Systems of Toda Type, Inverse Spectral Problems, and Representation Theory.
The algebraic Bethe Ansatz and vacuum vectors.
The Algebraic Integrability of Geodesic Flow on S0(4).
The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data.
The complex geometry of the Kowalewski-Painlevé analysis.
The cubic Szegő equation
We consider the following Hamiltonian equation on the Hardy space on the circle,where is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...
The family of isometric superconformal harmonic maps and the affine Toda equations.
The finite discrete KP hierarchy and the rational functions.
The four positive vortices problem : regions of chaotic behavior and the non-integrability
The generalized de Rham-Hodge theory aspects of Delsarte-Darboux type transformations in multidimension
The differential-geometric and topological structure of Delsarte transmutation operators and their associated Gelfand-Levitan-Marchenko type eqautions are studied along with classical Dirac type operator and its multidimensional affine extension, related with selfdual Yang-Mills eqautions. The construction of soliton-like solutions to the related set of nonlinear dynamical system is discussed.
The geometry of dented pentagram maps
We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension there are such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps...
The Lax integrable differential-difference dynamical systems on extended phase spaces.
The Novikov-Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in .
The Painlevé III equation and the Iwasawa decomposition.
The Painlevé test and reducibility to the canonical forms for higher-dimensional soliton equations with variable-coefficients.
The real-analytic solutions of the Abel functional equation
For the Abel equation on a real-analytic manifold a dynamical criterion of solvability in real-analytic functions is proved.
The solution of general KdV equation in a class of steplike functions.
The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI
Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension , time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomonodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give...
The Theory of differential invariance and infinite dimensional Hamiltonian evolutions
In this paper we describe the close relationship between invariant evolutions of projective curves and the Hamiltonian evolutions of Adler, Gel'fand and Dikii. We also show how KdV evolutions are related as well to invariant evolutions of projective surfaces.
The theory of differential invariants and KDV hamiltonian evolutions