computational methods for the dynamics of nonlinear...
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Computational Methods for the Dynamics of Nonlinear Schrodinger / Gross-Pitaevskii Equations
Weizhu Bao
Department of Mathematics
& Center for Computational Science and Engineering National University of Singapore
Email: [email protected] URL: http://www.math.nus.edu.sg/~bao
& Beijing Computational Science Research Center (CSRC)
URL: http://www.csrc.ac.cn
Outline
Nonlinear Schrodinger / Gross-Pitaevskii equations Dynamical properties – Conserved quantities – Center-of-mass & an analytical solution – Specific solutions – soliton in 1D
Numerical methods – Finite difference time domain (FDTD) methods – Time-splitting spectral (TSSP) method – Applications – collapse & explosion of a BEC, vortex lattice dynamics
Extension to -- rotation, nonlocal interaction
Conclusions
NLSE / GPE
The nonlinear Schrodinger equation (NLSE) --1925
– t : time & : spatial coordinate – : complex-valued wave function – : real-valued external potential – : given interaction constant
• (=0: linear; >0: repulsive & <0: attractive)
– : scaled Planck constant • ( : standard; : semiclassical)
22 2( , ) ( ) | |
2ti x t V xεε ψ ψ ψ β ψ ψ∂ = − ∇ + +
( )dx ∈
( , )x tψ
( )V x
β
0 1ε< ≤0 1& 1ε β< = ±1ε =
Model for BEC
Bose-Einstein condensation (BEC): – Bosons at nano-Kevin temperature – Many atoms occupy in one obit -- at quantum mechanical ground state – Form like a `super-atom’, New matter of wave --- fifth state
Theoretical prediction – S. Bose & E. Einstein 1924’
Experimental realization – JILA 1995’
2001 Noble prize in physics – E. A. Cornell, W. Ketterle, C. E. Wieman
Mean-field approximation – Gross-Pitaevskii equation (GPE) :
• E.P. Gross 1961’; L.P. Pitaevskii 1961’
BEC@ JILA
Laser beam propagation
Nonlinear wave (or Maxwell) equations Helmholtz equation – time harmonic
In a Kerr medium Paraxial (or parabolic) approximation -- NLSE
Other applications
In plasma physics: wave interaction between electrons and ions – Zakharov system, …..
In quantum chemistry: chemical interaction based on the first principle – Schrodinger-Poisson system
In materials science: – First principle computation – Semiconductor industry
In nonlinear (quantum) optics In biology – protein folding In superfluids – flow without friction
Conservation laws
Dispersive Time symmetric: Time transverse (gauge) invariant Mass (or wave energy) conservation
Energy (or Hamiltonian) conservation
22 2( , ) ( ) | |
2ti x t V xεε ψ ψ ψ β ψ ψ∂ = − ∇ + +
& take conjugate unchanged!!t t→ − ⇒
2/( ) ( ) --unchanged!!i tV x V x e α εα ψ ψ ρ ψ−→ + ⇒ → ⇒ =
2 2( ) : ( ( , )) ( , ) ( ,0) 1, 0
d d
N t N t x t d x x d x tψ ψ ψ= = ≡ = ≥∫ ∫
22 2 4( ) : ( ( , )) ( ) (0), 0
2 2d
E t E t V x d x E tε βψ ψ ψ ψ
= = ∇ + + ≡ ≥ ∫
Dynamics with no potential
Momentum conservation Dispersion relation Soliton solutions in 1D: – Bright soliton when ---- decaying to zero at far-field
– Dark (or gray) soliton ---- nonzero &oscillatory at far-field
( ) 0, dV x x≡ ∈
( ) : Im (0) 0d
J t d x J tψ ψ= ∇ ≡ ≥∫
2( ) 2( , )2
i k x tx t Ae k Aω ε βψ ωε
−= ⇒ = +
0β <
0β >
2 20
1( ( ) )2
0( , ) sech( ( )) , , 0i vx v a tax t a x vt x e x t
θψ
β− − +
= − − ∈ ≥−
[ ]2 2 2
01( ( 2 2( ) ) )2
01( , ) tanh( ( )) ( ) , , 0
i kx k a v k tx t a a x vt x i v k e x t
θψ
β− + + − +
= − − + − ∈ ≥
1ε =
Dynamics with harmonic potential
Harmonic potential
Center-of-mass: An analytical solution if
2 2
2 2 2 2
2 2 2 2 2 2
11( ) 22
3
x
x y
x y z
x dV x x y d
x y z d
γγ γ
γ γ γ
== + = + + =
2( ) ( , )
dcx t x x t d xψ= ∫
2 2 2( ) diag( , , ) ( ) 0, 0 each component is periodic!!c x y z cx t x t tγ γ γ+ = > ⇒
0 0( ) ( )sx x xψ φ= −
( , )0
2 2
( , ) ( ( )) , (0) & ( , ) 0
( , ) : | ( , ) | | ( ( )) | -- moves like a particle!!
