David V. Chudnovsky

Painleve property and multicomponent isospectral deformation equations

Abstract The Painleve property for partial differential equations proposed by Weiss, Tabor and Carnevale is studied for two- and three-dimensional multicomponent and matrix isospectral deformation systems. Applications to pole dynamics are presented.

Bäcklund transformation as a method of decomposition and reproduction of two-dimensional nonlinear systems

Abstract General definitions of Backlund transformations (BT) in terms of monodromy data and spectral data are reviewed. BT is applied to solutions of the matrix Dirac equation and new exact lattice equivalents of the nonlinear Schrodinger equation and similar equations are presented in classical and quantum cases.

Number theory : New York seminar, 1991-1995

1 Sums of Four Squares.- 2 On the Number of Co-Prime-Free Sets.- 3 The Primary Role of Modular Equations.- 4 Approximation Methods in Transcendental Function Computations and Some Physical Applications.- 5 Diophantine Approximation Problem Arising From VLSI Design.- 6 Linear Diophantine Problems.- 7 On the Sum of the Reciprocals of the Differences Between Consecutive Primes.- 8 The Smallest Maximal Set of Pairwise Disjoint Partitions.- 9 Sum Set Cardinalities of Line Restricted Planar Sets.- 10 On Solvability of a System of Two Boolean Linear Equations.- 11 Brauer Number and Twisted Fermat Motives.- 12 A Remark on a Paper of Erdos and Nathanson.- 13 Towards a Classification of Hilbert Modular Threefolds.- 14 Special Theta Relations.- 15 Minimal Bases and g-adic Representations of Integers.- 16 Finite Graphs and the Number of Sums and Products.- 17 Hilberts Theorem 94 and Function Fields.- 18 Some Applications of Probability to Additive Number Theory and Harmonic Analysis.- 19 Quadratic Irrationals and Continued Fractions.- 20 Progression Bases for Finite Cyclic Groups.- 21 Sums of Finite Sets.- 22 Four Squares with Few Squares.

The Bäcklund transformation for isomonodromy deformation Schlesinger equations

We define the transformation of linear differential equations with rational function coefficients that fix monodromy data and change local multiplicities by any sequence of integers. This transformation that gives rise to Padé approximations, at the same time defines the Backlund transformation of Schlesinger equations.

Bäcklund transformation for two-dimensional nonlinear evolution equations and the decomposition theorem

Abstract The simplest Backlund transformations are derived for matrix spectral problems associated with the Dirac equation. Decomposition theorems reducing equations of the nonlinear Schrodinger type to the common action of commuting generalized Toda lattice flows are presented.

Completely X-symmetric S-matrices corresponding to theta functions and models of statistical mechanics

We consider general expressions of factorized S-matrices with Abelian symmetry expressed in terms of θ-functions. These expressions arise from representations of the Heisenberg group. New examples of factorized S-matrices lead to a large class of completely integrable models of statistical mechanics which generalize the XYZ-model of the eight-vertex model.

Hamiltonian structure of isospectral deformation equation and semi-classical approximation to factorizedS-matrices

We consider semi-classical approximation to factorizedS-matrices. We show that this new class of matrices, calleds-matrices, defines Hamiltonian structures for isospectral deformation equations. Concrete examples of factorizeds-matrices are constructed and they are used to define Hamiltonian structure for general two-dimensional isospectral deformation systems.

Interactions of bubbles in a fluid

Abstract Interactive force between bubbles in a fluid is described, following Bjerknes, in connection with its effects on dynamics of microbubbles in blood vessels.

Algebraic reductions of scalar three-dimensional systems

Abstract Two-dimensional multicomponent and matrix isospectral deformation equations are described as subsystems of three-dimensional scalar Zakharov-Shabat systems in terms of algebraic relations on (pseudo) differential operators. The evolution of poles of solutions of arbitrary two-dimensional systems is given by invariant manifolds and many-particle completely integrable hamiltonians of Calogero type.

The relationship between non-abelian matrix Toda lattices and Ising-type models of statistical mechanics

Abstract We show that the Ising model hamiltonian is a special case of a matrix Toda lattice hamiltonian with operator variables from a Clifford algebra of dimension equal to the size of the lattice.