Analysis & Computation for the Semiclassical Limits of the - - PowerPoint PPT Presentation

Analysis & Computation for the Semiclassical Limits of the Nonlinear Schrodinger Equations

Weizhu Bao

Department of Mathematics & Center of Computational Science and Engineering National University of Singapore Email: bao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~bao

Outline

Motivation Semiclassical limits of ground and excited states

– Matched asymptotic approximations – Numerical results

Semiclassical limits of the dynamics of NLS

– Formal limits – Efficient computation – Caustics & vacuum – Difficulties in rotating frame and system

Conclusions

Motivation: NLS

The nonlinear Schr dinger (NLS) equation

– t : time & : spatial coordinate – : complex-valued wave function – : real-valued external potential – : scaled Planck constant – : interaction constant

• =0: linear; =1: repulsive interaction
• = -1: attractive interaction

2 2 2

( , ) ( ) | | 2

t

i x t V x

ε ε ε ε ε

ε ε ψ ψ ψ β ψ ψ ∂ = − ∇ + + r r

( R )

d

x ∈ r

( , ) x t ψ r

( ) V x r

( 0, 1) β = ±

• &&

(0 1) ε ε < ฀

Motivation

In quantum physics & nonlinear optics:

– Interaction between particles with quantum effect – Bose-Einstein condensation (BEC): bosons at low temperature – Superfluids: liquid Helium, – Propagation of laser beams, …….

In plasma physics; quantum chemistry; particle physics; biology; materials science; ….

Conservation laws

2 2 2 2 R R R 2 2 2 4 2 R

( ) : ( , ) ( ,0) ( ) : ( ) ( 1), ( ) : ( , ) ( ) ( , ) ( , ) ( ) 2

d d d d

N x t d x x d x x d x N E x t V x x t x t d x E

ε ε ε ε ε ε β ε ε ε ε ε

ψ ψ ψ ψ ψ ψ ε ψ ψ ψ ψ ψ = = ≡ = = = ⎡ ⎤ = ∇ + + ≡ ⎢ ⎥ ⎣ ⎦

∫ ∫ ∫ ∫

r r r r r r r r r r

Semiclassical limits

Suppose initial data chosen as Semiclassical limits:

– Density: – Current: – Other observable:

Analysis: dispersive limits

– WKB method vs Winger transform

Efficient computation

– Highly oscillatory wave in space & time

0 ( )/

( , )/

( ,0) : ( ) ( ) ( , ) ( , )

iS x iS x t

x x A x e x t A x t e

ε ε

ε ε ε ε ε ε ε

ψ ψ ψ = = ⇒ =

r r

r r r r r

ε →

2

: ????

ε ε

ρ ψ = →

: ??? : ??? J v v S

ε ε ε ε ε

ρ = → = ∇ → r r r

For ground & excited states

For special initial data: Time-independent NLS:nonlinear eigenvalue problem

– Eigenvalue (or chemical potential) – Eigenfunctions are

• Orthogonal in linear case & Superposition is valid for dynamics!!
• Not orthogonal in nonlinear case !!!! No superposition for dynamics!!!

/

( ) ( ) & ( ) ( , ) ( )

i t

A x x S x x t x e

ε

ε ε ε ε ε μ ε

φ ψ φ

−

= = ⇒ = r r r r r

2 2 2 2 2

( ) ( ) | | , : ( ) 1 2

d

R

x V x x d x

ε ε ε ε ε ε ε ε

ε μ φ φ φ β φ φ φ φ = − ∇ + + = =

∫

r r r r

2 2 2 4 R

: ( ) : ( ) ( ) ( ) ( ) 2

d

x V x x x d x

ε ε ε ε ε

ε μ μ φ φ φ β φ ⎡ ⎤ = = ∇ + + ⎢ ⎥ ⎣ ⎦

∫

r r r r

For ground & excited states

Ground state: minimizer of the nonconvex minimization problem

– Existence: – Positive solution is unique

Excited states: eigenfunctions with higher energy Semiclassical limits

{ }

: ( ) min ( ), | 1, ( )

g g S

E E E S E

ε

ε ε ε φ

φ φ φ φ φ

∈

= = = = < ∞

| |

& lim ( )

x

V x β

→∞

≥ = ∞

r

r

ε →

1 2 1 2

??? ??? : ( ) ??? ??? ??? ??? ?????