si t iw x ts c c
s c
x t e x x t e x x w x tx t x t x x t
µψ φ
ρ ψ φ
−= − = ∆ =
⇒ = = −
1ε =
22 2( ) ( ) | |
2s s s s s sx V xεµ φ φ φ β φ φ= − ∇ + +
Numerical methods for dynamics
Dispersive & nonlinear Solution and/or potential are smooth but may oscillate wildly Keep the properties of NLSE on the discretized level – Time reversible & time transverse invariant – Mass & energy conservation – Dispersion relation
In high dimensions: many-body problems
Design efficient & accurate numerical algorithms – Explicit vs implicit (or computation cost) – Spatial/temporal accuracy, Stability – Resolution in strong interaction regime: 1& 1 or 0 1& 1β ε ε β= < = ±
22 2
0
( , ) ( ) | | , , 02
with ( ,0) ( )
dti x t V x x t
x x
εε ψ ψ ψ β ψ ψ
ψ ψ
∂ = − ∇ + + ∈ >
=
Numerical difficulties
Explicit vs implicit (or computation cost)
Spatial/temporal accuracy Stability Keep the properties of NLSE in the discretized level – Time reversible & time transverse invariant – Mass & energy conservation – Dispersion conservation
Resolution in the semiclassical regime:
0 1ε<
/ (solution has wavelength of ( ))i Se Oεψ ρ ε=
Crank-Nicolson finite difference (CNFD)
Crank-Nicolson finite difference (CNFD) method (Glassey, JCP, 92’; Chan, Guo & Jiang, Math. Comp., 94’; Chan& Shen, SINUM, 88’; Chang & Sun, JCP 03’; ....)
– Implicit: need solve a fully nonlinear system per time step – Time reversible: Yes – Time transverse invariant: No – Mass conservation: Yes
1 1 1 121 1 1 1
2 2
1 1 2 2 1
2 24
( )( ) (| | | | ) ( )
2 4
n n n n n n n nj j j j j j j j
j n n n n n nj j j j j j
ih h
V x
ψ ψ ψ ψ ψ ψ ψ ψεετ
βψ ψ ψ ψ ψ ψ
+ + + ++ − + −
+ + +
− − + − += − +
+ + + + +
22
0
( , ) ( ) | | , , 02
( , ) ( , ) 0, with ( ,0) ( ),
t xxi x t V x a x b t
a t b t x x a x b
εε ψ ψ ψ β ψ ψ
ψ ψ ψ ψ
∂ = − ∂ + + < < >
= = = ≤ ≤
CNFD for NLSE
– Stability: Yes – Energy conservation: Yes – nonlinear system must be solved very accurately
– Dispersion relation without potential: No – Accuracy
• Spatial: 2nd order • Temporal: 2nd order
– Resolution in semiclassical regime (Markowich, Poala & Mauser, SINUM, 02’)
– Error estimate in H^1-norm: • In 1D: R.T. Glassey, Q. Chang, B. Guo, H. Jiang, etc. • In 2D&3D: W. Bao and Y. Cai , Math. Comp. 13’
2 2( ) & ( ) ( ) & ( )h o o h O Oε τ ε ε τ ε= = ⇐ = =
2 2( )ne C h τ≤ +
Time-splitting spectral method (TSSP)
For , apply time-splitting technique – Step 1: Discretize by spectral method & integrate in phase space exactly
– Step 2: solve the nonlinear ODE analytically
Use 2nd or 4th order splitting (Bao,Jin & Markowich, JCP, 02’; citations >200 !!) – (Tarppent, SIAM 78’; Moris etc., JCP, 88’; Ablowitz etc. JCP, 84’,…..)
