On the continuous resonant equation for NLS: I. Deterministic analysis

0.34 MB·English

by  Pierre Germain

ON THE CONTINUOUS RESONANT EQUATION FOR NLS
I. DETERMINISTIC ANALYSIS
PIERREGERMAIN,ZAHERHANI,AND LAURENTTHOMANN
5 Abstract. Westudythecontinuousresonant(CR)equationwhichwasderivedin[7]asthelarge-box
1 limit of the cubic nonlinear Schr¨odinger equation in the small nonlinearity (or small data) regime.
0 We first show that the system arises in another natural way, as it also corresponds to the resonant
2
cubic Hermite-Schr¨odinger equation (NLS with harmonic trapping). We then establish that the basis

g of special Hermite functions is well suited to its analysis, and uncover more of the striking structure
u of theequation. Westudyin particularthedynamicson afewinvariantsubspaces: eigenspaces ofthe
A harmonic oscillator, of the rotation operator, and the Bargmann-Fock space. We focus on stationary
waves and theirstability.
2
2

]
P 1. Introduction
A
1.1. Presentation of the equation. The purpose of this manuscript is to study the so-called con-
.
h
tinuous resonant equation which was introduced by Faou-Germain-Hani [7] as the large-box limit of
t
a the cubic nonlinear Schro¨dinger equation in the small nonlinearity regime. This equation reads
m
i∂ u = (u,u,u), (t,x) R R2,
[ t
(CR) T ∈ ×
2 (u(0,x) = f(x),
v
where the nonlinearity is defined by
0
6
def
7 (1.1) (f1,f2,f3)(z) = f1(x+z)f2(λx⊥+z)f3(x+λx⊥+z)dxdλ
3 T R R2
Z Z
0
1. = f1(x⊥ +z)f2(λx+z)f3(x⊥ +λx+z)dxdλ,
R R2
0 Z Z
5 for any z R2 (if x = (x1,x2) we set x⊥ = ( x2,x1)). While the above formula seems mysterious at
∈ −
1 this stage, it can be thought of as an integration over all rectangles for which z is a vertex. Indeed,
v: the points z, x+z, λx⊥+z, x+λx⊥ +z form a rectangle in R2, and this yields a parameterization
i of all rectangles which count z as a vertex1.
X
r
a 2000 Mathematics Subject Classification. 35BXX ; 37K05 ; 35Q55 ; 35C07.
Key words and phrases. Nonlinear Schr¨odinger equation, resonant equation, harmonic oscillator, Lowest Landau
Level, stationary solutions.
P. G. is partially supported by NSF grant DMS-1101269, a start-up grant from the Courant Institute, and a Sloan
fellowship.
L.T. is partially supported bythe grant “ANAE´”ANR-13-BS01-0010-03.
Z.H.ispartiallysupportedbyNSFGrantDMS-1301647,andastart-upfundfromtheGeorgiaInstituteofTechnology.
1This isrelated tothewell-known fact thatfourfrequenciesξ ,ξ ,ξ ,ξ areresonant for NLS (say,on the2-torus),
1 2 3 4
if ξ +ξ =ξ +ξ and |ξ |2+|ξ |2 =|ξ |2+|ξ |2, which is equivalent to these frequencies forming a rectangle.
1 2 3 4 1 2 3 4
1
2 PIERREGERMAIN,ZAHERHANI,AND LAURENTTHOMANN
This expression can also be reformulated using the unitary group eit∆, as was observed in [7].
Note that the definition of above is slightly different but equivalent to that in [7] as we explain
T
in Section 1.3. Here we will show that can also be reformulated using the semigroup eit( ∆+x2)
− | |
T
(see Lemma 2.2 below). The key ingredient is the so-called lens transform which is an explicit for-
mula (given in (2.5)) which links e it∆ to eit( ∆+x2). This suggests that the harmonic oscillator
− − | |
H = ∆+ x 2 will play a central role in the study of (CR).
