 # Explicit solutions for integrable systems and applications

- English
52 Pages

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

Subjects

##### Minimal surface

Informations

Report a problem 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.
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