Search

Improved uniform error bounds on time-splitting methods are rigorously proven for the long-time dynamics of the weakly nonlinear Dirac equation (NLDE), where the nonlinearity strength is characterized by a dimensionless parameter |$\varepsilon \in (0, 1]$|⁠. We adopt a second-order Strang splitting method to discretize the NLDE in time, and combine with the Fourier pseudospectral method in space for the full-discretization. By employing the regularity compensation oscillation (RCO) technique, where the high frequency modes are controlled by the regularity of the exact solution, and the low frequency modes are analyzed by phase cancellation and energy method, we establish improved uniform error bounds at |$O(\varepsilon ^{2}\tau ^{2})$| and |$O(h^{m-1}+ \varepsilon ^{2}\tau ^{2})$| for the second-order Strang splitting semidiscretization and full-discretization up to the long-time |$T_{\varepsilon } = T/\varepsilon ^{2}$| with |$T>0$| fixed, respectively. Furthermore, the numerical scheme and error estimates are extended to an oscillatory NLDE, which propagates waves with |$O(\varepsilon ^{2})$| wavelength in time and at |$O(\varepsilon ^{-2})$| wave speed in space. Finally, numerical examples verifying our analytical results are given. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and multiplicity of solutions for the following class of nonlinear Dirac equations [Display omitted] where V : R 3 → R and f : [ 0 , + ∞) → R are continuous functions. It is proved that the number of solutions is at least the number of global minimum points of V when ε is small enough. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $ \mathbf{SU}(1,1) $ symmetry: the first preserving and the second breaking the parity symmetry. We show that a particular perturbation which breaks the $ \mathbf{SU}(1,1) $ symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a particular perturbation which breaks both the $ \mathbf{SU}(1,1) $ symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. This instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $ \pm 2\omega \mathrm{i} $. [ABSTRACT FROM AUTHOR]

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of O ε - 2 , where 0 < ε ≪ 1 is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of O ε 2 by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the explicit exponential midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy significantly. [ABSTRACT FROM AUTHOR]

We construct two SAV/CN–Hermite schemes for one and two dimensional nonlinear Dirac equation in this paper. Both SAV/CN methods preserve the energy and mass and an efficient and accurate scheme is proposed by Hermite approximation. The numerical analysis of error estimates is established on SAV/CN scheme. Numerical experiments confirm the expected convergence order. In addition, they are given to show the simulations of binary and ternary collisions. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear Dirac equation - i ħ ∂ t ψ = i c ħ ∑ k = 1 3 α k ∂ k ψ - m c 2 β ψ + K (x) | ψ | p - 2 ψ , (N D E) where ψ : R × R 3 → C 4 , K ∈ L ∞ (R 3) , c denotes the speed of light, m > 0 , the mass of the electron, ħ is Planck's constant, ∂ k = ∂ ∂ x k , α 1 , α 2 , α 3 , β are 4 × 4 Pauli–Dirac matrices and p ∈ (2 , 8 3) . We present a new approach which is based on a perturbation argument and show the existence of L 2 -normalized solitary wave solutions of (NDE) using variational methods. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to look for solutions of the following critical nonlinear Dirac equation - i ε α · ∇ u + a β u + V (x) u = K (x) | u | p - 2 u + Q (x) | u | u x ∈ R 3 , where ε > 0 is a small parameter, a > 0 is a constant, p ∈ (5 / 2 , 3) , α = (α 1 , α 2 , α 3) is triplets of matrices, α 1 , α 2 , α 3 and β are 4 × 4 Pauli-Dirac matrices. The potential V(x) may attain ± a at somewhere or at infinity, K , Q ∈ C 1 (R 3 , R +) are two functions. When ε > 0 small, we will prove the existence and concentration of the solutions by using variational methods under some mild assumptions on the potentials V, K and Q. [ABSTRACT FROM AUTHOR]

We study the properties of the mass gap for monopole solutions in SU(2) Yang-Mills theory with a source of the non-Abelian gauge field in the form of a spinor field described by the nonlinear Dirac equation. Different types of nonlinearities parameterized by the parameter λ are under consideration. It is shown that for the range of values of this parameter studied in the present paper the value of the mass gap depends monotonically on this parameter, and the position of the mass gap does not practically depend on λ. In the present paper, we study the dependence of the size of a mass gap, its position, etc., on the value of a parameter describing the nonlinear self-interaction potential of the spinor field. By the term a position of the mass gap, we mean the value of the frequency ẼΔ, for which the energy spectrum has a minimum. It is of interest to note that the position of the mass gap ẼΔ, does not practically depend on the nonlinearity parameter λ. One can assume that in fact this is the case, and deviations from this value are related to errors in numerical calculations. It is of interest that the position of the mass gap does not practically depend on the value of the nonlinearity parameter, at least in the range of values of the spinor field nonlinearity parameter considered here. [ABSTRACT FROM AUTHOR]