− | |
Defining
def
(f ,f ,f ,f ) = (f ,f ,f ), f
1 2 3 4 1 2 3 4 L2
E hT i
= f (x+z)f (λx +z)f (x+λx +z)f (z)dzdxdλ,
1 2 ⊥ 3 ⊥ 4
R R2 R2
Z Z Z
it is easy to check that the (CR) equation derives from the Hamiltonian
def
(f) = (f,f,f,f)
E E
given the symplectic form ω(f,g) = 4Im f , g L2(R2) on L2(R2) (this follows easily from the symme-
− h i
tries of ). In other words, (CR) can also be written
E
1∂ (f)
i∂ f = E .
t 2 ∂f¯
Important quantities conserved by the flow of the above equation (we shall come back to them) are
the mass M and angular momentum P:
M d=ef u2 and P d=ef i(x )uu,
R2| | R2 ×∇
Z Z
where x = x ∂ x ∂ .
×∇ 2 x1 − 1 x2
1.2. Physical and mathematical relevance. The (CR) equation has rich dynamics and can be
studied in its own right, but it also plays a role in the description of the dynamics of the usual cubic
NLS – with or without potential – in various situations, which we summarize here:
It was derived in [7] as a weakly nonlinear, big box limit of the cubic NLS
•
i∂ u ∆u= u 2u
t
− | |
(here, we consider the focusing case, but the defocusing case leads to the same picture) on
a periodic box of size L; equivalently, it appears as the limiting equation for high frequency
envelopes ofsolutionsofNLSontheunittorusT2. Tobemorespecific,settingtheaboveequa-
tion onthe2-dimensionaltorusofsize L,andprescribingdataof sizeε, itiswell-approximated
by (CR) on very long time scales (much longer than L2/ε2).
Wewillproveinthepresentpaperthat(CR)canalsobederivedasasmalldataapproximation
•
of the 2-dimensional cubic NLS equation with harmonic trapping, a.k.a. Hermite-Schro¨dinger
equation
(1.2) i∂ u ∆u+ x 2u = u2u
t
− | | | |
This model is widely used in several areas of physics from nonlinear optics to Bose-Einstein
condensates [16], but its relation to the dynamics of (CR) seems to be new. To be more
3
specific: set H = ∆+ x 2, Π the projector on the n-th eigenspace of H, and f = e itHu.
n −
− | |
Keeping only the totally resonant part of the nonlinearity in (1.2) gives the equation
i∂ f = Π (Π f )(Π f )(Π f ) .
t n4 n1 1 n2 2 n3 3
nn11+,nn2X2,n=3n,n34+≥n04 (cid:16) (cid:17)
We will prove that the above right-hand side is, up to a multiplicative constant, equal to
(f,f,f). Thus the totally resonant part of NLS with harmonic trapping is identical to (CR).
T
This implies that (CR) approximates the dynamics of (1.2) for large times in the small data
regime, as we illustrate in Theorem 3.1.
Theequation(CR)appearsasamodifiedscatteringlimitofthecubicNLSonR3withharmonic
•
trapping in two directions. Therefore any information on the asymptotic dynamics of (CR)
directly gives the corresponding behaviour for NLS. We refer to Hani-Thomann [14] for more
details and concrete applications.
Whenrestricted totheBargmann-Fock space, whichisgiven byL2(R2)functionswhichcanbe
•
|z|2
written as the product of a holomorphic function with the Gaussian e− 2 , the equation (CR)
coincides with the model known as Lowest Landau Level [1, 2, 18], used in the description of
rotating Bose-Einstein condensates.
