No Arabic abstract
Four 4-dimensional Painleve-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painleve system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painleve-type equations. If we only consider Painleve-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painleve-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as Noumi-Yamada systems are included in this list. They are written as Hamiltonian systems, and their Hamiltonians are neatly written using Hamiltonians of the classical Painleve equations.
We investigate the structure of $tau$-functions for the elliptic difference Painleve equation of type $E_8$. Introducing the notion of ORG $tau$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $tau$-functions associated with the configuration of generic nine points in the projective plane.
We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painleve equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonovs elliptic beta integral.
We argue the integrability of the generalized KdV(GKdV) equation using the Painleve test. For $d( le 2)$ dimensional space, GKdV equation passes the Painleve test but does not for $d geq 3$ dimensional space. We also apply the Ablowitz-Ramani-Segurs conjecture to the GKdV equation in order to complement the Painleve test.
A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from $({mathbb P}^1)^4$ by blowing-up along sixteen 2-dimensional subvarieties. The symmetry groups, the invariants and the degree growth rates are computed from the linearisation on the corresponding Neron-Severi bilattices. It turns out that the deautonomised version of one of the mappings is a Backlund transformation of a direct product of the fourth Painleve equation which has $A_2^{(1)}+A_2^{(1)}$ type affine Weyl group symmetry, while that of the other mapping is of Noumi-Yamadas $A_5^{(1)}$ Painleve equation.
In our previous works, a relationship between Hermites two approximation problems and Schlesinger transformations of linear differential equations has been clarified. In this paper, we study tau-functions associated with holonomic deformations of linear differential equations by using Hermites two approximation problems. As a result, we present a determinant formula for the ratio of tau-functions (tau-quotient).