Analysis and Computation for Analysis and Computation for Nonlinear - PowerPoint PPT Presentation

Analysis and Computation for Analysis and Computation for Nonlinear Eigenvalue Eigenvalue Problems Problems Nonlinear Under Constraints Under Constraints Fong Yin Lim Fong Yin Lim Department of Mathematics Department of Mathematics and and Center for Computational Science & Engineering Center for Computational Science & Engineering National University of Singapore National University of Singapore Email: fongyin.lim@nus.edu.sg fongyin.lim@nus.edu.sg Email: Joint work: Weizhu Bao (National University of Singapore) Joint work: Weizhu Bao (National University of Singapore) I- -Liang Liang Chern Chern (National Taiwan University) (National Taiwan University) I Ming Huang Chai Chai, , Hanquan Hanquan Wang, Wang, Yanzhi Yanzhi Zhang Zhang Ming Huang

Outline Outline Introduction Introduction Bose- -Einstein condensation (BEC) Einstein condensation (BEC) Bose Gross- Gross -Pitaevskii Pitaevskii equation equation Singularly perturbed nonlinear eigenvalue eigenvalue Singularly perturbed nonlinear problems problems Asymptotic approximations Asymptotic approximations Numerical methods and numerical results Numerical methods and numerical results Extension to other BEC systems Extension to other BEC systems Conclusions Conclusions

Introduction Introduction JILA (95’, 87 Rb) Bose- Bose -Einstein condensation Einstein condensation -- Bose -- Bose- -Einstein statistical mechanics Einstein statistical mechanics for bosons (spin 0, for bosons (spin 0, 1, 2, … …) ) 1, 2, 1 ε = ( ) f ( ) ε − µ − i / k T 1 e i B -- -- below transition temperature below transition temperature T T c c , a significant fraction of , a significant fraction of bosonic particles will be in their single particles will be in their single- -particle lowest energy particle lowest energy bosonic state state Experiments: Cooling of magnetically trapped dilute Experiments: Cooling of magnetically trapped dilute atomic vapor to nanokelvins nanokelvins temperature temperature atomic vapor to

Gross- -Pitaevskii Pitaevskii equation equation Gross Gross- Gross -Pitaevskii Pitaevskii equation (GPE) equation (GPE) r ∂ ψ r r r r r r ( ) 1 x = − ∇ ψ + ψ + β ψ 2 ψ ∈ ℜ 2 d ( ) ( ) ( ) ( ) ( ), i x V x x x x x ∂ 2 t V(x) ) – – trapping potential trapping potential V(x β -- -- mean mean- -field interaction field interaction β 2 = r = ψ Normalization conservation ( , ) 1 Normalization conservation N x t Energy conservation Energy conservation β = ∫ r r r r r [ ] 1 ( ) ( ) ( ) ( ) ( ) ψ ∇ ψ 2 + ψ 2 + ψ 4 , , , E t x t V x x t x t d x 2 2 r r ψ = φ − µ i t ( , ) ( ) Stationary state Stationary state x t x e µ -- µ -- chemical potential chemical potential r r ψ = φ Probability density of finding a particle 2 2 Probability density of finding a particle | ( , ) | | ( ) | x t x

Stationary States Stationary States Nonlinear eigenvalue eigenvalue problem (time problem (time- -independent GPE) independent GPE) Nonlinear r r r r r r r 1 µ φ = − ∇ φ + φ + β φ 2 φ ∈ ℜ 2 d ( ) ( ) ( ) ( ) ( ) ( ), x x V x x x x x 2 = ∫ r r 2 φ 2 φ = ( ) 1 x d x ℜ d Eigenvalue (chemical potential) (chemical potential) Eigenvalue β r r ∫ µ = µ φ = φ + φ 4 ( ) ( ) ( ) E x d x β β ℜ d 2 Energy Energy β ⎡ ⎤ r r r r r 1 ∫ 2 2 4 φ = ∇ φ + φ + φ ( ) ( ) ( ) ( ) ( ) E x V x x x d x ⎢ ⎥ β ⎣ ⎦ ℜ d 2 2 Other applications Other applications Nonlinear optics Nonlinear optics Complex quantum system Complex quantum system

Stationary States Stationary States φ g convex minimization problem: ground state φ Non- -convex minimization problem: ground state Non g { } φ = φ = φ φ = φ = φ < ∞ ( ) min ( ) | 1, | 0, ( ) E E S E r β ∈Γ β g x φ ∈ S φ φ φ L Excited states , , , Excited states 1 2 3 Open question Open question φ φ φ L , , , 1 2 g φ < φ < φ < L ( ) ( ) ( ) E E E β β 1 2 g µ φ < µ φ < µ φ < L ( ) ( ) ( ) ??????? β β β 1 2 g (Bao & W. Tang, JCP , 03’; Bao, F. Lim & Y. Zhang, Bull Int. Math , 06’)