g g g g j j j g j g j

E E E E E E

ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε

φ μ μ φ φ μ μ μ μ μ → → = → → → → < < < < < ⇒ < < < < < L L L L

1 2

, , , , : ( ), : ( )

j j j j j

E E

ε ε ε ε ε ε ε

φ φ φ φ μ μ φ = = L L

For ground state: Box Potential in 1D

The potential: The nonlinear eigenvalue problem (Bao,Lim,Zhang,Bull. Inst. Math., 05’) Leading order approximation, i.e. drop the diffusion term

– Boundary condition is NOT satisfied, i.e. – Boundary layer near the boundary

0, 1, ( ) 1, 1 ,

• therwise.

x V x d β ≤ ≤ ⎧ = = = ⎨∞ ⎩

2 2 1 2

( ) ( ) ( ) | ( ) | ( ), 1, 2 (0) (1) with | ( ) | 1 x x x x x x dx

ε ε ε ε ε ε ε ε

ε μ φ φ φ φ φ φ φ ′′ = − + < < = = =

∫

TF TF TF 2 TF TF TF 1 TF 2 TF TF TF g

( ) | ( ) | ( ), 1, ( ) | ( ) | 1 1 (x) ( ) 1, 1, E E 2

g g g g g g g g g g g g

x x x x x x dx x

ε ε ε

μ φ φ φ φ μ φ φ φ μ μ = < < ⇒ = ⇓ = ≈ = ≈ = ≈ =

∫

TF TF

(0) (1) 1

g g

φ φ = = ≠

1 ε < ฀

For ground state: Box Potential in 1D

Matched asymptotic approximation

– Consider near x=0, rescale – We get – The inner solution – Matched asymptotic approximation for ground state

, ( ) ( )

g g g

x X x x

ε ε ε

ε φ μ μ = = Φ

3

1 ( ) ( ) ( ), ; (0) 0, lim ( ) 1 2

X

X X X X X

→∞

′′ Φ = − Φ + Φ ≤ < ∞ Φ = Φ =

( ) tanh( ), ( ) tanh( ), (1)

g g g

X X X x x x

• ε

μ φ μ ε Φ = ≤ < ∞ ⇒ ≈ ≤ =

MA MA MA MA MA 1 MA 2 MA 2 2 TF 2 2

( ) ( ) tanh( ) tanh( (1 )) tanh( ) , 1 1 | ( ) | 1 2 1 2 2 1 2 , 1.

g g g g g g g g g g

x x x x x x dx

ε ε

μ μ μ φ φ μ ε ε ε φ μ μ ε ε ε μ ε ε ε ε ⎡ ⎤ ⎢ ⎥ ≈ = + − − ≤ ≤ ⎢ ⎥ ⎣ ⎦ = ⇒ ≈ = + + + = + + + <

∫

฀

For ground state: Box Potential in 1D

– Approximate energy – Asymptotic ratios: – Width of the boundary layer:

Semiclassical limits

MA 2 2

1 4 1 2 2 3

g g

E E

ε

ε ε ε ≈ = + + +

( ) O ε

1 lim , 2

g g

Eε

ε ε

μ

→

=

1 1 1 ( ) 1 0,1 2

g g g g

x x E x

ε ε ε

φ φ μ < < ⎧ → = → → ⎨ = ⎩

ε →

For excited states: Box Potential in 1D

Matched asymptotic approximation for excited states

– Approximate chemical potential & energy – Boundary & interior layers

Semiclassical limits

MA MA MA [( 1)/2] [ /2] MA MA

2 2 1 ( ) ( ) [ tanh( ( )) tanh( ( )) tanh( )] 1 1

j j g g g j j j j l l

l l x x x x C j j

ε

μ μ μ φ φ μ ε ε ε

+ = =

+ ≈ = − + − − + +

∑ ∑

MA 2 2 2 2 MA 2 2 2 2

1 2( 1) 1 ( 1) 2( 1) , 1 4 ( 1) 1 ( 1) 2( 1) , 2 3

j j j j

j j j E E j j j

ε ε

μ μ ε ε ε ε ε ε ≈ = + + + + + + ≈ = + + + + + +

( ) O ε

1 2 1 2

1 /( 1) 1 ( ) 1 /( 1) 2

j j j j g j g j

x l j x E x l j E E E E

ε ε ε ε ε ε ε ε ε ε ε

φ φ μ μ μ μ μ ± ≠ + ⎧ → = → → ⎨ = + ⎩ < < < < < ⇒ < < < < < L L L L

ε →

Extension & numerical computation

Extension

– High dimension – Nonzero external potential

Numerical method & results

– Normalized gradient flow – Backward Euler finite difference method

For dynamics: Formal limits

WKB analysis

– Formally assume – Geometrical Optics: Transport + Hamilton-Jacobi

2 2 2 ( )/

( , ) ( ) | | 2 ( ,0) : ( ) ( )

