B. G. Konopelchenko

Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies

An analytic-bilinear approach for the construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. A generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy, and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. The resolution of functional equations also leads to the τ function and addition formulas to it.

Analytic-bilinear approach to integrable hierarchies. I. Generalized KP hierarchy

An analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is proposed. It starts with the generalized Hirota identity for the Cauchy–Baker–Akhiezer (CBA) function and leads to a generalized KP hierarchy in the form of compact functional equations containing a special shift operator. A generalized KP hierarchy incorporates the basic KP hierarchy, modified KP hierarchy, KP singularity manifold equation hierarchy, and corresponding hierarchies of linear problems. Different “vertical” levels of the generalized KP hierarchy are connected via invariants of the Combescure symmetry group. The resolution of functional equations also gives rise to the τ function and the addition formulas for it.

Grassmannians Gr(N − 1, N + 1), closed differential N − 1-forms and N-dimensional integrable systems*

Integrable flows on the Grassmannians Gr(N − 1, N + 1) are defined by the requirement of closedness of the differential N − 1-forms ΩN − 1 of rank N − 1 naturally associated with Gr(N − 1, N + 1). Gauge-invariant parts of these flows, given by the systems of the N − 1 quasi-linear differential equations, describe coisotropic deformations of (N − 1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as the Liouville equation (N = 2), dispersionless Kadomtsev–Petviashvili equation (N = 3), dispersionless Toda equation (N = 3), Plebanski second heavenly equation (N = 4) and others. Gauge-invariant part of the forms ΩN − 1 provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2, N + 1) are defined via the requirement of the closedness of the dual forms ΩN − 1. It is shown that at N = 3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form Ω2, coincides with the Maxwell equations for orthogonal electric and magnetic fields.

Critical points, Lauricella functions and Whitham-type equations

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is emphasised. In particular, it is shown that the classical g-phase Whitham equations for the KdV and NLS equations are obtained via a g-fold iterated Darboux-type transformation generated by appropriate Lauricella functions.

A novel generalization of Clifford's classical point–circle configuration. Geometric interpretation of the quaternionic discrete Schwarzian Kadomtsev–Petviashvili equation

The algebraic and geometric properties of a novel generalization of Cliffords classical 4 point–circle configuration are analysed. A connection with the integrable quaternionic discrete Schwarzian Kadomtsev–Petviashvili equation is revealed.

Projective differential geometry of multidimensional dispersionless integrable hierarchies

We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential m-form Ωm with the coefficients satisfying the Plucker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invariant closedness equations.

Confluence of hypergeometric functions and integrable hydrodynamic-type systems

We construct a new class of integrable hydrodynamic-type systems governing the dynamics of the critical points of confluent Lauricella-type functions defined on finite-dimensional Grassmannian Gr(2, n), i.e., on the set of 2×n matrices of rank two. These confluent functions satisfy certain degenerate Euler–Poisson–Darboux equations. We show that in the general case, a hydrodynamic-type system associated with the confluent Lauricella function is an integrable and nondiagonalizable quasilinear system of a Jordan matrix form. We consider the cases of the Grassmannians Gr(2, 5) for two-component systems and Gr(2, 6) for three-component systems in detail.

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalized) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham-type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.

Geometry of basic statistical physics mapping

Geometry of hypersurfaces defined by the relation which generalizes classical formula for free energy in terms of microstates is studied. Induced metric, Riemann curvature tensor, Gauss-Kronecker curvature and associated entropy are calculated. Special class of ideal statistical hypersurfaces is analyzed in details. Non-ideal hypersurfaces and their singularities similar to those of the phase transitions are considered. Tropical limit of statistical hypersurfaces and double scaling tropical limit are discussed too.