Singularly Perturbed NEP Singularly Perturbed NEP Three typical parameter regimes: Three typical parameter regimes: β = Linear: Linear: 0 β = Weak interaction: ( 1 ) Weak interaction: o β >> Strong repulsive interaction: Strong repulsive interaction: 1 β >> , reformulate GPE into singularly perturbed NEP , reformulate GPE into singularly perturbed NEP 1 ε 2 r r r r r r r ε ε ε ε ε ε µ φ = − ∇ φ + φ + φ φ ∈ Ω ⊂ ℜ 2 d ( ) ( ) ( ) ( ) ( ) ( ), x x V x x x x x 2 r r r 2 ∫ φ ε = ∈ ∂ Ω φ ε = φ ε = ( ) 0 , ; : ( ) 1 x x x d x Ω r r 1 ∫ 4 ε ε ε ε µ = µ φ = φ + φ ( ) ( ) ( ) E x d x ε ε Ω 2 ⎡ ⎤ ε 2 r r r r r 1 2 2 4 ∫ φ ε = ∇ φ ε + φ ε + φ ε ( ) ⎢ ( ) ( ) ( ) ( ) ⎥ E x V x x x d x ε Ω ⎣ 2 2 ⎦

Singularly Perturbed NEP Singularly Perturbed NEP r ∈ Ω ⎧ 0 , r x = Bounded Ω Ω with box potential with box potential Bounded ⎨ ( ) V x ∞ ⎩ , otherwise r r 1 ε = µ ε = ε µ φ ε = φ 2 , , ( ) ( ) x x β Rescaled energy and eigenvalue eigenvalue Rescaled energy and ⎡ ⎤ ε = ∫ 2 r r r 1 2 4 ε ∇ φ ε + φ ε = ⎢ ( ) ( ) ⎥ ( 1 ) E x x d x O Ω ⎣ 2 2 ⎦ r r 1 ∫ 4 ε ε ε µ = + φ = < ε << ( ) ( 1 ), 0 1 E x d x O Ω 2 Leading asymptotic approximation Leading asymptotic approximation ε ε = β = β µ = βµ = β β >> ( ), ( ), 1 E E O O

Singularly Perturbed NEP Singularly Perturbed NEP Whole space with harmonic potential Whole space with harmonic potential r ( ) r r 1 ( ) W x = γ + + γ + = 2 2 2 2 L ( ) ( ), lim 0 V x x x W x r r 1 1 d d → ∞ 2 ( ) V x x Semiclassical scaling scaling Semiclassical r r r r − + ′ − ε − ε ′ ε = β = ε µ = ε µ φ = φ 2 /( 2 ) 1 / 2 1 d , , , ( ) ( ) x x x x Rescaled energy and eigenvalue eigenvalue Rescaled energy and ⎡ ⎤ ε ( ) = ∫ 2 r r r r r r 1 2 2 4 ε ε ε ε ∇ φ + + φ + φ = ( ) ( ) ( ) ( ) ( ) ( 1 ) ⎢ ⎥ E x V x W x x x d x O Ω ⎣ 2 2 ⎦ r r 1 ∫ 4 ε ε ε µ = + φ = < ε << ( ) ( 1 ), 0 1 E x d x O Ω 2 Leading asymptotic approximation Leading asymptotic approximation ε + ε + = β = β µ = βµ = β β >> 2 /( 2 ) 2 /( 2 ) d d ( ), ( ), 1 E E O O

Asymptotic Approximation Asymptotic Approximation Thomas- Thomas -Fermi regime Fermi regime -- -- drop the diffusion term in the drop the diffusion term in the GPE for β β >>1 ( >>1 ( ε ε << 1 ) << 1 ) GPE for r r r r r r µ φ = φ + φ 2 φ ∈ Ω ( ) ( ) ( ) ( ) ( ), x V x x x x x Ground state in box potential Ground state in box potential < < = ⎧ 0 , 0 1 , 1 to r x i d r = φ = µ i ⎨ TF TF ( ) V x ( ) x ∞ g g ⎩ , otherwise r r r ∫ Ω 2 φ = ⇒ µ = φ = TF TF TF ( ) 1 1 , ( ) 1 x d x x g g g 1.5 Boundary conditions not satisfied Boundary conditions not satisfied Existence of boundary layers Existence of boundary layers 1 φ g (x) 0.5 0 0 0.2 0.4 0.6 0.8 1 x

Matched Asymptotic Method Matched Asymptotic Method ε For boundary layer at x For boundary layer at x = 0, rescale = 0, rescale = φ = µ Φ , ( ) ( ) x X x X µ g g 1 Φ = − Φ + Φ ≤ < ∞ 3 ( ) ( ) ( ), 0 X X X X XX 2 Φ = Φ = ( 0 ) 0 , lim ( ) 1 X → ∞ X Solution Solution µ φ ≈ µ ≤ < g ( ) tanh( ), 0 1 / 2 x x x ε g g Match the solutions at intermediate region Match the solutions at intermediate region ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ µ µ µ MA MA MA ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ φ = µ g + g − − g MA MA ( ) tanh tanh ( 1 ) tanh x x x ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ε ε ε g g ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ 1 2 ∫ φ = ⇒ MA ( ) 1 µ = + ε + ε + ε x dx MA 2 2 1 2 1 2 g g 0 1 4 = + ε + ε + ε MA 2 2 1 2 E g 2 3