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Explicit solutions for integrable systems and applications

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52 pages
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


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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 IA vector fieldx˙=f(x)on a smooth manifoldM IA PDEut=F(u,ux,uxx, . . .)
Introduction : two theorems
Two types of“integrable” systemsin this talk IA vector fieldx=˙f(x)on a smooth manifoldM IA PDEut=F(u,ux,uxx, . . .)
“Explicit” solutions I ; theta functions ; Schur polynomialsRational solutions IFormal solutions Laurent series ;
IUnivalent, periodic, quasi-periodic solutions ISolitons, ∙ ∙ ∙
Introduction : two theorems
Two types of“integrable” systemsin this talk IA vector fieldx=˙f(x)on a smooth manifoldM IA PDEut=F(u,ux,uxx, . . .)
“Explicit” solutions I ;Rational solutions ; theta functions Schur polynomials I ;Formal solutions Laurent series
IUnivalent, periodic, quasi-periodic solutions ISolitons,∙ ∙ ∙
Applications IAbelian varieties, moduli spaces IRandom permutations, brownian motions IMinimal surfaces,∙ ∙ ∙
The Liouville theorem
I(M, ω)a symplectic manifold of dimension 2n I(F1, . . . ,Fn)independent functions in involution Then for a generic pointm0inM, the integral curve (solution) of eachXFistarting frommcan be determined by quadratures.