Sur une équation mi onde dégénérée Workshop Handdy île de Berder septembre

Sandrine Grellier

Description

The equation Compared to the Szego solution Integrable effective dynamics for a non linear wave equation Sandrine Grellier Universite d'Orleans- Federation Denis Poisson WORSHOP HANDDY SEPT. 2011 jointwork with P. Gerard (Universite Paris sud) Sandrine Grellier Integrable effective dynamics for a non linear wave equation

• half wave


• szego solution


• universite de paris sud


• universite d'orleans- federation


• toy model


• integrable effective


• ck eikx

Sujets

• Grellier
• Poisson
• Universite de Paris Sud
• Toy model

Informations

• Publié par: pefav
• Nombre de lectures: 28
• Langue: Français

Extrait

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Integrable effective dynamics for a non linear wave equation

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Universite ´ d’Orle ´ ans-Fe ´ d ´ erationDenisPoisson W ORSHOP HANDDY SEPT . 2011

jointwork with rard P. Ge´ is ´ tvire(nUd)issuePar

Sandrine Grellier

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Consider the half wave equation

i∂ t u− |D|u=|u| 2 u,t∈R,x∈T

where

= X |D|
k∈ X Z c k e ikx ! : k∈Z c k |k|e ikx .

•Toy model for NLS on degenerate geometries leading to lack of dispersion. •Admits the same conservation laws as NLS : H(u) = 1 2 (|D|u,u) L 2 + 1 4 kuk L 4 4 , Q(u) =kuk L 2 2 , M(u) = (Du,u) L 2 .

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i∂ t u− |D|u=|u| 2 u,t∈R,x∈T

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•Toy model for NLS on degenerate geometries leading to lack of dispersion. •Admits the same conservation laws as NLS : H(u) = 1 2 (|D|u,u) L 2 + 4 1 kuk L 4 4 , 2 Q(u) =kuk L 2 , M(u) = (Du,u) L 2 .

where

|D|
k X ∈Z c k e ikx ! := k X ∈Z c k |k|e ikx .

Consider the half wave equation

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i∂ t u− |D|u=|u| 2 u,t∈R,x∈T

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•Toy model for NLS on degenerate geometries leading to lack of dispersion. •Admits the same conservation laws as NLS : 4 H(u) = 1 2 (|D|u,u) L 2 + 1 4 kuk L 4 , Q(u) =kuk 2 L 2 , M(u) = (Du,u) L 2 .

a

where

u

D| X c |
k∈Z k e ikx ! := k∈ X Z c k |k|e ikx .

i

Consider the half wave equation

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whereu ± := Π ± uand Π + ( X c k e ikx ) := X c k e ikx , k∈Zk≥0 Π − ( X ce ikx ) := X c k e ikx k . n∈Z k<0

Compared to the ”usual” 1D-NLS

i∂ t u+∂ x 2 u=|u| 2 u,

this is anon dispersiveequation : ( i(∂ t +∂ x )u + = Π + (|u| 2 u), i(∂ t −∂ x )u − = Π − (|u| 2 u),

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i∂ t u+∂ 2 u|u| 2 u, x =

this is anon dispersiveequation : ( i(∂ t +∂ x ) ) u u + − =Π Π= + − ( ( | | u u | | 2 2 u u ) ) , , i(∂ t −∂ x

whereu ± := Π ± uand Π + ( X c k e ikx ) := X c k e, ikx k∈Zk≥0 Π − ( X c k e ikx ) := X c k e ikx . n∈Z k<0

Compared to the ”usual” 1D-NLS

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