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Register a new userA new Lax pair for the sixth Painleve equation associated with $hat{mathfrak{so}}(8)$
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A new Lax pair for the sixth Painleve equation $P_{VI}$ is constructed in the framework of the loop algebra $mathfrak{so}(8)[z,z^{-1}]$. The whole affine Weyl group symmetry of $P_{VI}$ is interpreted as gauge transformations of the corresponding linear problem.
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In this paper, we consider the discrete power function associated with the sixth Painleve equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $widetilde{W}(3A_1^{(1)})$ as a subgroup of $widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{rm VI}$, we show how to relate $widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $tau$ function.
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 systematic construction of a Lax pair and an infinite set of conservation laws for the Ernst equation is described. The matrix form of this equation is rewritten as a differential ideal of gl(2,R)-valued differential forms, and its symmetry condition is expressed as an exterior equation which is linear in the symmetry characteristic and has the form of a conservation law. By means of a recursive process, an infinite collection of such laws is then obtained, and the conserved charges are used to derive a linear exterior equation whose components constitute a Lax pair.
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.
We study the spectrum of the linear operator $L = - partial_{theta} - epsilon partial_{theta} (sin theta partial_{theta})$ subject to the periodic boundary conditions on $theta in [-pi,pi]$. We prove that the operator is closed in $L^2([-pi,pi])$ with the domain in $H^1_{rm per}([-pi,pi])$ for $|epsilon| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in $H^1_{rm per}([-pi,pi])$.
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