A stochastic NLS equation arising in Bose

A stochastic NLS equation arising in Bose-Einstein
condensation
A. de Bouard, R. Fukuizumi
Laboratoire de Mathématiques
CNRS et Université de Paris-Sud
Orsay
(stochastic NLS)
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Optically confined Bose-Einstein condensates
Stamper-Kurn et al., Phys. Rev. Lett, 1998
Advantages
Obtain different geometrical configurations
study magnetic properties of atoms (trapping not limited to specific
magnetic states)
(stochastic NLS)
2 / 26
Drawbacks
ex : fluctuations of the laser intensity
introduce stochasticity in the dynamical behavior of the condensate,
which has to be taken into acccount in real situations
Mean field theory : fluctuations of laser field intensity regarded as
modulations of the harmonic trap
Dynamics of BEC under regular variations of trap parameters :
widely studied
Castin and Dum, Phys. Rev. Lett. 1996
Kagan, et. al ; Phys. Rev. A, 1996
Ripoll, Perez-Garcia, Phys. Rev. A, 1999, etc...
(stochastic NLS)
3 / 26
Randomly varying optical trap potential
Abdullaev, Baizakov, Konotop, in Nonlinearity and disorder, 2001
2D radially symmetric Gross-Pitaevskii equation :
∂ψ
1 ∂ ∂ψ
=−
r
+ (1 + ε(t))r 2 ψ + χ|ψ|2 ψ − iγψ
∂t
r ∂r ∂r
in dimensionless variables, with
i
χ = ±1 (related to the sign of the s-wave scattering length a of atoms)
γ damping coefficient (thermal cloud)
ε(t) =
E (t)−E0
E0
with E (t) = laser field intensity, mean value E0 .
Assume ε(t) is δ-correlated with zero mean : hε(t)i = 0,
hε(t)ε(t 0 )i = σ 2 δ(t − t 0 )
(stochastic NLS)
4 / 26
Our aim : study the preceding model from a mathematical point of view
in the attractive case (negative scattering length) in space dimensions 1, 2
or 3 (no result in 3-D for the moment)
Content :
Physical motivations of the model
Mathematical description of the equation
Local existence of solutions in 1-D or 2-D
Global existence (no collapse) in 1-D, or 2-D with small total number
of atoms
Collapse in 2-D (without damping)
Conclusion and open problems
(stochastic NLS)
5 / 26
Mathematical description of the equation
(Ω, F, P) probability space
W (t) real valued Brownian motion ; ε(t) = Ẇ (t) =
dW (t)
dt
Use of a Stratonovich product :
idψ + (∆ψ − |x|2 ψ)dt − χ|ψ|2 ψdt + iγψdt = |x|2 ψ ◦ dW
Conservation of the squared L2 norm (total number of atoms) in the
absence of damping (W is real valued)
limit case of processes with non vanishing correlation length
Equivalent equation in Itô form :
idψ + (∆ψ − |x|2 ψ)dt − χ|ψ|2 ψdt + iγψdt = |x|2 ψdW − i
(stochastic NLS)
σ2 4
|x| ψdt
2
6 / 26
More generally :
Consider the equation
idψ + (∆ψ − |x|2 ψ)dt − χ|ψ|2α ψdt + iγψdt + i
σ2 4
|x| ψdt = |x|2 ψdW
2
α > 0, γ ≥ 0, x ∈ Rd , d = 1 or 2.
We look for solutions with paths (realizations) having finite “energy”
almost surely :
Z
Z
1
χ
H(ψ) =
(|∇ψ|2 + |x|2 |ψ|2 )dx +
|ψ|2α+2 dx
2
2α + 2
H : hamiltonian for the corresponding deterministic equation (without
damping) : combination of the energy of the wave packet, and mean
square width of the atomic cloud
(stochastic NLS)
7 / 26
Local existence of solutions with a.s. finite energy
realizations
If σ = 0, existence of solutions is known
Oh, J. Diff. Equ. 1989 ; Cazenave, Rio de Janeiro 1993 ; Carles, SIAM J.
