Exactly Solvable And Integrable Systems

Analytic-bilinear approach is used to study continuous and discrete non-isospectral symmetries of the generalized KP hierarchy. It is shown that Möbius symmetry transformation for the singular manifold equation leads to continuous or discrete non-isospectral symmetry of the basic (scalar or multicomponent KP) hierarchy connected with binary Bäcklund transformation. A more general class of multicomponent Möbius-type symmetries is studied. It is demonstrated that symmetry constraints of KP hierarchy defined using multicomponent Möbius-type symmetries give rise to Calogero-Moser system.

The bilinear equations of the N -component KP and BKP hierarchies and a corresponding extended Miwa transformation allow us to generate quadrilateral and circular lattices from conjugate and orthogonal nets, respectively. The main geometrical objects are expressed in terms of Baker functions.

For the sl_2 Gaudin model (degenerated quantum integrable XXX spin chain) an exponential generating function of correlators is calculated explicitely. The calculation relies on the Gauss decomposition for the SL_2 loop group. From the generating function a new explicit expression for the correlators is derived from which the determinant formulas for the norms of Bethe eigenfunctions due to Richardson and Gaudin are obtained.

We shown that, if you have two planes in the Segal-Wilson Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund-Darboux transformations (BDT). The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from elementary BDT's and is given here in a closed form. The geometric description of elementary DBT's requires that one has a geometric interpretation of the dual wavefunctions involved. This is done here with the help of a suitable algebraic characterization of the wavefunction. The BDT's also induce transformations of the tau-function associated to a plane in the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric characterization of the BDT'ss that preserves these subsystems of the KP hierarchy. This generalizes the classical Darboux-transformations. we also determine an explicit expression for the squared eigenfunction potentials. Next a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy. It is shown that infinite flags in the Grassmannian yield solutions of the latter hierarchy. these flags can be constructed by means of BDT's, starting from some plane. Other applications of these BDT's are a geometric way to characterize Wronskian solutions of the m -vector k -constrained KP hierarchy and the construction of a vast collection of orthogonal polynomials, playing a role in matrix models.

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with m arbitrary time-dependent coefficients are obtained possessing symmetries involving m arbitrary functions of time. A particular graded symmetry algebra for the modified KP equations is derived in this connection homomorphic to the Virasoro algebras.

We study the group theoretical properties of the Lame equation and its relation to su(1,1) and su(2). We compute the band structure, dispersion relation and transfer matrix and discuss the dynamical symmetry limits.

A study of Hamiltonian structures associated with supersymmetric Lax operators is presented. Following a constructive approach, the Hamiltonian structures of Inami-Kanno super KdV hierarchy and constrained modified super KP hierarchy are investigated from the reduced supersymmetric Gelfand-Dickey brackets. By applying a gauge transformation on the Hamiltonian structures associated with these two nonstandard super Lax hierarchies, we obtain the Hamiltonian structures of generalized Manin-Radul super KdV and constrained super KP hierarchies. We also work out a few examples and compare them with the known results.

A manifestly N=2 supersymmetric coset formalism is applied to analyse the "fermionic" extensions of N=2 a=4 and a=−2 KdV hierarchies. Both these hierarchies can be obtained from a manifest N=2 coset construction. This coset is defined as the quotient of some local but non-linear superalgebra by a U(1) ^ subalgebra. Three superextensions of N=2 KdV hierarchy are proposed, among which one seems to be entirely new.

The real homogeneous Monge-Ampère equation in one space and one time dimensions admits infinitely many Hamiltonian operators and is completely integrable by Magri's theorem. This remarkable property holds in arbitrary number of dimensions as well, so that among all integrable nonlinear evolution equations the real homogeneous Monge-Ampère equation is distinguished as one that retains its character as an integrable system in multi-dimensions. This property can be traced back to the appearance of arbitrary functions in the Lagrangian formulation of the real homogeneous Monge-Ampère equation which is degenerate and requires use of Dirac's theory of constraints for its Hamiltonian formulation. As in the case of most completely integrable systems the constraints are second class and Dirac brackets directly yield the Hamiltonian operators. The simplest Hamiltonian operator results in the Kac-Moody algebra of vector fields and functions on the unit circle.

It is shown that all 3-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial in generators of an infinite-dimensional Lie algebra of differential operators. It leads to new families of the orthogonal polynomials in two variables.