Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
The cubic Szegő equation
and spectral multiplicity
Sandrine Grellier
Université d’Orléans- Fédération Denis Poisson
HANDDY/ june 2013
from joint works with Patrick Gérard (Université Paris sud)
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Introduction
The cubic Szegő equation : a toy model of a non dispersive
Hamiltonian system.
A complete integrable system which admits two Lax pairs
related to Hankel operators.
Action-angle variables constructed for data in a generic
open set corresponding to simple spectra of the Hankel
operators.
Main goal : consider non generic data corresponding to
multiple spectra of the Hankel operators and construct
generalized ”action-angle” variables.
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
The cubic Szegő equation
L2+ (T)
2
= {u ∈ L (T) : u =
∞
X
û(k)eikx } ,
k=0
Z
Symplectic form ω(u, v ) = Im(u|v ) , (u|v ) :=
uv
T
Π:
L2 (T)
→
L2+ (T)
the Szegő projector.
The cubic Szegő equation : (S) i∂t u = Π(|u|2 u) .
dx
,
2π
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
A Hamiltonian system
(S) i∂t u = Π(|u|2 u) .
A Hamiltonian system for
E(u) =
1
4
Z
T
|u|4
dx
.
2π
Other conservation laws :
Z
Z
dx
2
2 dx
2
kukL2 :=
|u|
, kukḢ 1/2 :=
−i∂x u · u
.
2π
2π
T
T
s (T) = H s (T) ∩ L2 (T), s ≥ 1 .
Global flow on H+
+
2
Extension to initial data in BMO+ (T) by P. Gérard and H. Koch.
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
The Lax pair structure
Hankel operator of symbol u : Hu (h) = Π(uh), h ∈ L2+ .
Hu is anti-linear, non self-adjoint but (k|Hu (h)) = (h|Hu (k)).
T|u|2 (f ) = Π(|u|2 f ).
P
P
S the shift operator S( n≥0 an einx ) = n≥0 an ei(n+1)x
Ku = Hu S = S ∗ Hu the shifted Hankel operator.
Theorem [P. Gérard & S.G. (2008/2011)]
(S) is equivalent to
d
d
Hu = [Bu , Hu ] and Ku = [Cu , Ku ].
dt
dt
1
Bu = 2i Hu2 − iT|u|2 , (anti-selfadjoint)
2
Cu = 2i Ku2 − iT|u|2 , (anti-selfadjoint)
(Hu , Bu ) and (Ku , Cu ) Lax pairs for (S).
Similar to KdV (Lax, 68), cubic NLS 1D(Zakharov-Shabat, 72).
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Isospectrality
d
U = Bu U , U(0) = I , then U(t) unitary operator
dt
(Bu = 2i Hu2 − iT|u|2 anti-selfadjoint)
Let
U(t)∗ Hu(t) U(t) = Hu0 , Hu(t) isospectral to Hu0 .
d
V = Cu V , V (0) = I , then V (t) unitary operator
dt
(Cu = 2i Ku2 − iT|u|2 anti-selfadjoint)
Similarly, let
V (t)∗ Ku(t) V (t) = Ku0 , Ku(t) isospectral to Ku0 .
In particular, the rank of Hu and the rank of Ku are preserved by
the flow.
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Explicit formula
Theorem [P. Gérard & S.G. (2013)]
1
1
Let u0 ∈ H+2 (T), and u ∈ C(R, H+2 (T)) be the solution of
equation (S) such that u(0) = u0 . Then
2
2
2
u(t, z) = ((I − ze−itHu0 eitKu0 S ∗ )−1 e−itHu0 u0 | 1) , z ∈ D .
u(z) =
X
û(n)z n = ((I − zS ∗ )−1 u|1) , z ∈ D .
n≥0
since û(n) = (u|S n 1) = ((S ∗ )n u|1) . Then write
u = Hu (1) = U(t)Hu0 U(t)∗ (1)...
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Finite rank case
Recall Ku = Hu S.
Key point : Hu S = S ∗ Hu : Ku2 = Hu2 − (·|u)u.
d +1
d
V(d) = u, rank(Hu ) =
, rank(Ku ) =
.
2
2
By Kronecker (1881), complex d-submanifold of rational
functions (as functions of eix ).
1
(ρ2j (u))1≤j≤[ d+1 ] non zero eigenvalues of Hu2 ,
2
2 (u))
2
(σm
1≤m≤[ d ] non zero eigenvalues of Ku ,
2
2
ρ21 (u) ≥ σ12 (u) ≥ ρ22 (u) ≥ . . .
Define
V(d)gen := V(d) ∩ {ρ21 > σ12 > ρ22 > . . .}
dense open subset of V(d).