Math. Anal. 2003
Use of “dispersive” estimates for the group U(t) generated by
i
du
+ (∆u − |x|2 u) = 0
dt
obtained with change of variables from the free equation
|U(t)v |Lp ≤ CT |t|n/p−n/2 |v |Lp0 ,
∀t ∈ [0, T ],
p ≥ 2,
1
p
+
1
p0
=1
allows to treat the nonlinear term in any dimension provided
2
⊂ L2α+2 (Rd ) i.e α < d−2
H 1 (Rd )
(stochastic NLS)
8 / 26
Here : Cannot use the preceding estimates on U(t) ; cannot treat the
stochastic integral as a perturbation ( de Bouard, Debussche, CMP, 1999)
restrictions on the dimension
Indeed
2 Z t
U(t − s)x 2 ψ(s)dW (s) 2
E L
Z t0
Z t Z
2
2
4
2
=
E |U(t − s)x ψ(s)|L2 ds =
E
x |ψ(s)| dx ds
0
0
Remark : Whether a dispersive estimate holds on Uω (t, s) generated by
idu + (∆u − |x|2 u)dt = |x|2 u ◦ dW
is open (a.s. unitary propagator in L2 (Rd ))
(stochastic NLS)
9 / 26
One dimensional case : rather easy
I - First solve the linear equation
idu + (∆u − |x|2 u)dt + iγudt + i
σ2 4
|x| udt = |x|2 udW
2
by using compactness method :
tightness of the sequence of laws of approximate solutions (ex :
cut-off in the quadratic potential) by using energy estimates (Itô
formula + martingale inequalities)
existence of a solution in law in C ([0, T ]; Σ)
Uniqueness of this last solution (linear equation)
cvgence of the original sequence in probability
strong solution in the probabilistic sense of the linear equation
with a.s. finite energy realizations
(stochastic NLS)
10 / 26
II - This gives a propagator Uω (t, s) : solution above starting at u(s) = us
fixed (depending only on {W (τ ), τ ≤ s})
Solve then the equation
Z
ψ(t) = Uω (t, 0)ψ0 + iχ
t
Uω (t, s)(|ψ|2α ψ(s))ds
0
locally in time : cut-off the nonlinearity + fixed point in L2 (Ω; C [0, T ]; Σ))
Result in d=1 : For any ψ0 ∈ Σ, there is a random time τω∗ (ψ0 ) and a
unique adapted solution ψ(t) which has continuous paths in C ([0, τω∗ [; Σ)
a.s.
Remark : The method would work in any d provided α <
dispersive estimate hold
(stochastic NLS)
2
2−d
if the
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Two dimensional case : More involved
Σ 6⊂ L∞
pb for uniqueness
restrictions on α
Use of a compactness method on the nonlinear equation
! expect only local-in time solutions (e.g. if α = 1)
I - Approximate solutions
Consider an aproximate equation, e.g.
idψn + (∆ψn − |x|2 ψn )dt + 2i θn (x)|x|4 ψn dt + iγψn dt − gn (ψn )dt
= θn (x)|x|2 ψn dW
with gn (s) =
χ|s|2α s, |s| ≤ n
χn2α s, |s| ≥ n
For all ψ0 ∈ Σ, there is a unique solution ψn ∈ L2 (Ω; C ([0, +∞[; Σ))
(stochastic NLS)
12 / 26
II - Estimates on the approximate solutions :
Let R > 0 and τnR = inf{t ≥ 0, |ψn (t)|L2α+2 > R} (random time)
Application of the Itô formula to the truncated energy :
Z
Z
1
2
2
|∇v | + |xv | dx − Gn (v )dx
Hn (v ) =
2
with Gn0 (v ) = gn (v ). This leads to
E
sup Hn (ψn (t)) ≤ C (T , |ψ0 |2Σ , R)
t≤τnR ∧T
for any T > 0. Now, for t ≤ τnR , one has a.s. |ψn (t)|L2α+2 ≤ R hence
E
sup |ψn (t)|2Σ ≤ C (T , |ψ0 |2Σ , R)
t≤τnR ∧T
+ estimates on the modulus of continuity, or fractional derivatives, using
the equation (estimates e.g. in L2 (Ω; C β (0; τnR ; Σ−4 )))
(stochastic NLS)
13 / 26
III - Tightness and convergence (first try) :
R > 0 fixed, α > 0. Consider unR (t) such that

 u R (t) = ψ R (t), t ≤ τ R
n
n
n
 du R (t) = ψ R (τ R )dt, t ≥ τ R
n
n
n
n
Then the sequence of laws of (unR , W ) is tight in
C ([0, T ]; L2α+2 ∩ L2 ) × C ([0, T ])
Prokhorov + Skorokhod theorems
there is a probability space (Ω̃, F̃, P̃), a sequence of processes ũnR with
paths a.s. in C ([0, T ]; Σ) and Brownian motions W̃n , satisfying the
preceding equation (with W replaced by W̃n ) for t ≤ τ̃nR , and
id ũnR = unR (τ̃nR )dt
for t ≥ τ̃nR ,
where τ̃nR = inf{t ≥ 0, |ũnR (t)|L2α+2 > R}
(stochastic NLS)
14 / 26
Moreover, ũnR converges to some ũ R a.s. in C ([0, T ]; L2 ∩ L2α+2 ) as n goes
to infinity
The topology is sufficient to pass to the limit in the equation for ũ R
τ̃nR converges a.s. to
τ̃ R = inf{t ≥ 0, |ũ R (t)|L2α+2 > R}
and ũ R satisfies the equation
idu + (∆u − |x|2 u)dt − χ|u|2α+2 udt + iγudt = |x|2 udW − i
σ2 4
|x| udt
2
for t ≤ τ̃ R , while
d ũ R (t) = ũ R (τ̃ R )dt,
for t ≥ τ R
Pb : The probability space (Ω̃, F̃, P̃) depends on R !