t iS x

i x t V x x x x e

ε

ε ε ε ε ε ε ε ε ε

ε ε ψ ψ ψ β ψ ψ ψ ψ ρ ∂ = − ∇ + + = =

r

r r r r r

/ ,

,

iS

e v S J v

ε

ε ε ε ε ε ε ε ε

ψ ρ ρ = = ∇ = r r r

2 2 t

( ) 0, 1 1 ( ) 2 2

t d

S S S V x

ε ε ε ε ε ε ε ε

ρ ρ ε β ρ ρ ρ ∂ + ∇• ∇ = ∂ + ∇ + + = Δ r

For dynamics: Formal limits

– Quantum Hydrodynamics (QHD): Euler +3rd dispersion – Formal Limits

Mathematical justification: G. B. Whitman, E. Madelung, E. Wigner,

P.L. Lious, P. A. Markowich, F.-H. Lin, P. Degond, C. D. Levermore, D. W. McLaughlin,

• E. Grenier, F. Poupaud, C. Ringhofer, N. J. Mauser, P. Gerand, R. Carles, P. Zhang,
• P. Marcati, J. Jungel, C. Gardner, S. Kerranni, H.L. Li, C.-K. Lin, C. Sparber, …..

– Linear case – NLS before caustics

2 2

( ) ( ) / 2 ( ) ( ) ( ) ( ln ) 4

t t

v P J J J P V

ε ε ε ε ε ε ε ε ε ε ε

ρ ρ ρ β ρ ε ρ ρ ρ ρ ρ ∂ + ∇• = = ⊗ ∂ + ∇• + ∇ + ∇ = ∇ Δ r r r r

2

( ) ( ) / 2 ( ) ( ) ( )

t t

v P J J J P V ρ ρ ρ β ρ ρ ρ ρ ∂ + ∇• = = ⊗ ∂ + ∇• + ∇ + ∇ = r r r r

Efficient Computation

Solve the limiting QHD system with multi-values

– Level set method: S. Osher, S. Jin, H.L. Liu, L.T. Cheng, …. – K-branch method: L. Goss, P.A. Markowich, …..

Solve the Liouville equation (obtained by Wigner transform): S. Jin,

• X. Wen, ……..

Directly solve NLS: J.C. Bronksi, D.W. McLaughlin, P.A. Markowich, P.

Pietra, C. Pohl, P.D. Miller, S. Kamvissis, H.D. Ceniceros, F.R. Tian, W. Bao, S. Jin, P. Degond, N. J. Mauser, H. P. Stimming, …….

– is small but finite, e.g. 0.01 to 0.1 in typical BEC setups – Provide benchmark results for other approaches – Hints for analysis after caustics and/or with vacuum

ε

NLS and its properties

Time reversible Time transverse invariant (gauge invariant) Mass (wave energy) conservation Energy ( or Hamiltonian) conservation Dispersion relation without external potential

/

( ) ( ) | | unchanged

it

V x V x e

ε ε α ε ε

α ψ ψ ψ

−

→ + ⇒ → ⇒ r r

2 2 2

( ) : ( , ) (0) : ( ,0) ( ) ,

d d d

N t x t d x N x d x x d x t

ε ε ε ψ ψ

ψ ψ ψ = ≡ = = ≥

∫ ∫ ∫

฀ ฀ ฀

r r r r r r

2 2 2 4 2

( ) : ( , ) ( ) ( , ) ( , ) (0), 2

d

E t x t V x x t x t d x E t

β ε ε ε ψ ψ

ε ψ ψ ψ ⎡ ⎤ = ∇ + + ≡ ≥ ⎢ ⎥ ⎣ ⎦

∫

฀

r r r r

2 ( ) 2 2

( , ) (plane wave solution) | | 2

i k x t

x t a e k a

ε ω

ε ψ ω β

−

= ⇒ = +

2 2 2 ( )/

( , ) ( ) | | 2 ( ,0) ( ) ( )

t iS x

i x t V x x x A x e

ε

ε ε ε ε ε ε ε ε ε

ε ε ψ ψ ψ β ψ ψ ψ ψ ∂ = − ∇ + + = =

r

r r r r r

Numerical difficulties

Explicit vs implicit (or computation cost) Spatial/temporal accuracy Stability Keep the properties of NLS in the discretized level

– Time reversible & time transverse invariant – Mass & energy conservation – Dispersion conservation