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Action angle coordinates in the generic case
Let u be in V(d)gen .
Let uj be the orthogonal projection of u on ker(Hu2 − ρ2j I) then
there exists ϕj so that
Hu (uj ) = eiϕj ρj uj .
Similarly, let uk0 be the orthogonal projection of u on
ker(Ku2 − σk2 I) then there exists θk so that
Ku (uk0 ) = e−iθk σk uk0 .
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
The action angle diffeomorphism
Theorem [P. Gérard & S.G. (2012)]
Let Ωd = {(λ1 , µ1 , λ2 , . . . ) ∈ (R∗+ )d ; λ1 > µ1 > λ2 > µ2 > . . .} .
2 2 2
ρ1 σ1 ρ2
The map χd : u 7→
,
,
,
.
.
.
,
(ϕ
,
θ
,
ϕ
,
.
.
.
)
is a
1
1
2
2 2 2
diffeomorphism from V(d)gen onto the manifold Ωd × Td , with
!
!
X
ρ2j
σj2
ω=
d
∧ dϕj + d
∧ dθj .
2
2
E=
X
j
ρ2j
2
!2
−
X
k
σk2
2
!2
.
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Link with cubic Szegő
Theorem (P. Gérard - S.G. 2012)
Let u0 ∈ Vgen (d) with
χd (u0 ) = ((
(ρ01 )2 (σ10 )2 (ρ02 )2
0 0
0
2 , 2 , 2 , . . . ), (ϕ1 , θ1 , ϕ2 , . . . )).
i∂t u = Π(|u|2 u),
ρ2 σ12 ρ22
2 , 2 , . . . ), (ϕ1 , θ1 , ϕ2 , . . . ))
= (ρ01 , σ10 , ρ02 , . . . )
, θk = −σk2 t + θk0 .
is equivalent to χd (u) = (( 21 ,
(ρ1 , σ1 , ρ2 , . . . )
ϕj = ρ2j t + ϕ0j ,
u(0, ·) = u0
with
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Extension to infinite dimension
Let u so that Hu and Ku are compact : u ∈ VMO+ (T) (Hartman
1958).
Generalization on VMO+,gen (T) subset of VMO+ (T)
corresponding to spectra so that ρ1 > σ1 > ρ2 > · · · → 0.
Same statement χ : VMO+,gen (T) 7→ Ω∞ × T∞ is a
homeomorphism.
Let u0 ∈ VMO+,gen (T) with χ(u0 ) = ((ρ0j , σj0 ), (ϕ0j , θj0 )).
i∂t u = Π(|u|2 u),
u(0, ·) = u0
is equivalent to χ(u) = ((ρj , σj ), (ϕj , θj )) with
(ρj , σj ) = (ρ0j , σj0 )
ϕj = ρ2j t + ϕ0j ,
, θk = −σk2 t + θk0 .
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
An example
u0ε (x) = eix + ε ∈ V(3), hence u ε ∈ V(3)
u ε (t, x) =
aε (t)eix + bε (t)
,
1 − pε (t)eix
For ε = 0, |u00 | = 1, hence u 0 (t, x) = e−it u00 (x) = e−it+ix .
M(Hu2ε )
0
=
1 + ε2 ε
ε
1
,
M(Ku2ε )
0
The eigenvalues of Hu2ε are ρ2± = 1 +
0
ε2
2
=
1 0
0 0
± ω, ω :=
u0ε ∈ Vgen (3) as u00 is non generic.
pε (t) = − √
2i
4+
2 /2
ε2
sin(ωt) e−itε
.
ε
2
.
√
4 + ε2
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
F IGURE : Though p0 (t) ≡ 0, pε (t) may visit small neighborhoods of
the unit circle at large times.
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Consequence :
No action angle neither Birkhoff coordinates around u00 !
Otherwise, for initial data in a compact neighborhood of u00 , the
solutions would stay in a compact set.
This is in strong contrast with KdV (e.g. Kappeler-Pöschel )
or with 1D-cubic NLS (e.g. Grébert-Kappeler-Pöschel).
However, every single trajectory of (S) in V(d) is quasi-periodic
(P. Gérard–SG 2012). This is not the case for some
perturbations of (S) (see Haiyan Xu (2013)).
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
The general case
Recall Ku2 = Hu2 − (·|u)u.
If ρ2 is an eigenvalue of Hu2 of multiplicity m, then the
eigenspace of Ku2 for ρ2 has multiplicity m − 1 or m + 1 :
|dim ker(H2u − ρ2 I) − dim ker(K2u − ρ2 I)| = 1.
In the first case, we say that ρ2 is a dominant eigenvalue of
Hu2 .