need uniqueness
(stochastic NLS)
15 / 26
Uniqueness : fails ! (even if α ≤ 1)
Indeed : Uniqueness known for NLS equation in 2-D, if ≤ 1, even if no
dispersive estimate
Vladimirov, Sov. Math. Dokl. 1984 ; Ogawa-Ozawa, J. Math. Anal. Appl.
1991
If u and v are solutions with paths a.s. in C ([0, τ [; Σ) of preceding
equation and w = u − v then Itô formula implies a.s.
Z t∧τ Z
2
|w (t ∧ τ )|L2 ≤ C
(1 + |u(s)|2 + |v (s)|2 )|w (s)|2 dx ds
0
+ Trudinger inequality
w = 0 on [0, τ [
However :
this does not imply uniqueness of ũ R because one may have τuR 6= τvR
(stochastic NLS)
16 / 26
Tightness + convergence : second try
Consider the whole sequence (unR )R∈N n∈N : tight in
(C ([0, T ]; L2α+2 ∩ L2 ))N
Application of Prokhorov + Skorokhod theorems
ũnR defined on (Ω̃, F̃, P̃) for all R ∈ N, ũnR converges to ũ R as before,
a.s. in C ([0, T ]; L2α+2 ∩ L2 ) for all R ∈ N and ũ R+1 = ũ R on [0, τ R ].
Now, we add a point ∆ (cemetery point) to L2α+2 (R2 ) with
O(L2α+2 ∪ {∆}) being the union of open sets in L2α+2 and complements
in L2α+2 ∪ {∆} of closed bounded sets of L2α+2
(stochastic NLS)
17 / 26
If we set
ũn∞ (t) =
ũnR (t) if t ≤ τnR for some R
∆ if t > lim inf R→∞ τnR
then ũn∞ converges in probability in C ([0, T ]; L2α+2 ∩ L2 ) to ũ ∞ defined by
R
ũ (t) if t ≤ τ R for some R
∞
ũ (t) =
∆ if t > lim inf R→∞ τ R
Moreover :
ũ ∞ is a solution of the equation on [0, τ ∗ [, with
τ ∗ = inf{t ≥ 0, ũ ∞ (t) = ∆} ;
lim supt→τ ∗ |ũ ∞ (t)|L2α+2 = +∞ a.s. by definition i.e. τ ∗ is the
blow-up time
ũ ∞ is unique
convergence in probability of the original sequence
(stochastic NLS)
18 / 26
Conclusion : For α ≤ 1, for any ψ0 ∈ Σ, there is a random time τ ∗ (ψ0 )
and a unique solution of
idψ + (∆ψ − |x|2 ψ)dt − χ|ψ|2α ψdt + iγψdt = |x|2 ψdW − i
σ2 4
|x| ψdt
2
with paths a.s. continuous in time with values in Σ for the weak topology ,
and with lim supt→τ ∗ |ψ(t)|L2α+2 = +∞ if τ ∗ < +∞, a.s.
Remark We do not have the energy equality for those solutions
(convergence only in L2α+2
only inequality )
But If 1/2 ≤ α ≤ 1 and ψ0 ∈ Σ2 , there are solutions with paths a.s. in
L∞ (0, τ ∗ ; Σ2 ) ∩ C ([0, τ ∗ [; Σ)
Brezis-Gallouët argument : Itô formula on
F (u) = (1 + ln(1 + |u(t)|2Σ2 ))
For those solutions we have the energy equality
(stochastic NLS)
19 / 26
Global existence
Conservation of the L2 -norm : ψ0 ∈ Σ, then a.s. for τ < τ ∗ (ψ0 ),
|ψ(τ )|2L2 = |ψ0 |2L2
Energy equality : ψ0 ∈ Σ if d = 1, ψ0 ∈ Σ2 if d = 2, and
Z
Z
1
χ
2
2
|∇v | + |xv | dx +
|v |2α+2 dx
H(v ) =
2
2α + 2
then by Itô formula, for all τ < τ ∗ (ψ0 ) a.s.