In this paper, we propose a time fourth-order exponential wave integrator (EWI) Fourier pseudo-spectral method for solving the nonlinear Dirac equation with periodic boundary value conditions. The new numerical method is designed by using Fourier pseudo-spectral method for spatial derivatives combined with a new fourth-order EWI for temporal derivatives. We give rigourously error analysis and establish error bounds for the numerical solutions without any CFL-type condition constraint. In more details, the proposed method has the fourth-order temporal accuracy and spectral spatial accuracy, respectively, in general H m -norm. Extensive numerical experiments are reported to confirm the theoretical analysis and show the superiority of the new method. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new exponential integrator Fourier pseudo-spectral schemes for nonlinear Dirac (NLD) equation. The proposed schemes are time symmetric, unconditionally stable and preserve the total energy in the discrete level. We give rigorously error analysis and establish error bounds in the general H m -norm for the numerical solutions of the new schemes applied to the NLD equation. In more details, the proposed schemes have the second-order temporal accuracy and spectral spatial accuracy, respectively, without any CFL-type condition constraint. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity to bound the numerical approximate solutions or the mathematical induction. Extensive numerical results are reported to confirm our error bounds and theoretical analysis. [ABSTRACT FROM AUTHOR]

We address the bound-state dynamics of relativistic spin-1/2 neutral particles (in this paper, Dirac neutrinos) with anomalous magnetic dipole moment in the presence of an electromagnetic (EM) field described by a generalized Dirac–Pauli equation. This equation of motion is derived including appropriate couplings between Lorentz scalar and pseudoscalar fields with the EM field in the Lagrangian of the system. Specifically, we exactly solve the bound-state problem of neutrinos in the presence of a homogeneous magnetic field in cylindrical coordinates. We comment on the relevance of this approach to study Dirac neutrino self-interactions. [ABSTRACT FROM AUTHOR]

We address the bound-state dynamics of a neutron with anomalous magnetic dipole moment in the presence of electromagnetic (EM) fields described by a generalized Dirac–Pauli equation. This generalization consists of including appropriate couplings between Lorentz scalar and pseudoscalar fields with EM fields in the Lagrangian of the system. We exactly solve two single-particle problems: first, a Hydrogen-like system; second, a relativistic Schrödinger-like equation for a linear confining potential. We comment on the relevance of this approach to explore fermion (e.g. neutron) self-interactions as solitonic models of the neutron. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonrelativistic limit and some properties of solutions for the following nonlinear Dirac equation - i c ∑ k = 1 3 α k ∂ k ψ + m c 2 β ψ - ω ψ = g (| ψ |) ψ , in R 3 where c denotes the speed of light, m > 0 is the mass of the electron. We show that solutions of nonlinear Dirac equation converge to the corresponding solutions of a coupled system of nonlinear Schrödinger equations as the speed of light tends to infinity for electrons with small mass. Moreover, we also prove the uniform boundedness and the exponential decay properties of the solutions for the nonlinear Dirac equation with respect to the speed of light c. [ABSTRACT FROM AUTHOR]

This paper aims to study the existence and concentration of solutions for the stationary Dirac equation in R 3 with critical nonlinearities: i ε ∑ k = 1 3 α k ∂ k u - a β u + V (x) u = P (x) f (| u |) u + Q (x) g (| u |) u , where ε > 0 is a small parameter and a > 0 is a constant. We also show the semiclassical solutions ω ε with maximum points x ε concentrating at a special set H P characterized by V(x), P(x) and Q(x), and for any sequence, x ε → x 0 ∈ H P , v ε (x) : = ω ε (ε x + x ε) converges in H 1 (R 3 , C 4) to a least energy solution u of i ∑ k = 1 3 α k ∂ k u - a β u + V (x 0) u = P (x 0) f (| u |) u + Q (x 0) g (| u |) u. [ABSTRACT FROM AUTHOR]

Given any and , we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation on , , that do not vanish, i.e. for all and all sufficiently small. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from [Commun. Contemp. Math. 19 (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze two conservative fourth-order compact finite difference schemes for the (1+1) dimensional nonlinear Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise forms of the proposed compact schemes into equivalent vector forms and analyze their conservative and convergence properties. We prove that the proposed schemes preserve the total mass and energy in the discrete level and the convergence rate of the schemes, without any restrictions on the grid ratio, are at the order of O (h 4 + τ 2) in l ∞ -norm, where h and τ are spatial and temporal steps, respectively. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. The numerical experiments are carried out to confirm our theoretical analysis. The research method in this paper can be easily extended to higher order compact schemes or other types of wave equations. [ABSTRACT FROM AUTHOR]

