Integrable systems

\(\bf Introduction\, to\, Padé \,method\)

\(\bf Abstract.\)

Isomonodromic equations such as Painlevé and Garnier equations are very important class of nonlinear differential equations. On the discrete analog of these equations, much progress has been made over the last decades. In this series of lectures, I will explain a very simple method to approach the isomonodromy equations, both differential and discrete, based on the Padé approximation.

\(\bf1. Padé\, approximation.\) For a given function \(\psi(x)\), the Padé approximation supply a rational function \(\frac{P(x)}{Q(x)}\) as an approximation of \(\psi(x)\). We consider the linear differential equations for \(y(x)\) satisfied by \(y(x)=P(x)\) and \(y(x)= \psi(x)Q(x)\), and explain how to compute them in explicit examples. By choosing the function \(\psi(x)\) suitably, the linear differential equations give the Lax pair for Painlevé type equations.

\(\bf 2. Schur\, function\, formula.\) In case of \(\psi(x)=\sum_{i=0}^{\infty}p_i x^i\), an explicit formula of the polynomials \(P(x), Q(x)\) is known. The polynomials are given in terms of the some determinants (Schur functions) with entries \(p_i\). This formula is useful to obtain special solutions of the Painlevé type equations.

\(\bf 3. Discrete\, case.\) We consider the Padé approximation where the function \(\psi(x)\) is given in terms of \(q\)-Pochhammer symbols. Then the \(q\)-difference Painlevé equations and their special solutions are obtained.

\(\bf 4. Padé\, interpolation.\) We study the discrete Painlevé equations by using the Padé interpolation. The Padé interpolation is a discrete version of the Padé approximation and it is older than the usual differential case. The explicit formula for \(P(x), Q(x)\) are known by Cauchy and Jacobi.

\(\bf 5. Generalizations.\) I will discuss various generalizations such as higher-order/higher-rank, \(q\)-difference/elliptic-difference, multiple Padé approximation (by Hermite) etc, as long as time permits.