497 research outputs found
Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions
Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV)
and \break Kadomtsev-Petviashvili (KP) equations were constructed for a given
curve whose genus is three. This study was based upon the fact
that about one hundred years ago (Acta Math. (1903) {\bf{27}}, 135-156), H. F.
Baker essentially derived KdV hierarchy and KP equation by using bilinear
differential operator , identities of Pfaffians, symmetric
functions, hyperelliptic -function and -functions; . The connection between his theory and the modern
soliton theory was also discussed.Comment: AMS-Tex, 12 page
Dispersionless Hirota equations and the genus 3 hyperelliptic divisor
Equations of dispersionless Hirota type have been thoroughly investigated in
the mathematical physics and differential geometry literature. It is known that
the parameter space of integrable Hirota type equations in 3D is 21-dimensional
and the action of the natural equivalence group Sp(6, R) on the parameter space
has an open orbit. However the structure of the `master-equation' corresponding
to this orbit remained elusive. Here we prove that the master-equation is
specified by the vanishing of any genus 3 theta constant with even
characteristic. The rich geometry of integrable Hirota type equations sheds new
light on local differential geometry of the genus 3 hyperelliptic divisor, in
particular, the integrability conditions can be viewed as local
differential-geometric constraints that characterise the hyperelliptic divisor
uniquely modulo Sp(6, C)-equivalence.Comment: amended version, to appear in Comm. Math. Phys., 15 page
Towards a classification of natural bi-hamiltonian systems
For construction and classification of the natural integrable systems we
propose to use a criterion of separability in Darboux--Nijenhuis coordinates,
which can be tested without an a priori explicit knowledge of these
coordinates.Comment: LaTeX 22 page
-function of the KdV hierarchy
In this paper we construct a family of commuting multidimensional
differential operators of order 3, which is closely related to the KdV
hierarchy. We find a common eigenfunction of this family and an algebraic
relation between these operators. Using these operators we associate a
hyperelliptic curve to any solution of the stationary KdV equation. A basic
generating function of the solutions of stationary KdV equation is introduced
as a special polarization of the equation of the hyperelliptic curve. We also
define and discuss the notion of a -function of a solution of the stationary
-KdV equation
Multi-Dimensional Sigma-Functions
In 1997 the present authors published a review (Ref. BEL97 in the present
manuscript) that recapitulated and developed classical theory of Abelian
functions realized in terms of multi-dimensional sigma-functions. This approach
originated by K.Weierstrass and F.Klein was aimed to extend to higher genera
Weierstrass theory of elliptic functions based on the Weierstrass
-functions. Our development was motivated by the recent achievements of
mathematical physics and theory of integrable systems that were based of the
results of classical theory of multi-dimensional theta functions. Both theta
and sigma-functions are integer and quasi-periodic functions, but worth to
remark the fundamental difference between them. While theta-function are
defined in the terms of the Riemann period matrix, the sigma-function can be
constructed by coefficients of polynomial defining the curve. Note that the
relation between periods and coefficients of polynomials defining the curve is
transcendental.
Since the publication of our 1997-review a lot of new results in this area
appeared (see below the list of Recent References), that promoted us to submit
this draft to ArXiv without waiting publication a well-prepared book. We
complemented the review by the list of articles that were published after 1997
year to develop the theory of -functions presented here. Although the
main body of this review is devoted to hyperelliptic functions the method can
be extended to an arbitrary algebraic curve and new material that we added in
the cases when the opposite is not stated does not suppose hyperellipticity of
the curve considered.Comment: 267 pages, 4 figure
Soliton self-modulation of the turbulence amplitude and plasma rotation
The space-uniform amplitude envelope of the Ion Temperature Gradient driven
turbulence is unstable to small perturbations and evolves to nonuniform,
soliton-like modulated profiles. The induced poloidal asymmetry of the
transport fluxes can generate spontaneous poloidal spin-up of the tokamak
plasma.Comment: Latex file, 66 pages, 24 postscript figures included. New section on
rotation five new figures, comparison with magnetic pumping dampin
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