411 research outputs found
Reductions of the Volterra and Toda chains
The Volterra and Toda chains equations are considered. A class of special
reductions for these equations are derived.Comment: LaTeX, 6 page
The Poisson geometry of SU(1,1)
We study the natural Poisson structure on the Lie group SU(1,1) and related
questions. In particular, we give an explicit description of the
Ginzburg-Weinstein isomorphism for the sets of admissible elements. We also
establish an analogue of Thompson's conjecture for this group.Comment: 11 pages, minor correction
The Complex Toda Chains and the Simple Lie Algebras - Solutions and Large Time Asymptotics
The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends
related to the classical series of simple Lie algebras are classified. It is
shown that the CTC models have much richer variety of asymptotic regimes than
the real Toda chain (RTC). Besides asymptotically free propagation (the only
possible regime for the RTC), CTC allow bound state regimes, various
intermediate regimes when one (or several) group(s) of particles form bound
state(s), singular and degenerate solutions. These results can be used e.g., in
describing the soliton interactions of the nonlinear Schroedinger equation.
Explicit expressions for the solutions in terms of minimal sets of scattering
data are proposed for all classical series B_r - D_r.Comment: LaTeX, article style, 16 pages; corrections of formulas and text
improvement
Canonical transformations of the time for the Toda lattice and the Holt system
For the Toda lattice and the Holt system we consider properties of canonical
transformations of the extended phase space, which preserve integrability. The
separated variables are invariant under change of the time. On the other hand,
mapping of the time induces transformations of the action-angles variables and
a shift of the generating function of the B\"{a}cklund transformation.Comment: LaTeX2e, +amssymb.cls, 8
Canonically conjugate variables for the periodic Camassa-Holm equation
The Camassa-Holm shallow water equation is known to be Hamiltonian with
respect to two compatible Poisson brackets. A set of conjugate variables is
constructed for both brackets using spectral theory.Comment: 10 pages, no figures, LaTeX; v. 2,3: references updated, minor
change
Direct delay reductions of the Toda hierarchy
We apply the direct method of obtaining reductions to the Toda hierarchy of
equations. The resulting equations form a hierarchy of ordinary differential
difference equations, also known as delay-differential equations. Such a
hierarchy appears to be the first of its kind in the literature. All possible
reductions, under certain assumptions, are obtained. The Lax pair associated to
this reduced hierarchy is obtained.Comment: 11 page
Singular normal form for the Painlev\'e equation P1
We show that there exists a rational change of coordinates of Painlev\'e's P1
equation and of the elliptic equation after which these
two equations become analytically equivalent in a region in the complex phase
space where and are unbounded. The region of equivalence comprises all
singularities of solutions of P1 (i.e. outside the region of equivalence,
solutions are analytic). The Painlev\'e property of P1 (that the only movable
singularities are poles) follows as a corollary. Conversely, we argue that the
Painlev\'e property is crucial in reducing P1, in a singular regime, to an
equation integrable by quadratures
The Liouville-type theorem for integrable Hamiltonian systems with incomplete flows
For integrable Hamiltonian systems with two degrees of freedom whose
Hamiltonian vector fields have incomplete flows, an analogue of the Liouville
theorem is established. A canonical Liouville fibration is defined by means of
an "exact" 2-parameter family of flat polygons equipped with certain pairing of
sides. For the integrable Hamiltonian systems given by the vector field
on where
is a complex polynomial in 2 variables, geometric properties of
Liouville fibrations are described.Comment: 6 page
Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background
We investigate trace formulas for Jacobi operators which are trace class
perturbations of quasi-periodic finite-gap operators using Krein's spectral
shift theory. In particular we establish the conserved quantities for the
solutions of the Toda hierarchy in this class.Comment: 7 page
A symplectic realization of the Volterra lattice
We examine the multiple Hamiltonian structure and construct a symplectic
realization of the Volterra model. We rediscover the hierarchy of invariants,
Poisson brackets and master symmetries via the use of a recursion operator. The
rational Volterra bracket is obtained using a negative recursion operator.Comment: 8 page
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