Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Singular values of Hankel operators and
weak turbulence of the cubic Szegő
equation
Sandrine Grellier
Université d’Orléans – Fédération Denis Poisson
Lyon – Colloque franco-roumain – 29 août 2014
from joint works with Patrick Gérard and Zaher Hani
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
The cubic Szegő equation
L2+ (S) = {u : u(eix ) =
∞
X
û(n)einx ,
n=0
' H2 (D) = {u : u(z) =
∞
X
|û(n)|2 < ∞} ⊂ L2 (S) ,
n=0
∞
X
∞
X
n=0
n=0
û(n)z n ,
|û(n)|2 < ∞} ,
Π : L2 (S) −→ L2+ (S) the Szegő projector .
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
The cubic Szegő equation
The cubic Szegő equation : u = u(t, z), z ∈ D, such that
i
∂
u = Π(|u|2 u) .
∂t
A Hamiltonian system on L2+ (S) for
E(u) =
1
4
Z
S
|u|4
dx
, ω(u, v ) := Im (u, v )L2 .
2π
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Motivation
Study the dynamic of some non dispersive Hamiltonian system
the cubic Szegő equation.
A toy model for large time behavior of solutions of some
non linear wave equation.
Simple equation containing the interaction of a
non-linearity and a non-local operator (the Szegő
projector).
An integrable equation : explicit formulae for the solutions.
But the conservation laws do not control high regularity !
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Welposedness
First conservation laws from the hamiltonian structure :
kuk4L4 , kuk2L2 , M := (u, Du)L2 .
P
2
2
In particular kuk2L2 + M = ∞
k=0 (1 + k )|û(k)| ' kuk 1/2 .
H+
Theorem
∀s ≥ 1/2, global wellposedness of the Cauchy problem
i
∂
u = Π(|u|2 u), u(0, ·) = u0
∂t
s (S).
for initial datum u0 in H+
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Large time dynamics
Theorem
1
( P. Gérard & S.G 2013)
1/2
∀u0 ∈ H+ (S), the solution to the cubic Szegő equation
1/2
t 7→ u(t, ·) ∈ H+ (S) is almost periodic
(uniform limit of quasi-periodic functions).
2
(P. Gérard, S.G & Z. Hani 2014)
s (S) such that,
Let s > 1/2. There exists a dense Gδ in H+
s (S).
for all u0 in this set, u is unbounded in H+
Motivation
Large time dynamics
Hankel operators
The Fourier transform
Hankel operators
Given u ∈ L2+ , define Hu on L2+ (S) by
Hu (h) = Π uh .
On the Fourier side : infinite matrix constant on the
anti-diagonals
( û(j + k) )j,k ≥0 .
The cubic Szegő dynamic
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Smoothness
Nehari (1957) : Hu is bounded iff u ∈ Π(L∞ )
(= BMO+ Fefferman (1971)).
Hu is Hilbert-Schmidt iff
X
X
|û(j + k)|2 '
(1 + `)|û(`)|2 = kuk2H 1/2 < ∞.
j,k≥0
`≥0
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Singular values
Theorem (P. Gérard & S.G.)
If u satisfies
∂
u = Π(|u|2 u), u(0, ·) = u0
∂t
then the singular values of Hu and of Ku are conservation laws.
i
The shifted Hankel operator : Ku := Hu S
Sh(eix ) := eix h(eix )
Ku := S ∗ Hu = Hu S = HS ∗ u .
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Singular values
Theorem (P. Gérard & S.G.)
If u satisfies
∂
u = Π(|u|2 u), u(0, ·) = u0
∂t
then the singular values of Hu and of Ku are conservation laws.
i
The shifted Hankel operator : Ku := Hu S
Sh(eix ) := eix h(eix )
