dimensional Dirac equations with power type and Hartree type nonlinearity derived from honeycomb structure in $H^s$ for $s>\frac78$ and $s>\frac38$, respectively. We also provide the smoothness failure of flows of Dirac equations.
For arbitrarily large initial data in an open set defined by an approximate Majorana condition, global existence and scattering results for solutions to the Dirac equation with Soler-type nonlinearity and the Dirac-Klein-Gordon system in critical spaces in spatial dimension three are established.
The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s > 1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.
Recently low-regularity behaviour of solutions to cubic Dirac equations with the Hartree-type nonlinearity has been extensively studied in somewhat a specific assumption on the structure of the nonlinearity. The key approach of previous results was to exploit the null structure in the nonlinearity and the decay of the Yukawa potential. In this paper, we aim to go beyond; we investigate the strong scattering property of cubic Dirac equations with quite a general class of the Hartree-type nonlinearity, which covers the Coulomb potential as well as the Yukawa potential, and the bilinear form, in …
In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long range potential as $|x|^{-1} $. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Ma…
We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential in two and three spatial dimensions. This paper improves our previous results \cite{chohlee,cholee}; we establish global well-posedness and scattering for large data with a certain condition. Firstly we investigate the long-time behavior of solutions to the Dirac equation satisfies good control provided that a particular dispersive norm of solutions is bounded. The key of our proof relies on modifying multilinear estimates obtained in our previous papers. Secondly, we obtain large data scatte…
We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_tψ= ic\hbar\sum_{k=1}^3α_k\partial_kψ- mc^2βψ- M(x)ψ+ f(|ψ|)ψ,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a bounded continuous potential function and the nonlinear term $f(|ψ|)ψ$ is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity $f(s)=…
In this article we will study the initial value problem for some Schrödinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schrödinger maps through the so called Hasimoto transformation.
We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.
We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by $V_b(x) = \frac1{4π}\frac{e^{-b|x|}}{|x|}\, (b \ge 0)$. In previous works, a standard argument is to utilise null form estimates in order to prove global well-posedness for $H^s$-data, $s>0$. However, the null structure inside the equations is not enough to attain the critical regularity. We impose an extra regularity assumption with respect to the angular variable. Firstly, we prove global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for $b>0$ for small $L^2_x$-d…
We apply the Fountain theorem to a class of nonlinear Dirac-Laplace equation on compact spin manifold. We show the standard Ambrosetti-Rabinowitz condition can be replaced by a more natural super-quadratic condition that is enough to obtain the Cerami condition under certain conditions. Multiple solutions of nonlinear Dirac-Laplace equation are obtained in this note.
We are interested in the cubic Dirac equation in two space dimensions. We establish the small data global existence and sharp pointwise decay results for general cubic nonlinearities without additional structure. We also prove the scattering of the Dirac equation for certain classes of nonlinearities. In all the above results we do not require the initial data to have compact support.
We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference $H=D_m+V$ has only two double eigenvalues and that degeneracies are due to a symmetry of $H$ (theorem of Kramers). In this case, we can build a small 4-dimensional manifold of stationary solutions tangent to the first eigenspace of $H$. Then we assume that a resonance condition holds and we build a center manifold of real codimension 8 around each stationary solution. Inside this center manifold any $H^{s}$ perturbation of stationary solutions, with $s>2$, stabi…
For a $n$-dimensional spin manifold $M$ with a fixed spin structure and a spinor bundle $ΣM$, we prove an $ε$-regularity theorem for weak solutions to the nonlinear Dirac equation of cubic nonlinearity. This, in particular, answers a regularity question raised by Chen-Jost-Wang when $n=2$.
We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy regularity and removable singularity theorems and energy identities for solutions.
We consider a nonlinear Schroedinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac type. The latter describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized near a Dirac point.
This paper studies the large time behavior of solution for a class of nonlinear massless Dirac equations in $R^{1+1}$. It is shown that the solution will tend to travelling wave solution when time tends to infinity.
We prove that for a Dirac operator with no resonance at thresholds nor eigenvalue at thresholds the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues close enough, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the…
This paper studies an initial boundary value problem for a class of nonlinear Dirac equations with cubic terms and moving boundary. For the initial data with bounded $L^2$ norm and the suitable boundary conditions, the global existence and the uniqueness of the strong solution are proved.
We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.
We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved by Cuccagna for the nonlinear Schrödinger equation, because of the strong indefiniteness of the energy.
This paper studies a class of nonlinear massless Dirac equations in one dimension, which include the equations for massless Thirring model and massless Gross-Neveu model. Under the assumptions that the initial data has small charge and is bounded, the global existence of the strong solution is established. The decay of the local charge is also proved.
In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation \[ i \sum_{j=0}^{3} \ga^j \pd_j ψ- mψ+ F(\barψψ)ψ=0 \] where $ m >0$ and $ F(s) = |s|^θ$ for $ 1\leq θ< 2.$ More precisely we show that there exists $\e_0 > 0$ such that for $ω\in(m - \e_0, m), $ there exists a solution $ ψ(t,x) = e^{-iωt}ϕ_ω(x), x_0 = t, x = (x_1, x_2, x_3),$ and the mapping from $ ω$ to $ ϕ_ω $ is continuous. We prove this result by relating the stationary solutions to the ground states of nonlinear Schrödinger equations.
We prove endpoint estimates with angular regularity for the wave and Dirac equations perturbed with a small potential. The estimates are applied to prove global existence for the cubic Dirac equation perturbed with a small potential, for small initial $H^{1}$ data with additional angular regularity. This implies in particular global existence in the critical energy space $H^{1}$ for small radial data.
We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.
