The Extended Bigraded Toda hierarchy

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series.Comment: 32 pages, corrected typo

Topological Phenomena in the Real Periodic Sine-Gordon Theory

The set of real finite-gap Sine-Gordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the $x$-derivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components.Comment: LaTeX, 18 pages, 3 figure

Algebraic varieties in Birkhoff strata of the Grassmannian Gr(2)\mathrm{^{(2)}}: Harrison cohomology and integrable systems

Local properties of families of algebraic subsets $W_g$ in Birkhoff strata $\Sigma_{2g}$ of Gr$^{(2)}$ containing hyperelliptic curves of genus $g$ are studied. It is shown that the tangent spaces $T_g$ for $W_g$ are isomorphic to linear spaces of 2-coboundaries. Particular subsets in $W_g$ are described by the intergrable dispersionless coupled KdV systems of hydrodynamical type defining a special class of 2-cocycles and 2-coboundaries in $T_g$. It is demonstrated that the blows-ups of such 2-cocycles and 2-coboundaries and gradient catastrophes for associated integrable systems are interrelated.Comment: 28 pages, no figures. Generally improved version, in particular the Discussion section. Added references. Corrected typo

Classical R-matrix theory of dispersionless systems: I. (1+1)-dimension theory

A systematic way of construction of (1+1)-dimensional dispersionless integrable Hamiltonian systems is presented. The method is based on the classical R-matrix on Poisson algebras of formal Laurent series. Results are illustrated with the known and new (1+1)-dimensional dispersionless systems.Comment: 23 page

On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy

We give a simple algebraic proof that the two different Lax pairs for the Kac-van Moerbeke hierarchy, constructed from Jacobi respectively super-symmetric Dirac-type difference operators, give rise to the same hierarchy of evolution equations. As a byproduct we obtain some new recursions for computing these equations.Comment: 8 page

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems

Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our results to linearly degenerate semi-Hamiltonian systems in Riemann invariants. Since all such systems are linearizable by appropriate (generalized) reciprocal transformations, our formulae provide an infinity of mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by arbitrary functions of one variable.Comment: 26 page

The Abelian/Nonabelian Correspondence and Frobenius Manifolds

We propose an approach via Frobenius manifolds to the study (began in math.AG/0407254) of the relation between rational Gromov-Witten invariants of nonabelian quotients X//G and those of the corresponding ``abelianized'' quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.Comment: 35 pages, no figure

On the water-bag model of dispersionless KP hierarchy

We investigate the bi-Hamiltonian structure of the waterbag model of dKP for two component case. One can establish the third-order and first-order Hamiltonian operator associated with the waterbag model. Also, the dispersive corrections are discussed.Comment: 19 page

SU(2) Gluodynamics and HP1 sigma-model embedding: Scaling, Topology and Confinement

We investigate recently proposed HP1 sigma-model embedding method aimed to study the topology of SU(2) gauge fields. The HP1 based topological charge is shown to be fairly compatible with various known definitions. We study the corresponding topological susceptibility and estimate its value in the continuum limit. The geometrical clarity of HP1 approach allows to investigate non-perturbative aspects of SU(2) gauge theory on qualitatively new level. In particular, we obtain numerically precise estimation of gluon condensate and its leading quadratic correction. Furthermore, we present clear evidences that the string tension is to be associated with global (percolating) regions of sign-coherent topological charge. As a byproduct of our analysis we estimate the continuum value of quenched chiral condensate and the dimensionality of regions, which localize the lowest eigenmodes of overlap Dirac operator.Comment: 22 pages, 18 ps figures, revtex4. Replaced to match published version (PRD, to appear

An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.Comment: LaTeX, submitted to Reviews in Mathematical Physic