178 research outputs found
Hamiltonian motions of plane curves and formation of singularities and bubbles
A class of Hamiltonian deformations of plane curves is defined and studied.
Hamiltonian deformations of conics and cubics are considered as illustrative
examples. These deformations are described by systems of hydrodynamical type
equations. It is shown that solutions of these systems describe processes of
formation of singularities (cusps, nodes), bubbles, and change of genus of a
curve.Comment: 15 pages, 12 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
Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems
Structure and properties of families of critical points for classes of
functions obeying the elliptic Euler-Poisson-Darboux equation
are studied. General variational and differential equations
governing the dependence of critical points in variational (deformation)
parameters are found. Explicit examples of the corresponding integrable
quasi-linear differential systems and hierarchies are presented There are the
extended dispersionless Toda/nonlinear Schr\"{o}dinger hierarchies, the
"inverse" hierarchy and equations associated with the real-analytic Eisenstein
series among them. Specific bi-Hamiltonian
structure of these equations is also discussed.Comment: 18 pages, no figure
Birkhoff strata of Sato Grassmannian and algebraic curves
Algebraic and geometric structures associated with Birkhoff strata of Sato
Grassmannian are analyzed. It is shown that each Birkhoff stratum
contains a subset of points for which each fiber of the
corresponding tautological subbundle is closed with respect to
multiplication. Algebraically is an infinite family of
infinite-dimensional commutative associative algebras and geometrically it is
an infinite tower of families of algebraic curves. For the big cell the
subbundle represents the tower of families of normal
rational (Veronese) curves of all degrees. For such tautological
subbundle is the family of coordinate rings for elliptic curves. For higher
strata, the subbundles represent families of plane
curves (trigonal curves at ) and space curves of genus .
Two methods of regularization of singular curves contained in
, namely, the standard blowing-up and transition to higher
strata with the change of genus are discussed.Comment: 31 pages, no figures, version accepted in Journal of Nonlinear
Mathematical Physics. The sections on the integrable systems present in
previous versions has been published separatel
On the fine structure and hierarchy of gradient catastrophes for multidimensional homogeneous Euler equation
Blow-ups of derivatives and gradient catastrophes for the -dimensional
homogeneous Euler equation are discussed. It is shown that, in the case of
generic initial data, the blow-ups exhibit a fine structure in accordance of
the admissible ranks of certain matrix generated by the initial data. Blow-ups
form a hierarchy composed by levels with the strongest singularity of
derivatives given by along certain critical directions. It is
demonstrated that in the multi-dimensional case there are certain bounded
linear superposition of blow-up derivatives. Particular results for the
potential motion are presented too. Hodograph equations are basic tools of the
analysis.Comment: 22 pages, 4 figures, 3 table
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
Birkhoff strata of the Grassmannian Gr: Algebraic curves
Algebraic varieties and curves arising in Birkhoff strata of the Sato
Grassmannian Gr are studied. It is shown that the big cell
contains the tower of families of the normal rational curves of all odd orders.
Strata , contain hyperelliptic curves of genus
and their coordinate rings. Strata , contain
plane curves for and and
curves in , respectively. Curves in the strata
have zero genus.Comment: 14 pages, no figures, improved some definitions, typos correcte
- …