Ku := S ∗ Hu = Hu S = HS ∗ u .
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Singular values
Theorem (P. Gérard & S.G.)
If u satisfies
∂
u = Π(|u|2 u), u(0, ·) = u0
∂t
then the singular values of Hu and of Ku are conservation laws.
i
The shifted Hankel operator : Ku := Hu S
Sh(eix ) := eix h(eix )
Ku := S ∗ Hu = Hu S = HS ∗ u .
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Remarks
We recover that the H 1/2 -norm is a conservation law since
it is equivalent to the Hilbert-Schmidt norm of Hu which is
also the `2 -norm of the singular values.
Finite rank is preserved by the cubic Szegő flow . From
Kronecker (1881), Hu is of finite rank off u is a rational
function with no pole in D. Hence,
”the cubic Szegő equation preserves rational function”.
It gives rise to a system of ODE’s which is integrable in the
sense of Liouville.
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Eigenspaces of Hu2 , Ku2
Eu (s) := ker(Hu2 − s2 I) , Fu (s) := ker(Ku2 − s2 I) .
Observe that Ku2 = Hu2 − (·, u)u hence Ku2 = Hu2 on u ⊥ .
Lemma (P. Gérard-S.G., 2013)
Let s > 0 such that Eu (s) + Fu (s) 6= {0}.
| dim Eu (s) − dim Fu (s)| = 1 .
Hence,
either Fu (s) = Eu (s) ∩ u ⊥ s is H-dominant
or Eu (s) = Fu (s) ∩ u ⊥ s is K -dominant.
From the min-max formula s1 > s2 > s3 > . . . with s1 , s3 , . . .
are H-dominant and s2 , s4 , . . . are K -dominant.
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Blaschke product
Let s > 0 be a H-dominant singular value. Let us = Proj⊥ (u)
onto Eu (s) of dimension m.
Lemma
us
= Ψs
vs
a Blaschke product of degree m − 1 :
iα
Ψs (z) = e
m−1
Y
j=0
z − pj
1 − pj z
!
.
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Blaschke product
If s0 is K -dominant, us0 = Proj⊥ (u) onto Fu (s), vs0 =
Ku (us0 )
s0 .
Lemma
vs 0
= Ψs0 a Blaschke product of degree m0 −1, m0 = dim(Fu (s0 ))
us0
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
The non linear Fourier transform
The mapping
Φ : u ∈ H 1/2 \ {0} 7−→ ((s1 > s2 > . . . ), (Ψ1 , Ψ2 , . . . )) .
Theorem ( P. Gérard & S.G 2013)
The map Φ is bijective.
Moreover, explicit formula for Φ−1 .
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Explicit formula
Finite rank case
s1 > s2 > · · · > s2N ≥ 0,
C(z) :=
Ψ1 , Ψ2 , . . . , Ψ2N .
!
s2j−1 − zs2k Ψ2j−1 (z)Ψ2k (z)
.
2
2
− s2k
s2j−1
1≤j,k≤N
The formula
∀z ∈ D, C(z) is invertible and




−1 
C(z) 


*
u(z) =
Ψ1
Ψ3
·
·
·
Ψ2N−1
 
 
 
 
,
 
 
 
1
1
·
·
·
1

+


 .



Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
The cubic Szegő equation on the Fourier
transform side
Theorem ( P. Gérard & S.G 2013)
The solution of the cubic Szegő equation satisfies
r
Ψr (t, z) = ei(−1) sr t Ψr (0, t).
”action-angle” correspondence.
Motivation
Large time dynamics
Hankel operators
The Fourier transform
The cubic Szegő dynamic
Proof of the main theorem
1/2
gives almost-periodicity in H+ by a limiting argument (Φ is
a diffeomorphism for fixed multiplicities).
From a Baire argument, it suffices to reduce to
large time instability
s
s n
, ∃{u0n } ∈ H+
u0 → u0 ; ∃tn , ku n (tn )kH s → ∞
∀u0 ∈ H+
(as for NLS on T2 from Colliander-Keel-Staffilani-TakaokaTao- 2010).