SMS scnews item created by Yang Shi at Tue 1 Nov 2016 1251
Type: Seminar
Distribution: World
Expiry: 30 Nov 2016
Calendar1: 28 Nov 2016 1000-1130
CalLoc1: New Law School Lecture Theatre 026
Calendar2: 28 Nov 2016 1400-1530
CalLoc2: New Law School Lecture Theatre 026
CalTitle2: Special Lecture series: Yamada--2. Schur function formula
Calendar3: 29 Nov 2016 1000-1130
CalLoc3: New Law School Lecture Theatre 026
CalTitle3: Special Lecture series: Yamada--3. Discrete case
Calendar4: 29 Nov 2016 1400-1530
CalLoc4: New Law School Lecture Theatre 026
Calendar5: 30 Nov 2016 1000-1230
CalLoc5: New Law School Lecture Theatre 026
CalTitle5: Special Lecture series: Yamada--5. Generalizations
Auth: yangshi@p633.pc (assumed)

# Integrable systems

Professor Yasuhiko Yamada of Kobe University, an expert in mathematical physics and integrable systems will be visiting the School for a period of 28th-30th Nov. 2016. He will give a series of lectures on the Padé approach for constructing isomonodromic equations for nonlinear differential/discrete integrable systems. The lectures should be suitable for staff, PG students and Honours students. The lectures will be 1.5-2.5 hours each, in the mornings and afternoons of 28th-30th Nov. hold in the New Law School Lecture Theatre 026. Titles and abstracts of the lectures below.

$$\bf Introduction\, to\, Padé \,method$$

$$\bf Abstract.$$

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.

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