Resolution in the semiclassical regime: 0

1 ε < ฀

/

(solution has wavelength of ( ))

i S

A e O

ε

ε ε ε

ψ ε =

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 order Strang splitting (or 4th order time-splitting)

1

[ , ]

n n

t t +

2 2

( , ) 2

t

i x t

ε ε

ε ε ψ ψ ∂ = − ∇ r

2

2 2 2 ( )[ ( ) | ( , )| ]/

( , ) ( ) ( , ) | ( , ) | ( , ) (| ( , ) | ) | ( , ) | | ( , ) | ( , ) ( ) ( , ) | ( , ) | ( , ) ( , )

n n

t t n t n i t t V x x t

i x t V x x t x t x t x t x t x t i x t V x x t x t x t x t e

ε

ε ε ε ε ε ε ε ε ε ε ε β ψ ε ε

ε ψ ψ β ψ ψ ψ ψ ψ ε ψ ψ β ψ ψ ψ

− − +

∂ = + ⇓ ∂ = ⇒ = ∂ = + ⇒ =

r

r r r r r r r r r r r r r r ( , )

n

x t

ε

ψ r

An algorithm in 1D for NLS

Choose time step: ; set Choose mesh size ; set The algorithm (10 lines code in Matlab!!!) (Bao,Jin,Markowich, JCP,02’)

– with

k t = Δ

, 0,1,

n

t n k n = = L

b a h x M − = Δ =

& ( , )

n j j j n

x a j h x t ψ ψ = + ≈

2 2 (2) 2

[ ( ) | | ]/2 (1) /2 1 ( ) /2 (2) (1) / 2 [ ( ) | | ]/ 2 1 (2)

1 ˆ , 0,1, , 1

n j j l j l j j

i k V x n j j M i x a i k j j l M i k V x n j j

e e e j M M e

β ψ ε μ ε μ β ψ ε

ψ ψ ψ ψ ψ ψ

− + − − − =− − + +

= = = − =

∑

L

1 ( ) (1) (1)

ˆ , , , 1, , 1 ( ) 2 2 2

l j

M i x a l l j j

l M M M e l b a

μ

π μ ψ ψ

− − − =

= = = − − + − −

∑

L

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

1 & scheme unchanged!!

n n j j

n n ψ ψ

+

+ ↔ ↔ ⇒

/

( ) ( ) (0 ) | | unchanged!!!

n n i n k n j j j

V x V x j M e

α ε

α ψ ψ ψ

−

→ + ≤ ≤ ⇒ → ⇒

2 2 2

1 1 2 2

: | | : | ( ) | , 0,1, for any &

M M n n j j l l l j j

h h x n h k ψ ψ ψ ψ ψ

− − = =

= ≡ = = =

∑ ∑

L

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

( ) 2 2 2

(0 ) (0 & 0) with | | if 2

j j n

i k x i k x t n j j

a e j M a e j M n k a M k

ω

ψ ψ ε ω β

−

= ≤ ≤ ⇒ = ≤ ≤ ≥ = + >

Properties of the method

Accuracy

– Spatial: spectral order – Temporal: 2nd or 4th order

Resolution in semiclassical regime (Bao, Jin & Markowich, JCP, 02’)

– Linear case: – Weakly nonlinear case: – Strongly repulsive case:

Error estimate: not available yet!!

β =

( ) & independent of h O k ε ε = −

( ) O β ε =

( ) & independent of h O k ε ε = −

(1) O β < =

( ) & ( ) h O k O ε ε = =

Numerical Results

Example 1. Linear Schrodinger equation

2

25( 0.5) 5( 0.5) 5( 0.5)

1 0, ( ) 10, ( ) , ( ) ln[ ] 5

x x x

V x A x e S x e e β

− − − − −

= = = = − +

Density at t=0.54 Current at t=0.54

0.0256 0.0064 0.0008 ε ε ε = = =

Crank-Nicolson finite difference method Crank-Nicolson spectral method

Numerical Results

Example 2. NLS with defocusing nonlinearity

1 | | | | 1 1, ( ) 0, ( ) , ( ) ln[ ]

• therwise

x x

x x V x A x S x e e β

−

− < ⎧ = = = = − + ⎨ ⎩

( , )

t

x s d s

ε

ρ

∫

Before caustics After caustics

Observations

– Before caustics: converge strongly & converge to QHD – After caustics: converge weakly but the location of discontinuity is

different to QHD!!

