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Register a new userDegeneration scheme of 4-dimensional Painleve-type equations
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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.
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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).
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