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 && o ε 2 r r ε ε ε ε ε ε ∂ ψ = − ∇ ψ + ψ + β ψ ψ 2 2 ( , ) ( ) | | i x t V x t 2 r x ∈ d ( R ) – t : time & : spatial coordinate ψ r ( , ) x t – : complex-valued wave function r – ( ) : real-valued external potential V x ε < ε – ฀ : scaled Planck constant (0 1) β = ± ( 0, 1) – : interaction constant • =0: linear; =1: repulsive interaction • = -1: attractive interaction

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 r r r r r r 2 2 2 2 ∫ ∫ ∫ ψ ε = ψ ε = ψ ε ≡ ψ ε = ψ ε = ψ ε = ( ) : ( , ) ( ,0) ( ) : ( ) ( 1), N x t d x x d x x d x N 0 0 d d d R R R ⎡ ⎤ ε 2 r r r r ∫ 2 2 4 ψ ε = ∇ ψ ε + ψ ε + β ψ ε ≡ ψ ε ( ) : ( , ) ( ) ( , ) ( , ) ( ) E ⎢ x t V x x t x t ⎥ d x E 2 0 ⎣ 2 ⎦ d R

Semiclassical limits Suppose initial data chosen as r r r r r r r ε ε ε ε ε ε ε ε ε ψ = ψ = ⇒ ψ = 0 ( )/ ( , )/ iS x iS x t ( ,0) : ( ) ( ) ( , ) ( , ) x x A x e x t A x t e 0 0 ε → 0 Semiclassical limits: 2 ε ε ρ = ψ → : ???? – Density: r r r ε = ρ ε ε → ε = ∇ ε → – Current: : ??? : ??? J v v S – Other observable: Analysis: dispersive limits – WKB method vs Winger transform Efficient computation – Highly oscillatory wave in space & time

For ground & excited states For special initial data: r r r r r ε ε = φ ε ε = ⇒ ψ ε = φ ε − μ ε / i t ( ) ( ) & ( ) 0 ( , ) ( ) A x x S x x t x e 0 0 Time-independent NLS: nonlinear eigenvalue problem ε 2 r r r r 2 2 ∫ μ φ ε ε = − ∇ φ ε + φ ε + β φ ε φ ε φ ε = φ ε = 2 2 ( ) ( ) | | , : ( ) 1 x V x x d x 2 d R – Eigenvalue (or chemical potential) ⎡ ⎤ ε 2 r r r r ∫ 2 2 4 μ ε = μ φ ε = ∇ φ ε + φ ε + β φ ε : ( ) : ( ) ( ) ( ) ( ) ⎢ ⎥ x V x x x d x ⎣ 2 ⎦ d R – Eigenfunctions are • Orthogonal in linear case & Superposition is valid for dynamics!! • Not orthogonal in nonlinear case !!!! No superposition for dynamics!!!

For ground & excited states Ground state: minimizer of the nonconvex minimization problem { } ε = φ ε = φ ε = φ φ = φ < ∞ : ( ) min ( ), | 1, ( ) E E E S E g g ε φ ∈ S r β ≥ = ∞ – Existence: 0 & lim ( ) V x r →∞ | | x – Positive solution is unique Excited states: eigenfunctions with higher energy ε ε ε ε ε ε ε φ φ φ = φ μ = μ φ L L , , , , : ( ), : ( ) E E 1 2 j j j j j ε → Semiclassical limits 0 ε ε ε ε ε ε ε φ → → μ = μ φ → φ → → μ → ??? ??? : ( ) ??? ??? ??? ??? E E g g g g j j j ε < ε < ε < < ε < ⇒ μ ε < μ ε < μ ε < < μ ε < L L L L ????? E E E E 1 2 1 2 g j g j

For ground state: Box Potential in 1D ≤ ≤ ⎧ 0, 0 1, x = = β = ⎨∞ ( ) 1, 1 V x d The potential: ⎩ , otherwise. The nonlinear eigenvalue problem (Bao,Lim,Zhang,Bull. Inst. Math., 05’) ε 2 ε ε ε ′′ ε ε μ φ = − φ + φ φ < < 2 ( ) ( ) ( ) | ( ) | ( ), 0 1, x x x x x 2 1 ∫ φ ε = φ ε = φ ε = 2 (0) (1) 0 with | ( ) | 1 x dx 0 Leading order approximation, i.e. drop the diffusion term < ε ฀ 0 1 μ φ = φ φ < < ⇒ φ = μ TF TF TF 2 TF TF TF ( ) | ( ) | ( ), 0 1, ( ) x x x x x g g g g g g 1 ∫ ⇓ φ = TF 2 | ( ) | 1 x dx g 0 1 φ ε ≈ φ = μ ε ≈ μ = ε ≈ = TF TF TF (x) ( ) x 1, 1, E E g g g g g g 2 – Boundary condition is NOT satisfied, i.e. φ = φ = ≠ TF TF (0) (1) 1 0 g g – Boundary layer near the boundary