Numerical Results

Example 2. NLS with focusing nonlinearity (Bao,Jin,Markowich, SISC,03’)

2

, ( ) 0, ( ) , ( )

x

V x A x e S x β ε

−

= − = = =

( , )

t

x s d s

ε

ρ

∫

Before caustics After caustics

Observations

– Before caustics: converge strongly & converge to QHD – After caustics: converge weakly but the location of discontinuity is

different to QHD!!

Initial data with vacuum

Vacuum at a point Vacuum at a region

2 2

(1 ) 1 1, ( ) 0, ( ) , ( )

• therwise

x x x V x A x S x β ⎧ − < < = = = = ⎨ ⎩

2 2

2 2

• ( -1)

1, ( ) 0, ( ) 1, ( ) ( -1) 1

x x

x e x V x A x x S x x e x β

−

⎧ < ⎪ = = = ≤ ≤ = ⎨ ⎪ > ⎩

Observations

– For fixed : the location of vacuum moves and interact – When : compare the motion of vacuum with Euler system,

compressible Navier-Stokes equations (with Z.P. Xin & H.L. Li,ongoing)

ε

ε →

Difficulties in rotating frame

GPE in a rotational frame

– Angular momentum rotation

– Formal WKB analysis – Analysis & critical threshhold: E. Tadmor, H.L Liu, C. Sparber, ….

2 2 2

( , ) [ ( ) | | ] 2

z

i x t V x L t

ε ε ε

ε ε ψ ε ψ ψ ∂ = − ∇ + − Ω + ∂ r r

: ( ) , ,

z y x y x

L xp yp i x y i L x P P i

θ

= − = − ∂ − ∂ ≡ − ∂ = × = − ∇ r r r r

2 2

ˆ ˆ ( ) 0, : ( ) ˆ ( ) ( ) ( ) ( ln ) 4 1 ( ) / 2 , J , A 1

t z z y x t d z

v L L x y J J J P V L J A J P v

ε ε ε ε θ ε ε ε ε ε ε ε ε ε ε ε ε ε

ρ ρ ρ ε ρ ρ ρ ρ ρ ρ ρ ρ ∂ + ∇• + Ω = = ∂ − ∂ ≡ ∂ ⊗ ∂ + ∇• + ∇ + ∇ + Ω + Ω = ∇ Δ ⎛ ⎞ = = = ⎜ ⎟ − ⎝ ⎠ r r r r r r r r

Difficulties in rotational frame

Efficient computation for rotating GPE

– Polar (2D) & cylindrical (3D) coordinates (Bao,Du,Zhang,SIAP,05’) – ADI techniques (Bao, Wang, JCP, 06’) – Generalized Laguerre-Fourier-Hermite function (Bao, Shen,08’)

Typical initial data: vacuum+two-scale in phase Grenier’s approach (Grenier, 98’; Carles, CMP, 07’)

2 2

( )/ 2 ( , )/ ( , )/ 0( , )

( ) ( , ) ( , ) ,

x y iS x y iS x t

x y x i y e e x t A x t e A a ib

ε

ε ε ε ε ε ε ε

ψ ψ

− + −

= + ⇒ = = +

r

r r r r

2 2 t

1 2 2 1 ( ) | | 2

t z d z

i A S A A S L A A S S V x A L S

ε ε ε ε ε ε ε ε ε ε ε

ε ∂ + ∇ ∇ + Δ − Ω = Δ ∂ + ∇ + + −Ω = r r r r r ) ฀ r ) r

Difficulties in system

Couple GPE system Analysis

– Formal WKB doesn’t work!!!! No techniques are needed! – For linear case, Wigner transform can be applied; for nonlinear case??? – V(x) is periodic and depends on

Efficient computation for coupled GPE:

– Numerical method (Bao, MMS, 04’; Bao, Li, Zhang, Physica D, 07’)

2 2 2 2 1 2 2 2 2 2

( , ) ( ) [ | | | | ] 2 ( , ) ( ) [ | | | | ] 2

t t

i x t V x i x t V x

ε ε ε ε ε ε ε ε ε ε ε ε ε ε

ε ε ψ ψ ψ β ψ δ φ ψ λφ ε ε φ φ φ δ ψ β φ φ λψ ∂ = − ∇ + + + + ∂ = − ∇ + + + + r r r r

ε

Conclusions

For time-independent NLS

– Matched asymptotic approximation – Boundary/interior layers – Semiclassical limits of ground and excited states

For dynamics of NLS

– Formal WKB analysis – Time-splitting spectral method (TSSP) for computation – Semiclassical limits: convergence, caustics, QHD, vacuum – Difficulties in rotational frame and system – Analysis and efficient computation are two important tools