The (CR) equation can be considered as a model of NLS-like equation without dispersion. A lot
of attention has recently been paid to such equations, at least starting with the work of Colliander,
Keel, Staffilani, Takaoka, and Tao [5] and subsequent works on the growth of Sobolev norms [12,
13]. There, the dynamics of resonant systems like (CR) arise either as approximating or asymptotic
dynamics for the original NLS model (see also [14]). Another important instance of zero-dispersion
Hamiltonian equations comes from the work of G´erard and Grellier [9, 8] who studied the so-called
cubic Szego¨ equation on S1 (this latter equation is obtained by replacing Π in (see Lemma 2.3) by
n
the projection on einx for x S1). We refer to Pocovnicu [19, 20] for the studyTof Szego¨ on R.
∈
Despite the absence of a linear part, and of the corresponding dispersive effects, (CR) is well-
posed in L2(R2) (see [7]) which is remarkable for a zero-dispersion equation with a zero-order trilinear
nonlinearity (in comparison, the Szego¨ equation is well-posed in Hs for s 1/2 and the result is
≥
sharp). In fact, the nonlinearity has a hidden smoothing property coming from Strichartz estimates
of the original NLS model, and this compensates the lack of linear dispersion (see Lemma 2.2). In
this direction, let us also mention the dispersion managed Schro¨dinger equation, which is obtained by
averaging a nonlinear Schro¨dinger equation with varying dispersion. Thislatter equation has a similar
structure as (CR). We refer to [25, 11] and references therein for more details.
In [10] we undertake the study of (CR) with random initial conditions. We exhibit global rough
dynamics (for initial data less regular than L2), and we construct Gibbs measures which are invariant
by this flow. This is related to weak turbulence theory, which is often considered to occur in the
statistical regime where the phases of the Fourier coefficients are initially uncorrelated.
1.3. Known results. Here we recall some important properties of the (CR) and the operator ,
T
which were proved in [7]. We first clarify the slightly different formulation of the operator we use
T
hereandthatderivedin[7]. Inadditiontoreversingtherolesoff andf , themaindifferencebetween
2 3
the two definitions is that the λ integral in [7] is over [ 1,1] instead of R. Denoting by the
λ [ 1,1]
− T ∈−
4 PIERREGERMAIN,ZAHERHANI,AND LAURENTTHOMANN
same operator as with the λ integral taken over [ 1,1] instead, one can easily show by a change of
T −
variable that:
1
(f ,f ,f ) = (f ,f ,f ).
Tλ∈[−1,1] 1 1 3 2T 1 1 3
In particular, one can use either formulation to define the (CR) equation in which f = f = f .
1 2 3
Now we recall thatthe operator is boundedfrom L2 L2 L2 to L2, and also from L˙ ,1 L˙ ,1
∞ ∞
L˙ ,1 to L˙ ,1, where L˙ ,1 is givenTby the norm f ×= x×f . × ×
∞ ∞ ∞ k kL˙∞,1 k| | kL∞
This implies immediately that (CR) is locally well-posed for data in L2 or L˙ ,1. Using the conser-
∞
vation of the L2 norm (the mass M), one obtains global well-posedness for data in L2 (and afterwards
in any Sobolev and weighted L2 space).
Gaussians play the role of a ground state for the equation, since they minimize the Hamiltonian
E
for fixed M. This variational characterization leads to orbital stability in L2 (up to the symmetry
group of the equation, which will be recalled in Section 2.1); it also holds in L2,1 H1 (where L2,1 is
∩
given by the norm f = x f ) by a different argument.
