322 research outputs found
On the deformation theory of structure constants for associative algebras
Algebraic scheme for constructing deformations of structure constants for
associative algebras generated by a deformation driving algebras (DDAs) is
discussed. An ideal of left divisors of zero plays a central role in this
construction. Deformations of associative three-dimensional algebras with the
DDA being a three-dimensional Lie algebra and their connection with integrable
systems are studied.Comment: minor corrections and references adde
Confluence of hypergeometric functions and integrable hydrodynamic type systems
It is known that a large class of integrable hydrodynamic type systems can be
constructed through the Lauricella function, a generalization of the classical
Gauss hypergeometric function. In this paper, we construct novel class of
integrable hydrodynamic type systems which govern the dynamics of critical
points of confluent Lauricella type functions defined on finite dimensional
Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent
functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is
also shown that in general, hydrodynamic type system associated to the
confluent Lauricella function is given by an integrable and non-diagonalizable
quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5)
for two component systems and Gr(2,6) for three component systems are
considered in details.Comment: 22 pages, PMNP 2015, added some comments and reference
Nonlinear Dynamics on the Plane and Integrable Hierarchies of Infinitesimal Deformations
A class of nonlinear problems on the plane, described by nonlinear
inhomogeneous -equations, is considered. It is shown that the
corresponding dynamics, generated by deformations of inhomogeneous terms
(sources) is described by Hamilton-Jacobi type equations associated with
hierarchies of dispersionless integrable systems. These hierarchies are
constructed by applying the quasiclassical -dressing method.Comment: 30 pages, tcilate
Tropical Limit in Statistical Physics
Tropical limit for macroscopic systems in equilibrium defined as the formal
limit of Boltzmann constant k going to 0 is discussed. It is shown that such
tropical limit is well-adapted to analyse properties of systems with highly
degenerated energy levels, particularly of frustrated systems like spin ice and
spin glasses. Tropical free energy is a piecewise linear function of
temperature, tropical entropy is a piecewise constant function and the system
has energy for which tropical Gibbs' probability has maximum. Properties of
systems in the points of jump of entropy are studied. Systems with finite and
infinitely many energy levels and phenomena of limiting temperatures are
discussed.Comment: 16 pages, 6 figure
Cohomological, Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of Sato Grassmannian
Cohomological and Poisson structures associated with the special tautological
subbundles for the Birkhoff strata of Sato Grassmannian
are considered. It is shown that the tangent bundles of
are isomorphic to the linear spaces of coboundaries with vanishing
Harrison's cohomology modules. Special class of 2-coboundaries is provided by
the systems of integrable quasilinear PDEs. For the big cell it is the dKP
hierarchy. It is demonstrated also that the families of ideals for algebraic
varieties in can be viewed as the Poisson ideals. This
observation establishes a connection between families of algebraic curves in
and coisotropic deformations of such curves of zero and
nonzero genus described by hierarchies of hydrodynamical type systems like dKP
hierarchy. Interrelation between cohomological and Poisson structures is noted.Comment: 15 pages, no figures, accepted in Theoretical and Mathematical
Physics. arXiv admin note: text overlap with arXiv:1005.205
Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations
We obtain new gauge-invariant forms of two-dimensional integrable systems of
nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the
generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov
system. We show how these forms imply both new and well-known two-dimensional
integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt
equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and
modified Nizhnik-Veselov-Novikov equation. We consider Miura-type
transformations between nonlinear equations in different gauges.Comment: Talk given at the Workshop "Nonlinear Physics: Theory and Experiment.
V", Gallipoli (Lecce, Italy), 12-21 June, 200
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