Explicit solutions for integrable systems and applications

Pol Vanhaecke

Description

Explicit solutions for integrable systems and applications Pol Vanhaecke Université de Poitiers Lyon, November 27, 2009

• ∂g ∂pi


• laurent series


• abelian varieties


• theta functions


• ∂f ∂qi


• dqi ?


• minimal surface

Sujets

• Université de Poitiers
• Laurent series
• Abelian variety
• Theta function
• Minimal surface

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 17
• Langue: English

Extrait

Explicit solutions for integrable systems and applications

Pol Vanhaecke

Université de Poitiers

Lyon, November 27, 2009

Introduction : two theorems

Two types of“integrable” systemsin this talk I A vector fieldx ˙ =f(x)on a smooth manifoldM I A PDEu t =F(u,u x ,u xx , . . .)

Introduction : two theorems

Two types of“integrable” systemsin this talk I A vector fieldx= ˙ f(x)on a smooth manifoldM I A PDEu t =F(u,u x ,u xx , . . .)

“Explicit” solutions I  ; theta functions ; Schur polynomialsRational solutions I Formal solutions Laurent series ;

I Univalent, periodic, quasi-periodic solutions I Solitons, ∙ ∙ ∙

Introduction : two theorems

Two types of“integrable” systemsin this talk I A vector fieldx= ˙ f(x)on a smooth manifoldM I A PDEu t =F(u,u x ,u xx , . . .)

“Explicit” solutions I  ;Rational solutions ; theta functions Schur polynomials I  ;Formal solutions Laurent series

I Univalent, periodic, quasi-periodic solutions I Solitons,∙ ∙ ∙

Applications I Abelian varieties, moduli spaces I Random permutations, brownian motions I Minimal surfaces,∙ ∙ ∙

The Liouville theorem

I (M, ω)a symplectic manifold of dimension 2n I (F 1 , . . . ,F n )independent functions in involution Then for a generic pointm 0 inM, the integral curve (solution) of eachX F i starting frommcan be determined by quadratures.