L2,1 L2
k k kh i k
1.4. Obtained results.
1.4.1. The basis of special Hermite functions. Recall that there exists a Hilbertian basis of L2(R2)
knownasthespecialHermitefunctions ϕ ,wheren N,m n,2 n,...,n 2,n whichdiago-
n,m
{ } ∈ ∈ {− − − }
nalizesjointlytheharmonicoscillator H = ∆+ x 2 andtheangularmomentumoperatorL = ix :
− | | ×∇
Hϕ = 2(n+1)ϕ , Lϕ = mϕ
n,m n,m n,m n,m
(seeSection5foramoredetailedpresentation). WeshowinSection5thatthisbasisisverywell-suited
to decomposing the trilinear operator : this follows from the formula (proved in Proposition 5.2)
T
(ϕ ,ϕ ,ϕ ) = (ϕ ,ϕ ,ϕ ,ϕ )ϕ ,
T n1,m1 n2,m2 n3,m3 E n1,m1 n2,m2 n3,m3 n4,m4 n4,m4
where n = n +n n and m = m +m m . This formula has the immediate consequence that
4 1 2 3 4 1 2 3
− −
all the special Hermite functions are stationary waves. Furthermore, it also implies the dynamical
invariance of all the subspaces of the form
Span ϕ , such thatαn+βm = γ mod δ or Span ϕ , such that αn+βm = γ
L2{ n,m } L2{ n,m }
where α, β, γ and δ are natural numbers.
1.4.2. Invariant subspaces. We investigate further the dynamics on particularly relevant or natural
examples of the invariant subspaces which were just described:
Sections 6.1–6.4 are dedicated to the analysis of the dynamics on the eigenspaces of the har-
• monicoscillatorH: forsomen N,Span ϕ ,wherem n ,2 n ,...,n 2,n .
Section 7 focuses on the eigens0p∈aces of thLe2r{otant0i,omn operator∈L:{−for0som−e m0 Z,0− 0}}
0
• ∈
Span ϕ ,wheren m , m +2, m +4,... .
L2{ n,m0 ∈ {| 0| | 0| | 0| }}
Finally, in Section 8, we study the equation on the Bargmann-Fock space which can be seen
• as Span ϕ ,wheren N .
L2 n,n
{ ∈ }
5
1.4.3. Stationary waves. The most classical notion of a stationary wave is given by a solution of the
typee iωtϕ,whereϕisafixedfunction,andω R. MoreelaboratewavesareofthetypeR e iωtϕ,
− αωt −
∈ −
where R is the rotation operator in space around 0 of angle θ, and α, ω real numbers. For reasons
θ
that will become clear, we call the former M-stationary waves (or simply stationary waves), and the
latter M +αP-stationary waves.
In the invariant subspaces detailed above, we give examples of stationary solutions, and try to
investigate their stability. Understanding the full picture - even finding all stationary solutions -
seems a daunting task, but we obtain first results in this direction.
Finally, inSection4,weprovegeneraltheoremsondecayandregularityofstationarywavesof (CR).
More precisely, we show that any M-stationary wave in L2 is analytic and exponentially decreasing,
while, roughly speaking, M +αP stationary waves belong to the Schwartz class as soon as they are a
little more localized, and smooth, than L2.
1.5. Notations. We set
For x R2, x is the rotation of x by π around the origin.
• ∈ ⊥ 2
For x R2, x = 1+ x 2; similarly if x R.
• ∈ h i | | ∈
def
• hf , giL2(R2) = R2pfg.
f(ξ) = f(ξ) d=eRf 1 f(x)e−ixξdx.
• F 2π R2
N is the set of all nonZ-negative integers (including 0).
• b
H d=ef ∆+ x 2 is the quantum harmonic oscillator on R2.
• − | |
def
L = i(x ∂ x ∂ ) is the angular momentum operator.
• 2 x1 − 1 x2
Hs is the Sobolev space given by the norm f Hs d=ef ξ sf L2.
• k k kh i k
L2,s is the weighted L2-space given by the norm f d=ef x sf .
L2,s L2
• k k bkh i k
s = Hs L2,s is the weighted Sobolev space given by the norm f s d=ef x sf L2 +
• H ∩ k kH kh i k
ξ sf . It is classical (see (3.1)) that s = u L2, s.t. Hs/2u L2 .
L2
kh i k H ∈ ∈
R is the counter-clockwise rotation of angle θ around the origin.
