10 research outputs found
Coisotropic deformations of algebraic varieties and integrable systems
Coisotropic deformations of algebraic varieties are defined as those for
which an ideal of the deformed variety is a Poisson ideal. It is shown that
coisotropic deformations of sets of intersection points of plane quadrics,
cubics and space algebraic curves are governed, in particular, by the dKP,
WDVV, dVN, d2DTL equations and other integrable hydrodynamical type systems.
Particular attention is paid to the study of two- and three-dimensional
deformations of elliptic curves. Problem of an appropriate choice of Poisson
structure is discussed.Comment: 17 pages, no figure
Quantum deformations of associative algebras and integrable systems
Quantum deformations of the structure constants for a class of associative
noncommutative algebras are studied. It is shown that these deformations are
governed by the quantum central systems which has a geometrical meaning of
vanishing Riemann curvature tensor for Christoffel symbols identified with the
structure constants. A subclass of isoassociative quantum deformations is
described by the oriented associativity equation and, in particular, by the
WDVV equation. It is demonstrated that a wider class of weakly (non)associative
quantum deformations is connected with the integrable soliton equations too. In
particular, such deformations for the three-dimensional and
infinite-dimensional algebras are described by the Boussinesq equation and KP
hierarchy, respectively.Comment: Numeration of the formulas is correcte
Dispersionless integrable equations as coisotropic deformations. Extensions and reductions
Interpretation of dispersionless integrable hierarchies as equations of
coisotropic deformations for certain algebras and other algebraic structures
like Jordan triple systInterpretation of dispersionless integrable hierarchies
as equations of coisotropic deformations for certain algebras and other
algebraic structures like Jordan triple systems is discussed. Several
generalizations are considered. Stationary reductions of the dispersionless
integrable equations are shown to be connected with the dynamical systems on
the plane completely integrable on a fixed energy level. ems is discussed.
Several generalizations are considered. Stationary reductions of the
dispersionless integrable equations are shown to be connected with the
dynamical systems on the plane completely integrable on a fixed energy level.Comment: 21 pages, misprints correcte
Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
The paper is an inquiry of the algebraic foundations of the theory of
dispersionless integrable hierarchies, like the dispersionless KP and modified
KP hierarchies and the universal Whitham's hierarchy of genus zero. It stands
out for the idea of interpreting these hierarchies as equations of coisotropic
deformations for the structure constants of certain associative algebras. It
discusses the link between the structure constants and the Hirota's tau
function, and shows that the dispersionless Hirota's bilinear equations are,
within this approach, a way of writing the associativity conditions for the
structure constants in terms of the tau function. It also suggests a simple
interpretation of the algebro-geometric construction of the universal Whitham's
equations of genus zero due to Krichever.Comment: minor misprints correcte
Extension of Hereditary Symmetry Operators
Two models of candidates for hereditary symmetry operators are proposed and
thus many nonlinear systems of evolution equations possessing infinitely many
commutative symmetries may be generated. Some concrete structures of hereditary
symmetry operators are carefully analyzed on the base of the resulting general
conditions and several corresponding nonlinear systems are explicitly given out
as illustrative examples.Comment: 13 pages, LaTe
Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions
The usual dispersionless limit of the KP hierarchy does not work in the case
where the dependent variable has values in a noncommutative (e.g. matrix)
algebra. Passing over to the potential KP hierarchy, there is a corresponding
scaling limit in the noncommutative case, which turns out to be the hierarchy
of a `pseudodual chiral model' in 2+1 dimensions (`pseudodual' to a hierarchy
extending Ward's (modified) integrable chiral model). Applying the scaling
procedure to a method generating exact solutions of a matrix (potential) KP
hierarchy from solutions of a matrix linear heat hierarchy, leads to a
corresponding method that generates exact solutions of the matrix
dispersionless potential KP hierarchy, i.e. the pseudodual chiral model
hierarchy. We use this result to construct classes of exact solutions of the
su(m) pseudodual chiral model in 2+1 dimensions, including various multiple
lump configurations.Comment: 37 pages, 10 figures, 2nd version: some extensions (Fig 3, Appendix
A, additional references), 3rd version: some minor changes, additional
reference
A List of 1 + 1 Dimensional Integrable Equations and Their Properties
This paper contains a list of known integrable systems. It gives their recursion-, Hamiltonian-, symplectic- and cosymplectic operator, roots of their symmetries and their scaling symmetry