1[ , ]n nt t +
22( , )
2ti x t εε ψ ψ∂ = − ∇
2
2
2
2
( )[ ( ) | ( , )| ]/
( , ) ( ) ( , ) | ( , ) | ( , )
(| ( , ) | ) 0 | ( , ) | | ( , ) |
( , ) ( ) ( , ) | ( , ) | ( , )
( , ) ( , )n n
t
t n
t n
i t t V x x tn
i x t V x x t x t x t
x t x t x ti x t V x x t x t x t
x t e x tβ ψ ε
ε ψ ψ β ψ ψ
ψ ψ ψ
ε ψ ψ β ψ ψ
ψ ψ− − +
∂ = +
⇓ ∂ = ⇒ =
∂ = +
⇒ =
1:ε =
Properties of the method
Explicit & computational cost per time step: Time reversible: yes Time transverse invariant: yes Mass conservation: yes Stability: yes
( ln )O M M
1 & scheme unchanged!!n n τ τ+ ↔ ↔ − ⇒
/( ) ( ) (0 ) | | unchanged!!!n n i n nj j j j jV x V x j M e τ α εα ψ ψ ψ−→ + ≤ ≤ ⇒ → ⇒
22 2
1 12 0 2
0 00 0
: | | : | ( ) | , 0,1, for any &M M
n nj jll l
j jh h x n h kψ ψ ψ ψ ψ
− −
= =
= ≡ = = =∑ ∑
Properties of the method
Dispersion relation without potential: yes
– Exact for plane wave solution
Energy conservation (Bao, Jin & Markowich, JCP, 02’): – cannot prove analytically – Conserved very well in computation
( / )0
22 2
(0 ) (0 & 0)
with | | if2
j j ni k x i k x tnj ja e j M a e j M n
k a M k
ω εψ ψ
εω β
−= ≤ ≤ ⇒ = ≤ ≤ ≥
= + >
Properties of the method
Accuracy----- Spatial: spectral order & Temporal: 2nd order
Resolution in semiclassical regime (Bao, Jin & Markowich, JCP, 02’)
– Linear case: – Weakly nonlinear case: – Strongly repulsive case:
Error estimate in L^2-norm and/or H^1:
– C. Besse, C. Lubich, O. Koch, M. Thalhammer, M. Caliari, C. Neuhauser, E. Faou, A. Debussche, L. Gauckler, E. Hairer, J. Shen&Z.Q. Wang, W. Bao&Y. Cai, etc.
0β =
( ) & independent of h O ε τ ε= −( )Oβ ε=
( ) & independent of h O ε τ ε= −0 (1)Oβ< =
( ) & ( )h O Oε τ ε= = 2( )n me C h τ≤ +
Comparison
Interaction of a lattice
3D collapse & explosion of BEC
Experiment (Donley et., Nature, 01’) – Start with a stable condensate (as>0) – At t=0, change as from (+) to (-) – Three body recombination loss
Mathematical model (Duine & Stoof, PRL, 01’)
ψψβδψψβψψψ 420
22 ||||)(21),( ixVtx
ti −++∇−=∂∂
4 s
s
N ax
πβ =
Numerical results (Bao et., J Phys. B, 04)
Jet formation
3D Collapse and explosion in BEC
4 s
s
N ax
πβ =
3D Collapse and explosion in BEC
GPE with angular rotation
GPE / NLSE with an angular momentum rotation
2 21( , ) [ ( ) | | ] , , 02
: ( ) , ,
dt z
z y x y x
i x t V x L x t
L xp yp i x y i L x P P iθ
ψ β ψ ψ∂ = − ∇ + −Ω + ∈ >
= − = − ∂ − ∂ ≡ − ∂ = × = − ∇
Vortex @MIT
Numerical methods
Time-splitting + polar (cylindrical) coordinates –Bao, Du & Zhang, SIAP, 05’
Time-splitting + ADI -- Bao & Wang, JCP, 06’
Time-splitting + Laguerre-Hermite functions –Bao, Li &Shen, SISC, 09’
2
2
1Step 1: ( , ) [ ]2
Step 2: ( , ) [ ( ) | | ]
t z
t
i x t L
i x t V x
ψ ψ
ψ β ψ ψ
∂ = − ∇ −Ω
∂ = +
2
1Step 1: ( , ) [ ]21Step 2: ( , ) [ ]2
Step 3: ( , ) [ ( ) | | ]
t xx x
t yy y
t
i x t i y
i x t i x
i x t V x
ψ ψ
ψ ψ
ψ β ψ ψ
