Hartree–Fock Theory for Pseudorelativistic Atoms

We study the Hartree–Fock model for pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudo-relativistic operator . We prove the existence of a Hartree–Fock minimizer, and prove regularity away from the nucleus and pointwise exponential decay of the corresponding orbitals. Submitted: August 1, 2007. Accepted: November 8, 2007.     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

Control of McKean–Vlasov dynamics versus mean field games

We discuss and compare two investigation methods for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which optimization and passage to the limit are performed. When optimizing first, the asymptotic problem is usually referred to as a mean-field game. Otherwise, it reads as an optimization problem over controlled dynamics of McKean–Vlasov type. Both problems lead to the analysis of forward–backward stochastic differential equations, the coefficients of which depend on the marginal distributions of the solutions. We explain the difference between the nature and solutions to the two approaches by investigating the corresponding forward–backward systems. General results are stated and specific examples are treated, especially when cost functionals are of linear-quadratic type.     

On mixed finite element approximations of shape gradients in shape optimization with the Navier–Stokes equation

For shape optimization of fluid flows governed by the Navier–Stokes equation, we investigate effectiveness of shape gradient algorithms by analyzing convergence and accuracy of mixed finite element approximations to both the distributed and boundary types of shape gradients. We present convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the distributed formulation has superconvergence property. Numerical results with comparisons are presented to verify theory and show that the shape gradient algorithm based on the distributed formulation is highly effective and robust for shape optimization.     

Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree–Fock equations: A sufficient condition

The multiconfiguration time-dependent Hartree–Fock (MCTDHF for short) system is an approximation of the linear many-particle Schrödinger equation with a binary interaction potential by nonlinear “one-particle” equations. MCTDHF methods are widely used for numerical calculations of the dynamics of few-electron systems in quantum physics and quantum chemistry, but the time-dependent case still poses serious open problems for the analysis, e.g. in the sense that global-in-time existence of solutions is not proved yet. In this letter we present the first result ever where global existence is proved under a condition on the initial datum that it has to be somewhat close to the “ground state”.     

Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.     

Kadomtsev–Petviashvili equation in relativistic fluid dynamics

The Kadomtsev–Petviashvili (KP) nonlinear wave equation is the three dimensional generalization of the Korteveg-de Vries (KdV) equation. We show how to obtain the cylindrical KP (cKP) and cartesian KP in relativistic fluid dynamics. The obtained KP equations describe the evolution of perturbations in the baryon density in a strongly interacting quark gluon plasma (sQGP) at zero temperature. We also show the analytical solitary wave solution of the KP equations in both cases.     

The Klein–Fock equation,proper-time formalism and symmetry of the hydrogen atom

In this talk we present main points of some of Fock papers which were not properly followed when published and review Fock's interpretation of the energy–time uncertainty relation, as well as his ideas on generalization of the concept of physical space published in his last paper.     

Mean field games with controlled jump–diffusion dynamics: Existence results and an illiquid interbank market model

We study a family of mean field games with a state variable evolving as a multivariate jump–diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and give conditions under which the optimal strategies are in fact Markovian, hence extending to a jump–diffusion setting previous results established in Lacker (2015). The proofs rely upon the notions of relaxed controls and martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results.     

Soliton dynamics for the nonlinear Schrödinger equation with magnetic field

The semiclassical regime of a nonlinear focusing Schrödinger equation in presence of non-constant electric and magnetic potentials V, A is studied by taking as initial datum the ground state solution of an associated autonomous stationary equation. The concentration curve of the solutions is a parameterization of the solutions of the second order ordinary equation \({\ddot x=-\nabla V(x)-\dot x\times B(x)}\), where \({B=\nabla\times A}\) is the magnetic field of a given magnetic potential A.     

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony-type equation with nonsmooth solutions

To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time-fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time-fractional Benjamin–Bona–Mahony-type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H¹-norm. Numerical examples are provided to justify the accuracy.     

Vortex dynamics in the presence of excess energy for the Landau–Lifshitz–Gilbert equation

We study the Landau–Lifshitz–Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for instantaneous changes in the strength of the gyrovector force due to bubbling events. The main tools are estimates of the Hodge decomposition of the supercurrent and an analysis of the defect measure of weak convergence of the stress energy tensor. Ginzburg–Landau equations with mixed dynamics in the presence of excess energy are also discussed.     

Simulation of collisionless ultrarelativistic electron–proton plasma dynamics in a self-consistent electromagnetic field

The evolution of a collisionless electron–proton plasma in the self-consistent approximation is investigated. The plasma is assumed to move initially as a whole in a vacuum with the Lorentz factor. The behavior of the dynamical system is analyzed by applying a three-dimensional model based on the Vlasov–Maxwell equations with allowance for retarded potentials. It is shown that the analysis of the solution to the problem is not valid in the “center-of-mass frame” of the plasmoid (since it cannot be correctly defined for a relativistic plasma interacting via an electromagnetic field) and the transition to a laboratory frame of reference is required. In the course of problem solving, a chaotic electromagnetic field is generated by the plasma particles. As a result, the particle distribution functions in the phase space change substantially and differ from their Maxwell–Juttner form. Computations show that the kinetic energies of the electron and proton components and the energy of the self-consistent electromagnetic field become identical. A tendency to the isotropization of the particle momentum distribution in the direction of the initial plasmoid motion is observed.     

Dark soliton dynamics under the complex Ginzburg–Landau equation

The dynamical properties of the complex Ginzburg–Landau equation are considered in the defocusing (normal dispersion) regime. It is found that under appropriate conditions stable evolution of dark solitons can occur. These conditions are derived using a newly developed perturbation theory that also reveals an important aspect of the dynamics: the formation of a shelf that accompanies the soliton and is an intricate part of its evolution. Further conditions to suppress this effect are also derived. These analytical predictions are found to be in excellent agreement with direct numerical simulations.     

Mean–Mean Multiple Comparison Displays for Families of Linear Contrasts

Traditional tabular and graphical displays of results of simultaneous confidence intervals or hypothesis tests are deficient in several respects. Expanding on earlier work, we present new mean–mean multiple comparison graphs that succinctly and compactly display the results of traditional procedures for multiple comparisons of population means or linear contrasts involving means. The MMC plot can be used with unbalanced, multifactor designs with covariates. After reviewing the construction of these displays in the S language (S-Plus and R), we demonstrate their application to four multiple comparison scenarios.     

Symmetry and Classification of the Dirac–Fock Equation

We consider the properties of the Dirac–Fock equation with differential operators of the first-order symmetry. For a relativistic particle in an electromagnetic field, we describe the covariant properties of the Dirac equation in an arbitrary Riemannian space V₄ with the signature (?1,?1,?1, 1). We present a general form of the differential operator with a first-order symmetry and characterize the pair of such commuting operators. We list the spaces where the free Dirac equation admits at least one differential operator with a first-order symmetry. We perform a symmetry classification of electromagnetic field tensors and construct complete sets of symmetry operators.     

Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation     

A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equations

In this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one-dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non-conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well-posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.