Action-Angle variables for the Gel'fand-Dikii flows

Using the scattering transform for $n^{th}$ 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 EXACTk,ln\rm{EXACT}_{k,l}^n

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 $k$ or $l$ of the $n$ 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 $u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ 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 $n$ 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