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Register a new userA Lax Formalism for the Elliptic Difference Painleve Equation
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Authors
Yasuhiko Yamada
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A Lax formalism for the elliptic Painleve equation is presented. The construction is based on the geometry of the curves on ${mathbb P}^1times{mathbb P}^1$ and described in terms of the point configurations.
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The 8-parameter elliptic Sakai difference Painleve equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrodinger equation for the $BC_1$ 8-parameter relativistic Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painleve-Calogero correspondence to the highest level in the two hierarchies.
A theoretical foundation for a generalization of the elliptic difference Painleve equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $tau$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $tau$-functions on the lattice.
Discrete Painleve equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painleve equations.
An interpolation problem related to the elliptic Painleve equation is formulated and solved. A simple form of the elliptic Painleve equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.
Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show that the general solution of the $q$-Painleve VI equation is a ratio of four tau functions, each of which is given by a combinatorial series arising in the AGT correspondence. We also propose conjectural bilinear equations for the tau functions.
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