θ
• (cid:8) (cid:9)
S u =bλu(λ ) is the L2 scaling.
λ
• Π is the or·thogonal projection on the eigenspace E = u L2(R2), Hu= 2(n+1)u .
n n
• ∈
In this paper c,C > 0 denote universal constants the value of(cid:8)which may change from line t(cid:9)o line.
For two quantities A and B, we denote A . B if A CB, and A B if A. B and A& B.
≤ ≈
2. Properties and symmetries of and
T E
2.1. Various formulations of and . As observed in [7] we have
T E
Lemma 2.1. The quantities and are invariant by Fourier transform:
T E
(2.1) (f ,f ,f ) = (f ,f ,f ) and (f ,f ,f ,f ) = (f ,f ,f ,f ).
1 2 3 1 2 3 1 2 3 4 1 2 3 4
F T T E E
(cid:0) (cid:1)
b b b b b b b
6 PIERREGERMAIN,ZAHERHANI,AND LAURENTTHOMANN
Proof. Using Fourier inversion and the identity 1 eixξdx = δ gives
4π2 ξ=0
R
1
(f,g,h)(ξ) = e izξf(x+z)g(λx +z)h(x+λx +z)dxdλdz
FT 2π R2 R R2 − ⊥ ⊥
Z Z Z
1
= eiz( ξ+α+β γ)eix(α λβ⊥+λγ⊥ γ)f(α)g(β)h(γ)dαdβdγdxdλdz
− − − −
16π4
Z Z Z Z Z Z
= f ξ+ 1ξ 1β g(β)h 1ξ +β 1β dbβ dbλ. b
λ ⊥− λ ⊥ λ ⊥ − λ ⊥ λ2
Z Z (cid:18) (cid:19) (cid:18) (cid:19)
Changing variables to ηb= 1(ξ β) gives b b
λ − ⊥
(f,g,h)(ξ) = f(ξ+η)g(ξ +λη )h(η+λη +ξ)dηdλ,
⊥ ⊥
FT R R2
Z Z
which is the desired result. b b b (cid:3)
The next result shows that can be related to the L4L4 Strichartz norm associated to the linear
E t x
flows eit∆ and e itH.
−
Lemma 2.2. The following formulations for hold
E
(2.2) (f ,f ,f ,f ) = 2π (eit∆f )(eit∆f )(eit∆f )(eit∆f )dxdt
1 2 3 4 1 2 3 4
E R R2
Z Z
π
(2.3) = 2π 4 (e itHf )(e itHf )(e itHf )(e itHf )dxdt.
− 1 − 2 − 3 − 4
π R2
Z−4 Z
Therefore we have
(f ,f ,f ) = 2π e it∆ (eit∆f )(eit∆f )(eit∆f ) dt
1 2 3 − 1 2 3
T R
Z π h i
(2.4) = 2π 4 eitH (e itHf )(e itHf )(e itHf ) dt.
− 1 − 2 − 3
π
Z−4 h i
Proof. Using Fourier inversion and the identity 1 eixξdx = δ gives
4π2 R2 ξ=0
R
Ad=ef (eit∆f )(eit∆f )(eit∆f )(eit∆f )dxdt
1 2 3 4
R R2
Z Z
1
= e it(α2+β2 γ2 δ2)eix(α+β γ δ)f (α)f (β)f (γ)f (δ)dαdβdγdδdxdt
(2π)4 − | | | | −| | −| | − − 1 2 3 4
Z Z Z Z Z Z
1
= e it(α2+β2 α+β δ2 δ2)f (α)f (β)f (bα+βb δ)bf (δ)bdαdβdδdt.