Superresolution of the Lie--Trotter splitting (S1) and Strang splitting (S2) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrel-ativistic regime with a small parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this regime, the solution highly oscillates in time with wavelength at O(ε²). The splitting methods surprisingly show superresolution, i.e., the methods can capture the solution accurately even if the time step size τ is much larger than the sampled wavelength at O(ε²). Similar to the linear case, S1 and S2 both exhibit 1/2 order convergence uniformly with respect to e. Moreover, if τ is nonreso-nant, i.e., τ is away from a certain region determined by ε, S1 would yield an improved uniform first order O(τ) error bound, while S2 would give improved uniform 3/2 order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that superresolution is still valid for higher order splitting methods. [ABSTRACT FROM AUTHOR]

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 = m c 2 , for any p > 2. In the Appendix we also discuss the nonrelativistic limit of the Dirac-Kirchhoff operator. [ABSTRACT FROM AUTHOR]

The possibility of introducing a nonlinear correction to the Dirac equation for graphene in order to adequately describe collective electronic phenomena is considered. In contrast to the other papers on this topic the interaction term includes the sum of the spinor components' squares instead of their difference. Particular attention is paid to the equality of the spatial coordinates. We investigate the properties of the obtained nonlinear equation, in order to describe the high-temperature ferromagnetism in graphene without making any assumptions on the key role of defects of the structure in providing the effect. The numerical simulation is carried out for the simple boundary and initial conditions with the Lax–Friedrichs scheme as a result of which information is obtained on the dynamics of the electron density for a number of the simplest initial and boundary conditions. [ABSTRACT FROM AUTHOR]

In this work, we consider the numerical integration of the nonlinear Dirac equation and the Dirac-Poisson system (NDEs) under rough initial data. We propose an ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in Hr for solutions in Hr, i.e., without requiring any additional regularity on the solution. In contrast to classical methods, a ULI overcomes the numerical loss of derivatives and is therefore more efficient and accurate for approximating low regular solutions. Convergence theorems and the extension of a ULI to second order are established. Numerical experiments confirm the theoretical results and underline the favourable error behaviour of the new method at low regularity compared to classical integration schemes. [ABSTRACT FROM AUTHOR]

Monopole solutions in SU(2) Yang-Mills theory which includes spinor fields described by the nonlinear Dirac equation are obtained. It is demonstrated that the energy spectrum of such a system possesses a global minimum whose appearance is brought about solely by the nonlinear spinor fields. It is shown that the monopole solution obtained differs in principle from the 't Hooft-Polyakov monopole in that it is topologically trivial.

The fine physical details for the quantum computation can easily be provided by high-order conservative schemes. By compensating the high-order difference operator (D μ p − 1) δ to the central difference operator δ, a time-dependent semi-discrete system, which conserves both charge and energy, is derived for the two-dimensional nonlinear Dirac equation (NLDE). Two kinds of fully discrete schemes are obtained by discretizing this semi-discrete system in time with the time-midpoint method and the time-splitting method. We prove theoretically that the former one conserves both charge and energy while the latter one only keeps the charge conservation. Some numerical experiments are given to verify the accuracy order, the stability and the conservative properties. In addition, the dynamics of the NLDE in one dimension and two dimensions are simulated. [ABSTRACT FROM AUTHOR]

Solitary waves of nonlinear Dirac, Maxwell–Dirac and Klein–Gordon–Dirac equations are considered. We deduce some virial identities and check that the energy-momentum relation for solitary waves coincides with the Einstein energy-momentum relation for point particles. [ABSTRACT FROM AUTHOR]

This paper studies an initial boundary value problem for a class of nonlinear Dirac equations with cubic terms, which include the equations for the massive Thirring model and the massive Gross-Neveu model. Under the assumptions that the initial data has bounded L 2 norm and the boundary satisfies suitable conditions, the global existence and the uniqueness of the strong solution are proved. [ABSTRACT FROM AUTHOR]

This article concerns the periodic solutions for a nonlinear Dirac equation. Under suitable assumptions on the nonlinearity, we show the existence of nontrivial and ground state solutions.