This paper studies a class of nonlinear Dirac equations with cubic terms in $R^{1+1}$, which include the equations for the massive Thirring model and the massive Gross-Neveu model. Under the assumption that the initial data has bounded $L^2$ norm, the global existence and the uniqueness of the strong solution in $C([0,\infty),L^2(R^1))$ are proved.
We prove sharp pointwise decay estimates for critical Dirac equations on $\mathbb{R}^n$ with $n\geq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreover, we provide some classification results both for ground states and for excited states.
We present some results obtained in collaboration with prof. Piero D'Ancona concerning global existence for the 3D cubic non linear massless Dirac equation with a potential for small initial data in $H^1$ with slight additional assumptions. The new crucial tool is given by the proof of some refined endpoint Strichartz estimates.
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. 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.
We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the subcritical and the critical case, and then on the NLDE, highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit, proving the convergence of bound states.
The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr-Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in L^\infty_loc at least at the rate t^{-5/6}. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p=0,1 or 0<p<1. The proofs are based on a refined analysis of the Dirac propagator constructed in gr-qc…
In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the $L^2$-subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit.
We consider the spectral stability of solitary wave solutions ϕ(x)e^{-iωt} to the nonlinear Dirac equation in any dimension. This equation is well-known to theoretical physicists as the Soler model (or, in one dimension, the Gross-Neveu model), and attracted much attention for many years. We show that, generically, at the values of where the Vakhitov-Kolokolov stability criterion breaks down, a pair of real eigenvalues (one positive, one negative) appears from the origin, leading to the linear instability of corresponding solitary waves. As an auxiliary result, we state the virial identities (…
In the present paper, we study the semi-classical approximation of a Yukawa-coupled massive Dirac-Klein-Gordon system with some general nonlinear self-coupling. We prove that for a constrained coupling constant there exists a family of ground states of the semi-classical problem, for all $\hbar$ small, and show that the family concentrates around the maxima of the nonlinear potential as $\hbar\to0$. Our method is variational and relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.
We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin $s=\pm \frac{1}{2}$ components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay $fv^{-3/2-s}τ^{-5/2+s}$ as both an upper and a lower bound outside the black hole, with function $f$ finite and explicitly expressed in terms of the initial data and the coordinates. This establ…
This paper studies the quantum lattice Boltzmann scheme for the nonlinear Dirac equations for Gross-Neveu model in $1+1$ dimensions. The initial data for the scheme are assumed to be convergent in $L^2$. Then for any $T\ge 0$ the corresponding solutions for the quantum lattice Boltzmann scheme are shown to be convergent in $C([0,T];L^2(R^1))$ to the strong solution to the nonlinear Dirac equations as the mesh sizes converge to zero. In the proof, at first a Glimm type functional is introduced to establish the stability estimates for the difference between two solutions for the corresponding qu…
This paper studies a class of nonlinear Dirac equations with cubic terms in $R^{1+1}$, which include the equations for the massive Thirring model and the massive Gross-Neveu model. Under the assumptions that the initial data has small charge, the global existence of the solution in $H^1$ are proved. The proof is given by introducing some Bony functional to get the uniform estimates on the nonlinear terms and the uniform bounds on the local smooth solution, which enable us to extend the local solution globally in time. Then $L^2$-stability estimates for these solutions are also established by a…
We use the perturbation theory to build solitary wave solutions $ϕ_ω(x)e^{-iωt}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\barψψ)βψ$, with $f(τ)=|τ|^k+o(|τ|^k)$, $k>0$, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit $ω\lesssim m$; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schrödinger charge-critical, one has $Q'(ω)<0$ for $ω\lesssim m$, i…
We study the dispersive properties of the wave equation and the massless Dirac equation in three space dimensions, perturbed with electromagnetic potentials. The potentials are assumed to be small but may be rough. For both equations, we prove a dispersive estimate of the form |u(t,x)|< C/t. The constant C can be estimated in terms of a weighted H^s norm of the data, for suitable values of s. As a consequence of our method of proof, we establish the limiting absorption principle for the massless Dirac operator perturbed with a small, rough matrix potential.
We give a rigorous deduction of the eigenvalue problem of the nonlinear Schrödinger equation (NLS) at Dirac Points for potential of honeycomb lattice symmetry. Based on a bootstrap method, we observe the bifurcation of the eigenfunctions into eight distinct modes from the two-dimensional degenerated eigenspace of the regressive linear Schrödinger equation. We give the existence, the way of construction, uniqueness in $H^2$ space and the $C^\infty$ continuity of these eigenfunctions.
The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $ψ^{p-2}ψ$) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite $N$-star graphs, the existence of standing waves bifurcating from the trivial solution at $ω=mc^2$, for any $p>2$. In the Appendix we also discuss the nonrelativistic limit of the Dirac-Kirchhoff operator.
We prove, by a shooting method, the existence of infinitely many solutions of the form $ψ(x^0,x) = e^{-iΩx^0}χ(x)$ of the nonlinear Dirac equation {equation*} i\underset{μ=0}{\overset{3}{\sum}} γ^μ\partial_μψ- mψ- F(\barψψ)ψ= 0 {equation*} where $Ω>m>0,$ $χ$ is compactly supported and \[F(x) = \{{array}{ll} p|x|^{p-1} & \text{if} |x|>0 0 & \text{if} x=0 {array}.] with $p\in(0,1),$ under some restrictions on the parameters $p$ and $Ω.$ We study also the behavior of the solutions as $p$ tends to zero to establish the link between these equations and the M.I.T. bag model ones.
The present paper studies concentration phenomena of semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: \[ -i\hbarα\cdot\nabla w+aβw+V(x)w=g(|w|)w \,. \] Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are $k$ bounded domains $Λ_j\subset\mathbb{R}^3$ such that $-a<\min_{Λ_j} V=V(x_j)<\min_{\partia…