(2π)2 − | | | | −| − | −| | 1 2 3 − 4
Z Z Z Z
b b b b
7
Changing variables to δ = z, α= z+x, β = z+λx +µx and resorting to the identity 1 eiyξdy =
⊥ 2π R
δ yields
ξ=0
R
1
A= e2itµx2f (z+x)f (z+λx +µx)f (z+x+λx +µx)f (z) x 2dxdzdλdµdt
(2π)2 | | 1 2 ⊥ 3 ⊥ 4 | |
Z Z Z Z Z
1
= f (x+z)f (z+bλx )f (zb+x+λx )f (z)bdxdzdλ b
2π 1 2 ⊥ 3 ⊥ 4
Z Z Z
1 1
= (f ,f ,fb,f ) = b (f ,f ,fb,f ), b
1 2 3 4 1 2 3 4
2πE 2πE
which givebs (b2.2b). bLet us now prove (2.3). Let f L2(R2), and denote v(t, ) = e itHf and
−
∈ ·
u(t, ) = eit∆f. Then the lens transform gives (see for instance [22])
·
1 arctan(2t) x i|x|2t
(2.5) u(t,x) = v , e1+4t2.
√1+4t2 2 √1+4t2
(cid:16) (cid:17)
x arctan(2t)
We first make the change of variables y = , then τ = . This gives
√1+4t2 2
1 arctan(2t)
(f ,f ,f ,f ) = 2π [v v v v ] ,y dydt
E 1 2 3 4 ZR 1+4t2 ZR2 1 2 3 4 (cid:18) 2 (cid:19)
π
= 2π 4 (e iτHf )(e iτHf )(e iτHf )(e iτHf )dydτ.
− 1 − 2 − 3 − 4
π R2
Z−4 Z
The relations for are obtained using that (f1,f2,f3),f4 L2(R2) = (f1,f2,f3,f4). (cid:3)
T hT i E
We are now able to prove the following result
Lemma 2.3. The following formulations for hold
E
(f ,f ,f ,f ) = π2 (Π f )(Π f )(Π f )(Π f )dx.
E 1 2 3 4 R2 n1 1 n2 2 n3 3 n4 4
n1+nX2=n3+n4Z
Therefore we have
(f ,f ,f )= π2 Π (Π f )(Π f )(Π f ) .
T 1 2 3 n4 n1 1 n2 2 n3 3
n1+nX2=n3+n4 (cid:16) (cid:17)
Proof. We compute in (2.3) for the eigenfunctions of H. Therefore we assume that Π f = f and
E nj j j
then
π
(f ,f ,f ,f )= 2π 4 e 2i(n1+n2 n3 n4)tdt f f f f dx.
1 2 3 4 − − − 1 2 3 4
E π R2
Z−4 Z
But now we use that Π f( x) = ( 1)njΠ f(x), thus f f f f dx = 0 unless n +n n n
nj − − nj R2 1 2 3 4 1 2− 3− 4
π
is even, which in turn implies −4π4 e−2i(n1+n2−n3−n4)tdt =Rπ2δ(n1+n2−n3−n4). (cid:3)
R
8 PIERREGERMAIN,ZAHERHANI,AND LAURENTTHOMANN
2.2. Symmetries of and conservation laws for (CR).
T
Lemma 2.4. The commutation relation
(2.6) Q (f ,f ,f ) = (Qf ,f ,f )+ (f ,Qf ,f ) (f ,f ,Qf )
1 2 3 1 2 3 1 2 3 1 2 3
T T T −T
holds for the (self-a(cid:0)djoint) operat(cid:1)ors
Q = 1, x , x 2 , i , ∆ , H , L = ix , i(x +1)
| | ∇ ×∇ ·∇
for all f ,f ,f (Q), where (Q) denotes the domain of Q.
1 2 3
∈ D D
Proof. Firstobservethatitsufficestoprovethecommutation relation forf ,f ,f sufficiently smooth,
1 2 3
and then argue by density and L2 boundedness of . For Q = 1, x, x 2, this follows easily from the
T | |
definition (1.1) of , in particular the fact that the arguments of f ,f ,f satisfy
1 2 3
T
z+(x+λx +z) = (x+z)+(λx +z)
⊥ ⊥
z 2 + x+λx +z 2 = x+z + λx +z 2.
