Inverse pb Cubic Szego Lax pair Action angle Applications Proof Traveling waves

Sandrine Grellier

Description

Inverse pb Cubic Szego Lax pair Action-angle Applications Proof Traveling waves Cubic Szego equation and inverse spectral problem Sandrine Grellier Universite d'Orleans- Federation Denis Poisson IHES/X Mai 2011 en collaboration avec P. Gerard (Universite Paris sud) Sandrine Grellier Cubic Szego equation and inverse spectral problem

• action-angle applications


• hankel operators


• iff ?f ?


• universite de paris sud


• adjoint hankel operators


• inverse spectral

Sujets

• Grellier
• Poisson
• Universite de Paris Sud

Informations

• Publié par: pefav
• Publié le: 01 mai 2011
• Nombre de lectures: 17
• Langue: English

Extrait

Inverse pb

Cubic

Szego ¨

Lax pair

Action-angle

Applicat

Cubic Szego¨  equation and inverse spectral problem

Sandrine Grellier

Universite ´ d’Orl ´ eans-F ´ ed ´ erationDenisPoisson IHES/X Mai 2011

en collaboration avecrd ´ eraP.G sud) (Universite´ Paris

Sandrine Grellier

CubicSzeg ¨ oequationandinversespectralproblem

Inverse pbxpLarAaiSzic ¨ oegbuCcatlgAeppiltcoi-nna Hankel operators

Givenc= (c n ) n≥0 a ”symbol”, define the operatorΓ c by ∞ ∀x= (x n ) n≥0 ∈` 2 (N),Γ c (x) n = X c n+p x p . p=0 Letu(z) := P n≥0 c n z n . 1 Γ c is offinite rankiffuis rational(Kronecker 1881). 2 Γ c isHilbert–Schmidtiffu∈H 1/2 (T). 3 Γ c belongs to the Schatten classS p , 0<p<∞iff u∈B p 1 , / p p (T)(Peller, Semmes 1984). ˆ 4 Γ c isboundediff∃f∈L ∞ (T);c n =f(n),n≥0 or iff u∈BMO(T)(Nehari 1957). ˆ 5 Γ c iscompactiff∃f∈ C(T);c n =f(n),n≥0 or iff u∈VMO(T)(Hartman 1958).

SandrineGrellierCubicSzeg ¨ oequationandinversespectralproblem

Inverse pb Action-angle Applicat pair Lax Cubic Szego¨ Inverse spectral problem for Hankel operators

Letu∈VMO(T)andcreal valuedso thatΓ c is compact and selfadjoint. Non zero eigenvalues ordered so that

|ρ 1 | ≥ |ρ 2 | ≥ ∙ ∙ ∙ ≥ |ρ j  →| ≥ ∙ ∙ ∙0.

Inverse spectral problem ? Assume the generic case

|ρ 1 |>|ρ 2 |>∙ ∙ ∙>|ρ j |>∙ ∙ ∙ →0.

Theorem (Megretskii, Peller, Treil 1995) ∃csuch that theρ j ’s are the non-zero eigenvalues ofΓ c .

Sandrine Grellier

CubicSzeg ¨ oequationandinversespectralproblem