984 research outputs found
Action-Angle variables for the Gel'fand-Dikii flows
Using the scattering transform for order linear scalar operators,
the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner
Poisson bracket for the KdV hierarchy, is computed on the scattering side.
Action-angle variables are then constructed. Using this, complete integrability
is demonstrated in the strong sense. Real action-angle variables are
constructed in the self-adjoint case
A Dirac Sea and thermodynamic equilibrium for the quantized three-wave interaction
The classical version of the three wave interaction models the creation and
destruction of waves; the quantized version models the creation and destruction
of particles. The quantum three wave interaction is described and the Bethe
Ansatz for the eigenfunctions is given in closed form. The Bethe equations are
derived in a rigorous fashion and are shown to have a thermodynamic limit. The
Dirac sea of negative energy states is obtained as the infinite density limit.
Finite particle/hole excitations are determined and the asymptotic relation of
energy and momentum is obtained. The Yang-Yang functional for the relative free
energy of finite density excitations is constructed and is shown to be convex
and bounded below. The equations of thermal equilibrium are obtained
Non-Linear Evolution Equations with Non-Analytic Dispersion Relations in 2+1 Dimensions. Bilocal Approach
A method is proposed of obtaining (2+1)-dimensional non- linear equations
with non-analytic dispersion relations. Bilocal formalism is shown to make it
possible to represent these equations in a form close to that for their
counterparts in 1+1 dimensions.Comment: 13 pages, to be published in J. Phys.
Hierarchy of general invariants for bivariate LPDOs
We study invariants under gauge transformations of linear partial
differential operators on two variables. Using results of BK-factorization, we
construct hierarchy of general invariants for operators of an arbitrary order.
Properties of general invariants are studied and some examples are presented.
We also show that classical Laplace invariants correspond to some particular
cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200
New multidimensional partially integrable generalization of S-integrable N-wave equation
This paper develops a modification of the dressing method based on the
inhomogeneous linear integral equation with integral operator having nonempty
kernel. Method allows one to construct the systems of multidimensional Partial
Differential Equations (PDEs) having the differential polynomial forms in any
dimension n. Associated solution space is not full, although it is parametrized
by a certain number of arbitrary functions of (n-1)-variables. We consider
4-dimensional generalization of the classical (2+1)-dimensional S-integrable
N-wave equation as an example.Comment: 38 page
Exact quantum query complexity of
In the exact quantum query model a successful algorithm must always output
the correct function value. We investigate the function that is true if exactly
or of the input bits given by an oracle are 1. We find an optimal
algorithm (for some cases), and a nontrivial general lower and upper bound on
the minimum number of queries to the black box.Comment: 19 pages, fixed some typos and constraint
A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator
\We consider an inverse scattering problem for Schr\"odinger operators with
energy dependent potentials. The inverse problem is formulated as a
Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for
two distinct symmetry classes. As an application we prove global existence
theorems for the two distinct systems of partial differential equations
for suitably restricted,
complementary classes of initial data
On the Caudrey-Beals-Coifman System and the Gauge Group Action
The generalized Zakharov-Shabat systems with complex-valued Cartan elements
and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their
gauge equivalent are studies. This includes: the properties of fundamental
analytical solutions (FAS) for the gauge-equivalent to CBC systems and the
minimal set of scattering data; the description of the class of nonlinear
evolutionary equations solvable by the inverse scattering method and the
recursion operator, related to such systems; the hierarchies of Hamiltonian
structures.Comment: 12 pages, no figures, contribution to the NEEDS 2007 proceedings
(Submitted to J. Nonlin. Math. Phys.
Constructive factorization of LPDO in two variables
We study conditions under which a partial differential operator of arbitrary
order in two variables or ordinary linear differential operator admits a
factorization with a first-order factor on the left. The factorization process
consists of solving, recursively, systems of linear equations, subject to
certain differential compatibility conditions. In the generic case of partial
differential operators one does not have to solve a differential equation. In
special degenerate cases, such as ordinary differential, the problem is finally
reduced to the solution of some Riccati equation(s). The conditions of
factorization are given explicitly for second- and, and an outline is given for
the higher-order case.Comment: 16 pages, to be published in Journal "Theor. Math. Phys." (2005
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