⊥ ⊥
| | | | | | | |
Using (2.1) and arguing similarly gives (2.6) for P = i and ∆. Combining x 2 and ∆ gives (2.6) for
∇ | |
Q = H. Finally, to obtain this commutation relation for Q = ix and i(x +1), define R to be
λ
×∇ ·∇
the rotation of angle λ around the origin, and S the scaling transformation S u = λu(λ ), observe
λ λ
·
that
R (u,u,u) = (R u,R u,R u)
λ λ λ λ
T T
S (u,u,u) = (S u,S u,S u)
λ λ λ λ
T T
and differentiate in λ. (cid:3)
Corollary 2.5. If Q is as in Lemma 2.4, and f ,f ,f L2(R2) are such that Qf = λ f , with
1 2 3 j j j
λ C and j = 1,2,3, then ∈
j
∈
Q (f ,f ,f ) = (λ +λ λ ) (f ,f ,f )
1 2 3 1 2 3 1 2 3
T − T
and for s R, (cid:0) (cid:1)
∈
eisQ (f ,f ,f ) = (eisQf ,eisQf ,eisQf ) and (eisQf)= (f).
1 2 3 1 2 3
T T E E
In particular, if f ,f , and f are eigenfunctions of Q as in Lemma 2.4, then so is (f ,f ,f ). This
1 2 3 1 2 3
T
corollary hints towards examining in a basis which simultaneously diagonalizes both H and the
T
angular momentum operator L = i(x )= i(x ∂ x ∂ ); this will be done in the next section.
×∇ 2 x1 − 1 x2
Lemma 2.6. If Q is an operator so that for all f (R2)
∈S
Q (f,f,f) = 2 (Qf,f,f) (f,f,Q⋆f),
T T −T
then (cid:0) (cid:1)
(Qu)u
R2
Z
is a conservation law for (CR). Applying this to
Q = 1 , x , x 2 , i , ∆ , H , L , i(x +1)
| | ∇ ·∇
9
gives the conserved quantities
M = u2 , x u2 , x 2 u2 , i uu , u2 , Huu , P = Luu , i(x +1)uu
| | | | | | | | ∇ |∇ | ·∇
Z Z Z Z Z Z Z Z
(which are real-valued since we chose Q self-adjoint).
Proof. We compute
d
(2.7) (iQu)u = Q (u),u Qu, (u) .
dt R2 h T i−h T i
Z
By assumption and the symmetries of
E
Q (u),u = 2 (Qu,u,u),u (u,u,Q⋆u),u
h T i hT i−hT i
= 2 Qu, (u) (u),Q⋆u ,
h T i−hT i
which implies that (u),Q⋆u = Qu, (u) , and yields the result by (2.7). (cid:3)
hT i h T i
The relation which has just been established between operators commuting with , symmetries
T
of and , and conserved quantities of (CR) is of course an instance of the Noether theorem. We
recTapitulatEe below the obtained results (with λ R).
∈
operator Q conserved quantity corresponding
commuting with Quu symmetry u eiλQu
T 7→
1 u2 u eiλu
R | | 7→
x x u2 u eiλx1u
1 R 1
| | 7→
x2 R x2 u2 u eiλx2u
| | 7→
x 2 R x 2 u2 u eiλ|x|2u
| | | | | | 7→
i∂ i∂ uu u u( +λe )
x1 R x1 7→ · 1
i∂ i∂ uu u u( +λe )
x2 R x2 7→ · 2
∆ u2 u eiλ∆u
R
|∇ | 7→
H Huu u eiλHu
R
7→
L = i(x ) Luu u u(R x)
R λ
×∇ 7→
i(x +1) i(x +1)uu u λu(λx)
R
·∇ ·∇ 7→
R
3. Approximation of NLS with harmonic trapping by (CR)
A first consequence of the findings of the previous section is the following theorem, which states
that (CR) approximates the dynamics of (1.2) in the small data regime. For s 0, we define the
≥
Sobolev space based on the harmonic oscillator s = u L2, s.t. Hs/2u L2 , endowed with the
H ∈ ∈
natural norm u = Hs/2u . By [24, Lemma 2.4], we have the following equivalence of norms
k kHs k kL2 (cid:8) (cid:9)
(3.1) u u + x su .
