Hamiltonian Structure of PI Hierarchy

Abstract

Abstract. The string equation of type (2, 2g + 1) may be thought of as a higher order analogue of the first Painlevé equation that corresponds to the case of g = 1. For g > 1, this equation is accompanied with a finite set of commuting isomonodromic deformations, and they altogether form a hierarchy called the PI hierarchy. This hierarchy gives an isomonodromic analogue of the well known Mumford system. The Hamiltonian structure of the Lax equations can be formulated by the same Poisson structure as the Mumford system. A set of Darboux coordinates, which have been used for the Mumford system, can be introduced in this hierarchy as well. The equations of motion in these Darboux coordinates turn out to take a Hamiltonian form, but the Hamiltonians are different from the Hamiltonians of the Lax equations (except for the lowest one that corresponds to the string equation itself).

Weierstrass’ Elliptic Function Solutions to the Autonomous Limit of the String Equation

“…the 4th order equation of the first Painlevé hierarchy. As proved by K. Takasaki [8], the string equation of type (2, 2 1) n + is equivalent to (6). So, in this article, we also call (6) the string equation of type (2, 2 1) n + .…”

“…for a couple of positive integers ( , ) q p . The above equation is called the string Equation (or Douglas equation) of type ( , ) q p , and appears in the string theory or the theory of 2D quantum gravity [1]- [8]. In the followings, we set ( , ) (2, 2 1) q p n = + for a positive integer n .…”

Pole-free solutions of the first Painlevé hierarchy and non-generic critical behavior for the KdV equation

“…In the above equations we have written q (j) for the j-th derivative of q with respect to s. We will call equation (1.2) the P m I equation, and we refer to [34,38,40,43,26] for more information about the first Painlevé hierarchy. Given t 1 , .…”

Systematic construction of nonautonomous Hamiltonian equations of Painlevé type. I. Frobenius integrability

“…It turns that this map is not surjective so that not all nonmagnetic systems have their magnetic counterparts. Finally, the coefficients 𝑐 𝛼 (𝑡 An elementary calculation using Lemma 1 shows that in the Viète coordinates (19), the transformation (111) takes the form Moreover, in accordance with (B.6)…”