69 research outputs found
Computer modeling of structure and calculation of the short-range order parameters disordered two-layers AB graphene
Short-range order and electronic properties of epitaxial graphene
One of the most rapidly developing areas of modern materials science is the study of graphene and materials on its basis. The experimental investigations have revealed different types of defects on the surface of graphene that form the ordered structures of atomic configurations. In the present work, the value of short-range order parameter for different configurations of foreign atoms in a graphene layer was calculated. The effect of various factors on the density of electronic states and electrical resistance in graphene was also investigated. The type of the ordering of foreign atoms in graphene rather than the concentration of impurities, was shown to be responsible for the change in the conductivity of graphene
Desargues maps and the Hirota-Miwa equation
We study the Desargues maps \phi:\ZZ^N\to\PP^M, which generate lattices
whose points are collinear with all their nearest (in positive directions)
neighbours. The multidimensional compatibility of the map is equivalent to the
Desargues theorem and its higher-dimensional generalizations. The nonlinear
counterpart of the map is the non-commutative (in general) Hirota--Miwa system.
In the commutative case of the complex field we apply the nonlocal
-dressing method to construct Desargues maps and the
corresponding solutions of the equation. In particular, we identify the
Fredholm determinant of the integral equation inverting the nonlocal
-dressing problem with the -function. Finally, we establish
equivalence between the Desargues maps and quadrilateral lattices provided we
take into consideration also their Laplace transforms.Comment: 17 pages, 5 figures; v2 - presentation improve
Hyperdeterminants as integrable discrete systems
We give the basic definitions and some theoretical results about
hyperdeterminants, introduced by A. Cayley in 1845. We prove integrability
(understood as 4d-consistency) of a nonlinear difference equation defined by
the 2x2x2-hyperdeterminant. This result gives rise to the following hypothesis:
the difference equations defined by hyperdeterminants of any size are
integrable.
We show that this hypothesis already fails in the case of the
2x2x2x2-hyperdeterminant.Comment: Standard LaTeX, 11 pages. v2: corrected a small misprint in the
abstrac
Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation
The BT of Adler's lattice equation is inherent in the equation itself by
virtue of its multidimensional consistency. We refer to a solution of the
equation that is related to itself by the composition of two BTs (with
different Backlund parameters) as a 2-cycle of the BT. In this article we will
show that such solutions are associated with a commuting one-parameter family
of rank-2 (i.e., 2-variable), 2-valued mappings. We will construct the explicit
solution of the mappings within this family and hence give the solutions of
Adler's equation that are 2-cycles of the BT.Comment: 10 pages, contribution to the NEEDS 2007 proceeding
Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings
Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model
(GN2), and its chiral cousin, the NJL2 model, have shown that there are phases
with inhomogeneous crystalline condensates. These (static) condensates can be
found analytically because the relevant Hartree-Fock and gap equations can be
reduced to the nonlinear Schr\"odinger equation, whose deformations are
governed by the mKdV and AKNS integrable hierarchies, respectively. Recently,
Thies et al have shown that time-dependent Hartree-Fock solutions describing
baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation,
and can be mapped directly to classical string solutions in AdS3. Here we
propose a geometric perspective for this result, based on the generalized
Weierstrass spinor representation for the embedding of 2d surfaces into 3d
spaces, which explains why these well-known integrable systems underlie these
various Gross-Neveu gap equations, and why there should be a connection to
classical string theory solutions. This geometric viewpoint may be useful for
higher dimensional models, where the relevant integrable hierarchies include
the Davey-Stewartson and Novikov-Veselov systems.Comment: 27 pages, 1 figur
Towards the theory of integrable hyperbolic equations of third order
The examples are considered of integrable hyperbolic equations of third order
with two independent variables. In particular, an equation is found which
admits as evolutionary symmetries the Krichever--Novikov equation and the
modified Landau--Lifshitz system. The problem of choice of dynamical variables
for the hyperbolic equations is discussed.Comment: 22
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