∂ = − ∂ − Ω ∂
∂ = − ∂ + Ω ∂
∂ = +
22
2
1Step 1: ( , ) [ / 2] :2
Step 2: ( , ) [ ( ) | | ]
t z
t
i x t L x L
i x t W x
ψ ψ ψ
ψ β ψ ψ
∂ = − ∇ −Ω + =
∂ = +
A simple & efficient method
Ideas – Bao, Marahrens,Tang & Zhang, 13’; Bao & Cai, KRM, 13’; ……
– A rotating Lagrange coordinate:
– GPE in rotating Lagrange coordinates – TSSP method
cos( ) sin( ) 0cos( ) sin( )
( ) for 2; ( ) sin( ) cos( ) 0 for 3sin( ) cos( )
0 0 1
t tt t
A t d A t t t dt t
Ω Ω Ω Ω = = = − Ω Ω = − Ω Ω
1( ) & ( , ) : ( , ) ( ( ) , )x A t x x t x t A t x tφ ψ ψ−= = =
2 21( , ) [ ( ( ) ) | | ] , , 02
dti x t V A t x x tφ β φ φ∂ = − ∇ + + ∈ >
2
2
1Step 1: ( , ) ,2
Step 2: ( , ) [ ( ( ) ) | | ] ,
t
t
i x t
i x t V A t x
φ φ
φ β φ φ
∂ = − ∇
∂ = +
Dynamics of a vortex lattice
Extension to dipolar quantum gas
Gross-Pitaevskii equation (re-scaled) – Trap potential – Interaction constants – Long-range dipole-dipole interaction kernel
References:
– L. Santos, et al. PRL 85 (2000), 1791-1797 – S. Yi & L. You, PRA 61 (2001), 041604(R); – D. H. J. O’Dell, PRL 92 (2004), 250401
( )2 2 2dip
1( , ) ( ) | | | |2t zi x t V x L Uψ β ψ λ ψ ψ ∂ = − ∇ + −Ω + + ∗
3( , )x t xψ ψ= ∈
( )2 2 2 2 2 21( )2 x y zV x x y zγ γ γ= + +
20 dip
2
4 (short-range), (long-range)3
s
s s
mNN ax x
µ µπβ λ= =
2 2 2
dip 3 33 1 3( ) / | | 3 1 3cos ( )( )
4 | | 4 | |n x xU x
x xθ
π π− ⋅ −
= =
A New Formulation
Using the identity (O’Dell et al., PRL 92 (2004), 250401, Parker et al., PRA 79 (2009), 013617)
Dipole-dipole interaction becomes
2
dip 3 2
2
dip 2
3 3( ) 1( ) 1 ( ) 3 4 4
3( ) ( ) 1| |
n nn xU x x
r r r
nU
δπ π
ξξξ
⋅ = − = − − ∂
⋅⇒ = − +
2 2dip
2 2 2
| | | | 3
1 | | | |4
n nU
r
ψ ψ ϕ
ϕ ψ ϕ ψπ
∗ = − − ∂
= ∗ ⇔ −∇ =
| | & & ( )n n n n nr x n= ∂ = ⋅∇ ∂ = ∂ ∂
BEC@Stanford
A New Formulation
Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10’)
– Energy
– Model in 2D
Numerical methods --- TSSP with sine basis instead of Fourier basis – Bao, Cai & Wang, JCP, 10’; Bao & Cai, KRM, 13’ – Bao, Marahrens,Tang & Zhang, 13’’; Bao & Cai, KRM, 13’; ……
2 2
2 2 3
| |
1( , ) ( ) ( ) | | 32
( , ) | ( , ) | , , lim ( , ) 0
t z n n
x
i x t V x L
x t x t x x t
ψ β λ ψ λ ϕ ψ
ϕ ψ ϕ→∞
∂ = − ∇ + −Ω + − − ∂ −∇ = ∈ =
3
2 2 4 21 3( ( , )) : | | ( ) | | | | | |2 2 2z nE t V x L d xβ λ λψ ψ ψ ψ ψ ψ ϕ− ⋅ = ∇ + −Ω + + ∂ ∇ ∫
21/2 2 2
| |( ) ( , ) | ( , ) | , , lim ( , ) 0
D
xx t x t x x tϕ ψ ϕ⊥ →∞
→ −∆ = ∈ =
Dynamics of a vortex lattice
Conclusions & Future Challenges
Conclusions: – NLSE / GPE – brief motivation
– Dynamical properties – Numerical methods
• Time-splitting spectral (TSSP) method & Applications – Extensions – rotations, nonlocal, system
Future Challenges – With random potential or high dimensions – Coupling GPE & Quantum Boltzmann equation (QBE) – NLSE / GPE coupled with other equations – BEC at finite temperature & quantum turbulence