In this work, a fourth-order in space and second-order in time compact scheme, a sixth-order in space and second-order in time compact scheme and two linearized compact schemes are proposed for the (1+1)-dimensional nonlinear Dirac equation. The iterative algorithm is used to compute the nonlinear algebraic system and the Thomas algorithm in the matrix form is adopted to enhance the computational efficiency. It is proved that all of the schemes are unconditionally stable in the linear sense. Numerical experiments are given to test the accuracy order of the presented schemes, record the error history for all of the schemes with respect to t, discuss the conservation laws of discrete charge and energy from the numerical point of view, study the stability of the solitary waves by adding a small random perturbation to the initial data, and simulate the collision of two and three solitary waves. [ABSTRACT FROM AUTHOR]

We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) and the Dirac--Klein--Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing. We show the relation of ±2wi eigenvalues of the linearization at a solitary wave, Bogoliubov SU(1; 1) symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves. [ABSTRACT FROM AUTHOR]

A high-order accuracy time discretization method is developed in this paper to solve the one-dimensional nonlinear Dirac (NLD) equation. Based on the implicit integration factor (IIF) method, two schemes are proposed. Central differences are applied to the spatial discretization. The semi-discrete scheme keeps the conservation of the charge and energy. For the temporal discretization, second-order IIF method and fourth-order IIF method are applied respectively to the nonlinear system arising from the spatial discretization. Numerical experiments are given to validate the accuracy of these schemes and to discuss the interaction dynamics of the NLD solitary waves. [ABSTRACT FROM AUTHOR]

Recently there has occurred a controversy between the semi-analytical prediction of linear stability of the soliton of the massive Gross–Neveu model and direct numerical observations of its instability for small values of the frequency. We revisit the problem of numerical computation of this soliton, find a mechanism behind the numerical instability observed in earlier studies, and propose methods to stably compute the soliton over long times. Thus, we confirm the semi-analytical prediction of the soliton's being linearly stable. [ABSTRACT FROM AUTHOR]

We use the perturbation theory to build solitary wave solutions φω(x)e-iωt to the nonlinear Dirac equation in ℝn, n ≥ 1, with the Soler-type nonlinear term f(ψ*βψ)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 ≲ 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 (k = 2/n), then one has Q'(ω) < 0 for ω ≲ m with Q(ω) being the charge of the corresponding solitary wave; this implies the absence of the degeneracy of zero eigenvalue of the linearization at this solitary wave. [ABSTRACT FROM AUTHOR]

In this work, four numerical time-splitting methods are proposed for the (1 + 1)-dimensional nonlinear Dirac equation. All of these methods (or schemes) are proved to satisfy the charge conservation in the discrete level. To enhance the computation efficiency, the block Thomas algorithm is adopted. Numerical experiments are given to test the accuracy order for these schemes, to simulate numerically the binary collision including two standing waves and two moving solitons, meanwhile, the dynamic properties for the nonlinear Dirac equation are discussed. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1582-1602, 2017 [ABSTRACT FROM AUTHOR]

We define and study the Cauchy problem for a one-dimensional (1-D) nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy-Šeba models and the concentrated versions of the Bragg resonance and 1-D Soler (also known as massive Gross-Neveu) type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable. [ABSTRACT FROM AUTHOR]

We consider the coupled nonlinear Dirac equations (NLDEs) in 1 + 1 dimensions with scalar–scalar self-interactions g 1 2 2 ( ψ ¯ ψ ) 2 + g 2 2 2 ( ϕ ¯ ϕ ) 2 + g 3 2 ( ψ ¯ ψ ) ( ϕ ¯ ϕ ) as well as vector–vector interactions of the form g 1 2 2 ( ψ ¯ γ μ ψ ) ( ψ ¯ γ μ ψ ) + g 2 2 2 ( ϕ ¯ γ μ ϕ ) ( ϕ ¯ γ μ ϕ ) + g 3 2 ( ψ ¯ γ μ ψ ) ( ϕ ¯ γ μ ϕ ) . Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ = e − i ω 1 t { R 1 cos ⁡ θ , R 1 sin ⁡ θ } , ϕ = e − i ω 2 t { R 2 cos ⁡ η , R 2 sin ⁡ η } , and assuming that θ ( x ) , η ( x ) have the same functional form they had when g 3 = 0 , which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for R i ( x ) which are valid for small values of g 3 2 / g 2 2 and g 3 2 / g 1 2 . In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ± ∞ . [ABSTRACT FROM AUTHOR]

For the nonlinear Dirac equation in ( 1 + 1 ) D with scalar self-interaction (Gross–Neveu model), with quintic and higher order nonlinearities (and within certain range of the parameters), we prove that solitary wave solutions are asymptotically stable in the “even” subspace of perturbations (to ignore translations and eigenvalues ± 2 ω i ). The asymptotic stability is proved for initial data in H 1 . The approach is based on the spectral information about the linearization at solitary waves which we justify by numerical simulations. For the proof, we develop the spectral theory for the linearized operators and obtain appropriate estimates in mixed Lebesgue spaces, with and without weights. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Dirac equation with a power nonlinearity where , . We prove the existence of the scattering operator in the neighborhood of the origin of the weighted Sobolev space , where . [ABSTRACT FROM AUTHOR]