Z τ Z
Z
2
H(ψ(τ )) = H(ψ0 )−2Im
∇ψ.xψdx dW (s)+2σ
0
0
τ
|xψ(s)|2L2 ds
+ martingale inequalities for the stochastic integrals
Z τ ∧T
2
|ψ(s)|4Σ ds
E sup H (ψ(t)) ≤ C (|ψ0 |Σ , T ) E
t≤τ ∧T
(stochastic NLS)
0
20 / 26
+ Gagliardo-Nirenberg inequality (if α ≤ 2/d)
Z
1
2α+2−αd
|v |2α+2 dx ≤ Cα |∇v |αd
L2 |v |L2
2α + 2
Assume that
ψ0 ∈ Σ
χ = +1 or χ = −1 and α < 2/d
4/d
or χ = −1, α = 2/d and |ψ0 |L2 < 1/Cα
then Gronwall Lemma implies
E sup |ψ(t)|4Σ ≤ C (T , |ψ0 |Σ
t≤τ ∧T
the solution in Σ exists for all t i.e. τ ∗ (ψ0 ) = +∞ a.s.
In particular true if α = 1 in 1-D for any ψ0 , and 2-D if the total number
of atoms is sufficiently small
(stochastic NLS)
21 / 26
Existence of collapsing states (case γ = 0)
R
Itô formula for I (ψ(t)) with I (v ) = |xv |2 dx, assuming ψ0 ∈ Σ (ψ0 ∈ Σ2
if d = 2) and 0 < t̄ < τ ∗ (ψ0 ) a.s. gives
I (ψ(t̄)) = I (ψ0 ) + 4G (ψ0 )t̄ + 8H(ψ0 )t̄ 2
Z t̄
Z
2
−16
(t̄ − s) Im
∇ψ.x ψ̄ dx dW (s)
Z t̄0
Z t̄
−8
(t̄ − s)I (ψ(s))dW (s) + 16σ 2
I (ψ(s))ds
0
0
Z
Z t̄
4χ(2 − αd)
−16
(t̄ − s)I (ψ(s))ds −
|ψ|2α+2 dx
α
+
1
0
Assume χ < 0, α ≥ 2/d ; choose ψ0 ∈ Σ (Σ2 if d = 2) such that
I (ψ0 ) + 4G (ψ0 )t̄ + 8H(ψ0 )t̄ 2 ≤ −δ < 0
for some t̄ > 0 with σ 2 t̄ ≤ 1 (always possible to find such a ψ0 )
(stochastic NLS)
22 / 26
Then a.s.
I (ψ(t̄)) ≤ I (ψ0 ) + 4G (ψ0 )t̄ + 8H(ψ0 )t̄ 2 +
Z
t̄
g (t̄, s)dW (s)
0
where
R t̄
0
g 2 (t̄, s)ds < +∞ a.s. Since I (ψ(s)) ≥ 0 a.s, this implies
Z
t̄
g (t̄, s)dW (s) ≥ δ > 0,
a.s.
0
but this is impossible for a local martingale starting at 0.
Hence, contradiction, i.e. collapse occurs before t̄ : the probability
P(τ ∗ (ψ0 )) ≤ t̄) is positive
(stochastic NLS)
23 / 26
Conclusion
No collapse for χ > 0 (positive atomic scattering length)
No collapse if χ < 0 (negative atomic scattering length) and
α < 2/d, or if α = 2/d and small total number of atoms :
Z
|ψ0 |2 dx < C0
Existence of collapsing states if χ < 0 and α ≥ 2/d :
At least those for which H(ψ0 ) < 0 or H(ψ0 ) ≥ 0 and
−2G (ψ0 ) ≥ (H(ψ0 )I (ψ0 ))1/2 and
8H(ψ0 ) + 4σ 4 I (ψ0 ) + 4σ 2 G (ψ0 ) < 0
True also in the presence of damping in the supercritical case
(α > 2/d)
(stochastic NLS)
24 / 26
Open problems
• Collapse for any initial state (supercritical focusing case) with positive
probability :
related to the determination of the support of the law of the solutions as
random variables with values in L2 (0, T ; Σ)
need to solve the approximate controllability problem
Find h ∈ L2 (0, T ; R) such that the solution at time T of
(
du
+ (∆u − |x|2 u) + |u|2α u = |x|2 uh
dt
u(0) = ψ0
i
satisfies I (u(T )) + 4G (u(T ))t̄ + 8H(u(T ))t̄ 2 < 0 for some T > 0, and
t̄ > 0 with σ 2 t̄ ≤ 1 then the solution would blow up before T + t̄
whatever ψ0 is
de Bouard, Debussche, Annals of Proba. 2005
(stochastic NLS)
25 / 26
Existence of solutions in the 3-D case : need dispersive estimates on
the linear equation
idψ + (∆ψ − |x|2 ψ)dt = |x|2 ψ ◦ dW
(no useful change of variable seems available)
Computation of the (bright) soliton parameters in the attractive case
(dark soliton in the repulsive case : much more difficult)
(stochastic NLS)
26 / 26