1,018 research outputs found
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
Lattice and q-difference Darboux-Zakharov-Manakov systems via -dressing method
A general scheme is proposed for introduction of lattice and q-difference
variables to integrable hierarchies in frame of -dressing
method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov
systems of equations are derived. Darboux, B\"acklund and Combescure
transformations and exact solutions for these systems are studied.Comment: 8 pages, LaTeX, to be published in J Phys A, Letters
Hydrodynamic type integrable equations on a segment and a half-line
The concept of integrable boundary conditions is applied to hydrodynamic type
systems. Examples of such boundary conditions for dispersionless Toda systems
are obtained. The close relation of integrable boundary conditions with
integrable reductions of multi-field systems is observed. The problem of
consistency of boundary conditions with the Hamiltonian formulation is
discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a
segment and a semi-line are presented
Algebro-Geometric Solutions of the Boussinesq Hierarchy
We continue a recently developed systematic approach to the Bousinesq (Bsq)
hierarchy and its algebro-geometric solutions. Our formalism includes a
recursive construction of Lax pairs and establishes associated
Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations
for analogs of Dirichlet and Neumann divisors. The principal aim of this paper
is a detailed theta function representation of all algebro-geometric
quasi-periodic solutions and related quantities of the Bsq hierarchy.Comment: LaTeX, 48 page
Fractional Quantum Hall Effect and vortex lattices
It is demonstrated that all observed fractions at moderate Landau level
fillings for the quantum Hall effect can be obtained without recourse to the
phenomenological concept of composite fermions. The possibility to have the
special topologically nontrivial many-electron wave functions is considered.
Their group classification indicates the special values of of electron density
in the ground states separated by a gap from excited states
Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions
Hamiltonian systems of hydrodynamic type occur in a wide range of
applications including fluid dynamics, the Whitham averaging procedure and the
theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the
integrability of such systems by the generalised hodograph transform implies
that integrable Hamiltonians depend on a certain number of arbitrary functions
of two variables. On the contrary, in 2+1 dimensions the requirement of the
integrability by the method of hydrodynamic reductions, which is a natural
analogue of the generalised hodograph transform in higher dimensions, leads to
finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we
classify integrable two-component Hamiltonian systems of hydrodynamic type for
all existing classes of differential-geometric Poisson brackets in 2D,
establishing a parametrisation of integrable Hamiltonians via
elliptic/hypergeometric functions. Our approach is based on the Godunov-type
representation of Hamiltonian systems, and utilises a novel construction of
Godunov's systems in terms of generalised hypergeometric functions.Comment: Latex, 34 page
The Gould-Hopper Polynomials in the Novikov-Veselov equation
We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov
(NV) equation. The root dynamics of the -flow in the NV equation is
studied using the GH polynomials and then the Lax pair is found. In particulr,
when , one can get the Gold-fish model. The smooth rational solutions
of the NV equation are also constructed via the extended Moutard transformation
and the GH polynomials. The asymptotic behavior is discussed and then the
smooth rational solution of the Liouville equation is obtained.Comment: 22 pages, no figur
Self-stabilization of extra dimensions
We show that the problem of stabilization of extra dimensions in Kaluza-Klein
type cosmology may be solved in a theory of gravity involving high-order
curvature invariants. The method suggested (employing a slow-change
approximation) can work with rather a general form of the gravitational action.
As examples, we consider pure gravity with Lagrangians quadratic and cubic in
the scalar curvature and some more complex ones in a simple Kaluza-Klein
framework. After a transition to the 4D Einstein conformal frame, this results
in effective scalar field theories with certain effective potentials, which in
many cases possess positive minima providing stable small-size extra
dimensions. Estimates made in the original (Jordan) conformal frame show that
the problem of a small value of the cosmological constant in the present
Universe is softened in this framework but is not solved completely.}Comment: 10 pages, 4 figures, revtex4. Version with additions and corrections,
accepted at Phys. Rev.
The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type
We obtain the necessary and sufficient conditions for a two-component
(2+1)-dimensional system of hydrodynamic type to possess infinitely many
hydrodynamic reductions. These conditions are in involution, implying that the
systems in question are locally parametrized by 15 arbitrary constants. It is
proved that all such systems possess three conservation laws of hydrodynamic
type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such
systems are proved to possess a scalar pseudopotential which plays the role of
the `dispersionless Lax pair'. We demonstrate that the class of two-component
systems possessing a scalar pseudopotential is in fact identical with the class
of systems possessing infinitely many hydrodynamic reductions, thus
establishing the equivalence of the two possible definitions of the
integrability. Explicit linearly degenerate examples are constructed.Comment: 15 page
- …