Exactly Solvable And Integrable Systems

Using the S.Lie's infinitesimal approach we establish the connection between integrability of a one-parameter family of the Riccati equations and the stationary KdV hierarchy.

Only the known integrable cases of the Kodama-Hasegawa higher-order nonlinear Schroedinger equation pass the Painleve test. Recent results of Ghosh and Nandy add no new integrable cases of this equation.

A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale perturbations can be taken and thus higher dimensional integrable couplings can be presented. The theory is applied to the KdV soliton hierarchy. Infinitely many integrable couplings are constructed for each soliton equation in the KdV hierarchy, which contain integrable couplings possessing quadruple Hamiltonian formulations and two classes of hereditary recursion operators, and integrable couplings possessing local 2+1 dimensional bi-Hamiltonian formulations and consequent 2+1 dimensional hereditary recursion operators.

We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized by the planarity of its elementary quadrilaterals. Therefore the linear property of planarity seems to be a basic geometric property underlying integrability. We present the geometric meaning of its tau-function, as the potential connecting its forward and backward data. We present the theory of transformations of the quadrilateral lattice, which is based on the discrete analogue of the theory of rectilinear congruences. In particular, we discuss the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. We also show how the sequence of Laplace transformations of a quadrilateral surface is described by the discrete Toda system. We finally show that these classical transformations are strictly related to the basic operators associated with the quantum field theoretical formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review the properties of quadrilateral hyperplane lattices, which play an interesting role in the reduction theory, when the introduction of additional geometric structures allows to establish a connection between point and hyperplane lattices. We present and fully characterize some geometrically distinguished reductions of the quadrilateral lattice, like the symmetric, circular and Egorov lattices; we review also basic geometric results of the theory of quadrilateral lattices in quadrics, and the corresponding analogue of the Ribaucour reduction of the fundamental transformation.

We preesent a new supersymmetric integrable extensions of the a=4,N=2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generate new hierarchy of integrable equations, for which we investigate the hamiltonian structure. In special case our system describes the interaction of the KdV equation with the two MKdV equations.

The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed as nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M, whose Poisson quotient, under a parametric family of Hamiltonian group actions, is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem method is used to identify the auxiliary space M with the data defining the integral kernel of the resolvent operator at the endpoints of the curve segments. A second associated isomonodromic family of covariant derivative operators D_z is derived, having rank n=2m, and r finite regular singular points at the values of the exponents defining the kernel of K. This family is similarly embedded into the algebra Lgl_R(n) through a dual parametric family of Poisson quotients of M. The operators D_z are shown to be analogously associated to the integral operator obtained from K through a Fourier-Laplace transform.

The recursion operator and bi-Hamiltonian formulation of the Drinfeld- Sokolov system are given

In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential equations. We choose existence of hierarchies of first integrals and/or symmetries as a criterion for integrability and justify it by examples. Using our componentless approach we have solved a number of classification problems for integrable equations on free associative algebras. Also, in the simplest case, we have listed all possible Hamiltonian operators of low order.

We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the n×n AKNS hierarchies and the Gel'fand-Dikii hierarchies. The third arises in string theory as the representation of the Heisenberg group by [( L k/n ) + ,L]=I where L is an n th order scalar differential operator. The monodromy data is constructed in each case; the inverse monodromy problem is solved as a Riemann-Hilbert problem; and a simple proof of the Painlevé property is given for the general case

We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite genus curves. Our approach applies to complex-valued periodic solutions of the KdV hierarchy and naturally identifies the Riemann surface familiar from standard Floquet theoretic considerations with a limit of Burchnall-Chaundy curves.