1,172 research outputs found
On some algebraic examples of Frobenius manifolds
We construct some explicit quasihomogeneous algebraic solutions to the
associativity (WDVV) equations by using analytical methods of the finite gap
integration theory. These solutions are expanded in the uniform way to
non-semisimple Frobenius manifolds.Comment: 14 page
Time-dependent deformation functional theory
We present a constructive derivation of a time-dependent deformation
functional theory -- a collective variable approach to the nonequalibrium
quantum many-body problem. It is shown that the motion of infinitesimal fluid
elements (i.e. a set of Lagrangian trajectories) in an interacting quantum
system is governed by a closed hydrodynamics equation with the stress force
being a universal functional of the Green's deformation tensor . Since
the Lagrangian trajectories uniquely determine the current density, this
approach can be also viewed as a representation of the time-dependent current
density functional theory. To derive the above theory we separate a
"convective" and a "relative" motions of particles by reformulating the
many-body problem in a comoving Lagrangian frame. Then we prove that a properly
defined many-body wave function (and thus any observable) in the comoving frame
is a universal functional of the deformation tensor. Both the hydrodynamic and
the Kohn-Sham formulations of the theory are presented. In the Kohn-Sham
formulation we derive a few exact representations of the exchange-correlation
potentials, and discuss their implication for the construction of new
nonadiabatic approximations. We also discuss a relation of the present approach
to a recent continuum mechanics of the incompressible quantum Hall liquids.Comment: RevTeX4, 15 page
Solution of the dispersionless Hirota equations
The dispersionless differential Fay identity is shown to be equivalent to a
kernel expansion providing a universal algebraic characterization and solution
of the dispersionless Hirota equations. Some calculations based on D-bar data
of the action are also indicated.Comment: Late
Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type
We consider the special type of the field-theoretical Symplectic structures
called weakly nonlocal. The structures of this type are in particular very
common for the integrable systems like KdV or NLS. We introduce here the
special class of the weakly nonlocal Symplectic structures which we call the
weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then
the connection of such structures with the Whitham averaging method and propose
the procedure of "averaging" of the weakly nonlocal Symplectic structures. The
averaging procedure gives the weakly nonlocal Symplectic Structure of
Hydrodynamic Type for the corresponding Whitham system. The procedure gives
also the "action variables" corresponding to the wave numbers of -phase
solutions of initial system which give the additional conservation laws for the
Whitham system.Comment: 64 pages, Late
Reciprocal transformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems
Reciprocal transformations of Hamiltonian operators of hydrodynamic type are
investigated. The transformed operators are generally nonlocal, possessing a
number of remarkable algebraic and differential-geometric properties. We apply
our results to linearly degenerate semi-Hamiltonian systems in Riemann
invariants. Since all such systems are linearizable by appropriate
(generalized) reciprocal transformations, our formulae provide an infinity of
mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by
arbitrary functions of one variable.Comment: 26 page
On the water-bag model of dispersionless KP hierarchy
We investigate the bi-Hamiltonian structure of the waterbag model of dKP for
two component case. One can establish the third-order and first-order
Hamiltonian operator associated with the waterbag model. Also, the dispersive
corrections are discussed.Comment: 19 page
Algebraic varieties in Birkhoff strata of the Grassmannian Gr: Harrison cohomology and integrable systems
Local properties of families of algebraic subsets in Birkhoff strata
of Gr containing hyperelliptic curves of genus are
studied. It is shown that the tangent spaces for are isomorphic to
linear spaces of 2-coboundaries. Particular subsets in are described by
the intergrable dispersionless coupled KdV systems of hydrodynamical type
defining a special class of 2-cocycles and 2-coboundaries in . It is
demonstrated that the blows-ups of such 2-cocycles and 2-coboundaries and
gradient catastrophes for associated integrable systems are interrelated.Comment: 28 pages, no figures. Generally improved version, in particular the
Discussion section. Added references. Corrected typo
On the numerical evaluation of algebro-geometric solutions to integrable equations
Physically meaningful periodic solutions to certain integrable partial
differential equations are given in terms of multi-dimensional theta functions
associated to real Riemann surfaces. Typical analytical problems in the
numerical evaluation of these solutions are studied. In the case of
hyperelliptic surfaces efficient algorithms exist even for almost degenerate
surfaces. This allows the numerical study of solitonic limits. For general real
Riemann surfaces, the choice of a homology basis adapted to the
anti-holomorphic involution is important for a convenient formulation of the
solutions and smoothness conditions. Since existing algorithms for algebraic
curves produce a homology basis not related to automorphisms of the curve, we
study symplectic transformations to an adapted basis and give explicit formulae
for M-curves. As examples we discuss solutions of the Davey-Stewartson and the
multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure
Topological Phenomena in the Real Periodic Sine-Gordon Theory
The set of real finite-gap Sine-Gordon solutions corresponding to a fixed
spectral curve consists of several connected components. A simple explicit
description of these components obtained by the authors recently is used to
study the consequences of this property. In particular this description allows
to calculate the topological charge of solutions (the averaging of the
-derivative of the potential) and to show that the averaging of other
standard conservation laws is the same for all components.Comment: LaTeX, 18 pages, 3 figure
New Examples of Systems of the Kowalevski Type
A new examples of integrable dynamical systems are constructed. An
integration procedure leading to genus two theta-functions is presented. It is
based on a recent notion of discriminantly separable polynomials. They have
appeared in a recent reconsideration of the celebrated Kowalevski top, and
their role here is analogue to the situation with the classical Kowalevski
integration procedure.Comment: 17 page
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