s Hs L2
k kH ≡ k k kh i k
10 PIERREGERMAIN,ZAHERHANI,AND LAURENTTHOMANN
Theorem 3.1. Let s > 1 and suppose that u(t) is a solution of (1.2) and f(t) a solution of (CR)
with the same initial data u . Assume that the following bound holds over an interval of time [0,T]
0
f(t) s B for all 0 t T.
k kH ≤ ≤ ≤
Then there exists a constant C > 1 such that
t
ku(t)−eitHf(π2)kHs(R2) ≤ C(B3+B5t)eCB2t for all 0 ≤ t ≤ T.
In particular, if B is sufficiently small and 0 t B 2logB 1, then
− −
≤ ≪
t
ku(t)−eitHf(π2)kHs(R2) ≤ B5/2.
We remark that B can be made small by taking the initial condition sufficiently small in s.
H
Proof. We start with some notation: we write ω = n +n n n and
1 2 3 4
− −
(f ,f ,f ) = Π (Π f )(Π f )(Π f ) ,
T′ 1 2 3 n4 n1 1 n2 2 n3 3
n1,n2ωX,n=30,n4≥0 (cid:16) (cid:17)
(f ,f ,f ) = eitωΠ (Π f )(Π f )(Π f ) = e itH(eitHf eitHf eitHf ),
Nt 1 2 3 n4 n1 1 n2 2 n3 3 − 1 2 3
n1,n2X,n3,n4≥0 (cid:16) (cid:17)
(f ,f ,f ) = eitωΠ (Π f )(Π f )(Π f ) = (f ,f ,f ) (f ,f ,f ).
Pt 1 2 3 n4 n1 1 n2 2 n3 3 Nt 1 2 3 −T′ 1 2 3
n1,n2ωX,n=30,n4≥0 (cid:16) (cid:17)
6
Note that only differs from by a constant multiplicative factor (see Lemma 2.3). Recall that
′
for s > 1, Twe have Hs(R2) LT (R2), and that Hs(R2) is an algebra. Then by (3.1), it is easy to
∞
check that s(R2) is also an⊂algebra when s > 1. Therefore, using the boundedness of the operator
H
eitH on s (uniformly in t), we obtain the boundedness of the trilinear operators , , and from
′
H T N P
s s s to s for any s > 1.
H ×H ×H H
Let g(t) =e itHu(t) and f(t) = f( t ). The equation satisfied by g(t) and f˜are the following:
− π2
i∂ g(t) = (g(t),g(t),g(t)); i∂ f = (f,f,f); g(0) = f(0) = u .
t Nt e t T′ 0
Now we write the equation for g(t) as follows:
e e e e
i∂ g = (g,g,g)+ eitωΠ (Π g)(Π g)(Π g)
t T′ n4 n1 n2 n3
n1,n2ωX,n=30,n4≥0 (cid:16) (cid:17)
6
eitω 1
(3.2) =T′(g,g,g)+∂t iω− Πn4 (Πn1g)(Πn2g)(Πn3g)
n1,n2ωX,n=30,n4≥0 (cid:16) (cid:17)
6
eitω 1
(3.3) − Π ∂ (Π g)(Π g)(Π g)
− iω n4 t n1 n2 n3
n1,n2ωX,n=30,n4≥0 (cid:16) (cid:17)
6
= (g,g,g)+∂ A+D,
′ t
T