In the second case, we say that ρ2 is a dominant
eigenvalue of Ku2 .
The number of dominant eigenvalues for Hu2 and for Ku2 is
always the same. We call it q(u).
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Angles and Blaschke products
Blaschke products of degree k :


k
Y

z − pj
Bk :=
, pj ∈ D .


1 − pj z
j=1
Inside a dominant eigenspace of Hu2 , E := ker(Hu2 − ρ2 I), the
action of Hu is described by an angle and a Blaschke product,
as follows
ρuρ = e−iϕ bm−1 Hu (uρ ) .
uρ the orthogonal projection of u onto E, m := dim(E),
bm−1 ∈ Bm−1
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Angles and Blaschke products
Similarly, inside a dominant eigenspace of Ku2 ,
F := ker(Ku2 − σ 2 I), the action of Ku is described by an angle
and a Blaschke product
Ku (uσ0 ) = e−iθ b̃`−1 σuσ0 .
uσ0 the orthogonal projection of u onto F , ` := dim(F ),
b̃`−1 ∈ B`−1
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
”Generalized action angle variables”
Denote by VM,L (d) the set of u ∈ V(d) such that Hu2 , Ku2 have a
given number q of dominant positive eigenvalues, with
prescribed multiplicities M = (m1 , . . . , mq ), L = (`1 , . . . , `q ).
We define a mapping χM,L on VM,L (d) giving
1
The dominant eigenvalues of Hu2 and Ku2 in the set Ω2q ,
2
The angles in T2q ,
3
The Blaschke products.
If VM,L (d) is non empty, VM,L (d) is a real manifold and this
mapping
χM,L : VM,L (d) → Tq := Ω2q × T2q ×
q
Y
j=1
is a diffeomorphism.
Bmj −1 × B`j −1
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Link with cubic Szegő Equation
Let u0 ∈ VM,L (d) with
(0)
(0) (0)
(0) (0)
(0)
(0) (0)
χM,L (u0 ) = ((ρ1 , σ1 , ρ2 , . . . ), (ϕ1 , θ1 , ϕ2 , . . . ), (bj , b̃j ))
The solution of
i∂t u = Π(|u|2 u), u(0) = u0
is characterized by
(0)
(0)
(0)
(ρ1 , σ1 , ρ2 , . . . ) = (ρ1 , σ1 , ρ2 , . . . ),
(0)
(0)
(bj , b̃j ) = (bj , b̃j )
ϕj = ρ2j t + ϕ0j , θj = −σj2 t + σj0 .
Extension to general data in VMO+ (T) : same statement.
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Application : traveling waves revisited
Solutions of (S) of the form
u(t, eix ) = e−iωt u0 (ei(x−ct) ) , ω, c ∈ R .
Theorem [P. Gérard & S.G. (2009)]
1/2
A function u0 in H+ is a traveling wave with c 6= 0 and ω ∈ R iff
there exist non negative integers k and N, k ≤ N − 1, α ∈ C so
that
αz k
u0 (z) =
1 − pz N
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Let u be a traveling wave so that u(t, eix ) = e−iωt u(ei(x−ct) ).
Then
2 t+ϕ )
0
ρuρ (t) = e−i(ρ
Ku (uσ0 (t))
=
bm−1 Hu (uρ (t))
2
σe−i(−σ t+θ0 ) b̃`−1 uσ0 (t)
.
gives
ρ2 = ω + c(m − 1) , σ 2 = ω − c(` − 1)
and
bm−1 (e
−ict
−ic(m−1)t
z) = e
m−1
Y
z − pj eict
j=1
1 − pj eict z
= bm−1 (z)
Hence
bm−1 (z) = z m−1 , b̃`−1 (z) = z `−1 .
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Assume q(u) ≥ 2.
ρ21 = ω + c(m1 − 1) > σ12 = ω − c(`1 − 1) > ρ22 = ω + c(m2 − 1) .
Impossible, hence q(u) = 1 and
ρu = e−iϕ z m−1 Hu (u), Ku (u) = σe−iθ z `−1 u
As SKu (u) = SS ∗ Hu (u) = Hu (u) − kuk2 = Hu (u) − (ρ2 − σ 2 ),
we get
z m−1
ρ2 − σ 2
.
u(z) =
σ
ρ
1 − ρ e−i(ϕ+θ) z m+`−1
Introduction
The cubic Szegő equation
Lax pair
Consequences
Action angle
The general case
Perspectives
Is VM,L (d) non empty for any M, L ?
Expected : the restriction of the symplectic structure on
VM,L (d) is
ω=
q
X
j=1
d
ρ2j
2
!
∧ dϕj + d
σj2
2
!
∧ dθj .