Bosonic symmetries of the massless Dirac equation

The results of spin 1 symmetries of massless Dirac equation [21] are proved completely in the space of 4-component Dirac spinors on the basis of unitary operator in this space connecting this equation with the Maxwell equations containing gradient-like sources. Nonlocal representations of conformal group are found, which generate the transformations leaving the massless Dirac equation being invariant. The Maxwell equations with gradient-like sources are proved to be invariant with respect to fermionic representations of Poincaré and conformal groups and to be the kind of Maxwell equations with maximally symmetrical properties. Brief consideration of an application of these equations in physics is discussed.     

Fourth-order schemes for the wave equation,Maxwell equations,and linearized elastodynamic equations

With simple finite-difference operators we construct fourth-order schemes in space and in time for the wave equation, Maxwell equations, and linearized elastodynamic equations using the modified equation approach. The schemes remain stable for arbitrary heterogeneous mediums because the relevant difference operators are always positive definite. We also present some dispersion curves to show the accuracy of the schemes. © 1994 John Wiley & Sons, Inc.     

Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds

We study some basic analytical problems for nonlinear Dirac equations involving critical Sobolev exponents on compact spin manifolds. Their solutions are obtained as critical points of certain strongly indefinite functionals defined on H1/2-spinors with critical growth. We prove the existence of a non-trivial solution for the Brezis-Nirenberg type problem when the dimension m of the manifold is larger than 3. We also prove a global compactness result for the associated Palais-Smale sequences and the regularity of -weak solutions.     

Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential

We study the effect of spatially-dependent mass function over the solution of the Dirac equation with the Coulomb potential in the (3+1)-dimensions for any arbitrary spin-orbit κ state. In the framework of the spin and pseudospin symmetry concept, the analytic bound state energy eigenvalues and the corresponding upper and lower two-component spinors of the two Dirac particles are obtained by means of the Nikiforov-Uvarov method, in closed form. This physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the Dirac equation. The special cases , the constant mass and the non-relativistic limits are briefly investigated.     

Simple connection between the geometric and the Hamiltonian representations of integrable nonlinear equations

One gives a simple and general derivation of the well-known connection between the geometric and the Hamiltonian approaches in the classical method of the inverse problem. Namely, for the case of a two-dimensional auxiliary problem and periodic boundary conditions it is explicitly shown how the existence of the classical -matrix for the integrable equations leads to their representation in the form of the condition of zero curvature.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 264–273, 1982.     

Bäcklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrable equations

P. N. Lebedev Physics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 1, pp. 151–154, April, 1989.     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Let be a connected semisimple Lie group with finite center. Let be the maximal compact subgroup of corresponding to a fixed Cartan involution . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary -module contains a -type with highest weight , then has infinitesimal character . Here is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary -modules with non-zero Dirac cohomology, provided has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant's cubic Dirac operator.

     

Reduction and quotient equations for differential equations with symmetries

The method of reduction previously known in the theory of Hamiltonian systems with symmetries is developed in order to obtain exact group-invariant solutions of systems of partial differential equations. This method leads to representations of quotient equations which are very convenient for the systematic analysis of invariant solutions of boundary value problems. In the case of partially invariant solutions, necessary and sufficient conditions of their invariance with respect to subalgebras of symmetry algebras are given. The concept of partial symmetries of differential equations is considered.     

Solutions of Dirac equations on compact spin manifolds via saddle point reduction

In this note, we apply the saddle point reduction to a class of nonlinear Dirac equation with a potential on a compact spin manifold. Two results are obtained, including a multiple theorem.     

Wave and Dirac operators,and representations of the conformal group

Let M be the flat Minkowski space. The solutions of the wave equation, the Dirac equations, the Maxwell equations, or more generally the mass 0, spin s equations are invariant under a multiplier representation U_s, of the conformal group. We provide the space of distributions solutions of the mass 0, spin s equations with a Hilbert space structure H_s, such that the representation U_s, will act unitarily on H_s. We prove that the mass 0 equations give intertwining operators between representations of principal series. We relate these representations to the Segal-Shale-Weil (or “ladder”) representation of U(2, 2).     

WKB method for the Dirac equation with a scalar-vector coupling

We outline a recursive method for obtaining WKB expansions of solutions of the Dirac equation in an external centrally symmetric field with a scalar-vector Lorentz structure of the interaction potentials. We obtain semiclassical formulas for radial functions in the classically allowed and forbidden regions and find conditions for matching them in passing through the turning points. We generalize the Bohr-Sommerfeld quantization rule to the relativistic case where a spin-1/2 particle interacts simultaneously with a scalar and an electrostatic external field. We obtain a general expression in the semiclassical approximation for the width of quasistationary levels, which was earlier known only for barrier-type electrostatic potentials (the Gamow formula). We show that the obtained quantization rule exactly produces the energy spectrum for Coulomb- and oscillatory-type potentials. We use an example of the funnel potential to demonstrate that the proposed version of the WKB method not only extends the possibilities for studying the spectrum of energies and wave functions analytically but also ensures an appropriate accuracy of calculations even for states with n_r 1.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 83–111, April, 2005.     

Soliton states of Maxwell’s equations and nonlinear Schrödinger equation

Similarities and fundamental differences between Maxwell’s equations and nonlinear Schrödinger equation in predicting a soliton evolution in a uniform nonlinear anisotropic medium are analyzed. It is found that in some cases, the soliton solutions to the nonlinear Schrödinger equation cannot be recovered from Maxwell’s equations while in others the soliton solutions to Maxwell’s equations are lost from the nonlinear Schrödinger equation through approximation, although there are cases where the soliton solutions to the two sets of the equations demonstrate only quantitative difference. The origin of the differences is also discussed.