Reduction Groups and Automorphic Lie Algebras

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have a useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts.Comment: 21 pages, standard LaTeX2e, corrected typos, accepted for publication in CMP - Communications in Mathematical Physic

Sine-Gordon-like action for the Superstring in AdS(5) x S(5)

We propose an action for a sine-Gordon-like theory, which reproduces the classical equations of motion of the Green-Schwarz-Metsaev-Tseytlin superstring on AdS(5) x S(5). The action is relativistically invariant. It is a mass-deformed gauged WZW model for SO(4,1) x SO(5) / SO(4) x SO(4) interacting with fermions.Comment: 19 pages, LaTeX; v2: added discussion of zero modes in Section 3; v3: improved presentatio

Theory of the giant plasmon enhanced second harmonic generation in graphene and semiconductor two-dimensional electron systems

An analytical theory of the nonlinear electromagnetic response of a two-dimensional (2D) electron system in the second order in the electric field amplitude is developed. The second-order polarizability and the intensity of the second harmonic signal are calculated within the self-consistent-field approach both for semiconductor 2D electron systems and for graphene. The second harmonic generation in graphene is shown to be about two orders of magnitude stronger than in GaAs quantum wells at typical experimental parameters. Under the conditions of the 2D plasmon resonance the second harmonic radiation intensity is further increased by several orders of magnitude.Comment: 9 pages, 2 figure

Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks

This is the pre-print version of the article. The official published version can be obtained from the link below - Copyright @ 2011 Wiley-BlackwellSegregated direct boundary-domain integral equation (BDIE) systems associated with mixed, Dirichlet and Neumann boundary value problems (BVPs) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed for domains with interior cuts (cracks). The main results established in the paper are the BDIE equivalence to the original BVPs and invertibility of the BDIE operators in the corresponding Sobolev spaces.This work was supported by the International Joint Project Grant - 2005/R4 ”Boundary- Domain Integral Equations: Formulation, Analysis, Localisation” of the Royal Society, UK, and the grant ”Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients” of the EPSRC, UK

Nonlinear Abel type integral equation in modelling creep crack propagation

Copyright @ 2011 Birkhäuser BostonA nonlinear Abel-type equation is obtained in this paper to model creep crack time-dependent propagation in the infinite viscoelastic plane. A finite time when the integral equation solution becomes unbounded is obtained analytically as well as the equation parameters when solution blows up for all times. A modification to the Nyström method is introduced to numerically solve the equation and some computational results are presented

Gauge-invariant nonlocal quark condensates in QCD: a new interpretation of the lattice results

We study the asymptotic short-distance behaviour as well as the asymptotic large-distance behaviour of the gauge-invariant quark-antiquark nonlocal condensates in QCD. A comparison of some analytical results with the available lattice data is performed.Comment: Talk given at the ``XVIIth International Symposium on Lattice Field Theory'', Pisa (Italy), June 29th - July 3rd, 1999 (LATTICE 99); 3 pages, LaTeX file, uses ``espcrc2.sty''; a mistake in Eq. (17) corrected plus other minor change

Bäcklund transformations, energy shift and the plane wave limit

We discuss basic properties of the Bäcklund transformations for the classical string in AdS space in the context of the null-surface perturbation theory. We explain the relation between the Bäcklund transformations and the energy shift of the dual field theory state. We show that the Bäcklund transformations can be represented as a finite-time evolution generated by a special linear combination of the Pohlmeyer charges. This is a manifestation of the general property of Bäcklund transformations known as spectrality. We also discuss the plane wave limit

Boundary-domain integro-differential equation of elastic damage mechanics model of stationary drilling

Copyright @ 2005 EC Lt

Localized boundary-domain integral formulations for problems with variable coefficients

Specially constructed localized parametrixes are used in this paper instead of a fundamental solution to reduce a boundary value problem with variable coefficients to a localized boundary-domain integral or integro-differential equation (LBDIE or LBDIDE). After discretization, this results in a sparsely populated system of linear algebraic equations, which can be solved by well-known efficient methods. This make the method competitive with the finite element method for such problems. Some methods of the parametrix localization are discussed and the corresponding LBDIEs and LBDIDEs are introduced. Both mesh-based and meshless algorithms for the localized equations discretization are described

Analysis of extended boundary-domain integral and integro-differential equations of some variable-coefficient BVP

For a function from the Sobolev space H1(Ω) definitions of non-unique external and unique internal co-normal derivatives are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain Ω, where they are prescribed, to the domain boundary, where they are not. The notions are then applied to formulation and analysis of direct boundary-domain integral and integro-differential equations (BDIEs and BDIDEs) based on a specially constructed parametrix and associated with the Dirichlet boundary value problems for the "Laplace" linear differential equation with a variable coefficient and a rather general right hand side. The BDI(D)Es contain potential-type integral operators defined on the domain under consideration and acting on the unknown solution, as well as integral operators defined on the boundary and acting on the trace and/or co-normal derivative of the unknown solution or on an auxiliary function. Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP are investigated in appropriate Sobolev spaces