For ground state: Box Potential in 1D Matched asymptotic approximation ε – Consider near x=0, rescale = φ ε = μ ε Φ , ( ) ( ) x X x x g g μ ε g – We get 1 ′′ Φ = − Φ + Φ ≤ < ∞ Φ = Φ = 3 ( ) ( ) ( ), 0 ; (0) 0, lim ( ) 1 X X X X X 2 →∞ X – The inner solution μ Φ = ≤ < ∞ ⇒ φ ε ≈ μ ≤ = g ( ) tanh( ), 0 ( ) tanh( ), 0 (1) X X X x x x o ε g g – Matched asymptotic approximation for ground state ⎡ ⎤ μ μ μ MA MA MA ⎢ ⎥ φ ε ≈ φ = μ + − − ≤ ≤ g g g MA MA ( ) ( ) tanh( ) tanh( (1 )) tanh( ) , 0 1 x x x x x ε ε ε ⎢ ⎥ g g g ⎣ ⎦ 1 ∫ = φ ⇒ μ ε ≈ μ = + ε + ε + ε = μ + ε + ε + ε < ε MA 2 MA 2 2 TF 2 2 1 | ( ) | 1 2 1 2 2 1 2 , 0 ฀ 1. x dx g g g g 0

For ground state: Box Potential in 1D – Approximate energy 1 4 ε ≈ = + ε + ε + ε MA 2 2 1 2 E E g g 2 3 E ε 1 = g lim , – Asymptotic ratios: ε μ ε → 2 0 g O ε ( ) – Width of the boundary layer: ε → 0 Semiclassical limits < < ⎧ 1 0 1 x 1 ε ε ε φ → φ = → μ → 0 ⎨ ( ) 1 x E = g g g g ⎩ 0 0,1 2 x

For excited states: Box Potential in 1D Matched asymptotic approximation for excited states μ μ μ MA MA MA + + [( 1)/2] [ /2] j j 2 2 1 ∑ l ∑ l φ ε ≈ φ = μ − + − − g g g MA MA ( ) ( ) [ tanh( ( )) tanh( ( )) tanh( )] x x x x C ε + ε + ε j j j j 1 1 j j = = 0 0 l l – Approximate chemical potential & energy μ ε ≈ μ = + + ε + + ε + + ε MA 2 2 2 2 1 2( 1) 1 ( 1) 2( 1) , j j j j j 1 4 ( ε ≈ = + + ε + + ε + + ε MA 2 2 2 2 1) 1 ( 1) 2( 1) , E E j j j j j 2 3 O ε ( ) – Boundary & interior layers ε → 0 Semiclassical limits ± ≠ + ⎧ 1 /( 1) x l j 1 φ ε → φ = ε → μ ε → 0 ⎨ ( ) 1 x E = + j j j j ⎩ 0 /( 1) 2 x l j ε ε ε ε ε ε ε ε < < < < < ⇒ μ < μ < μ < < μ < L L L L E E E E 1 2 1 2 g j g j

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 ε 2 r r ε ∂ ψ ε = − ∇ ψ ε + ψ ε + β ψ ε ψ ε 2 2 ( , ) ( ) | | i x t V x t 2 r r r r ε ψ ε = ψ ε = ρ ε ε ( )/ iS x ( ,0) : ( ) ( ) x x x e 0 0 0 – Formally assume r r r / , ε ψ ε = ρ ε ε ε = ∇ ε ε = ρ ε ε iS , e v S J v – Geometrical Optics: Transport + Hamilton-Jacobi ∂ ρ ε + ∇• ρ ε ∇ ε = ( ) 0, S t ε 2 r 1 1 2 ε ε ε ε ∂ + ∇ + + β ρ = Δ ρ ( ) S S V x t d ρ ε 2 2

For dynamics: Formal limits – Quantum Hydrodynamics (QHD): Euler +3 rd dispersion r ε ε ε ∂ ρ + ∇• ρ = ρ = β ρ 2 ( ) 0 ( ) / 2 v P t r r ε ε r ⊗ ε 2 J J ∂ ε + ∇• + ∇ ρ ε + ρ ε ∇ = ∇ ρ ε Δ ρ ε ( ) ( ) ( ) ( ln ) J P V ε ρ t 4 – Formal Limits r ∂ ρ + ∇• ρ = ρ = β ρ 0 0 0 2 ( ) 0 ( ) / 2 v P t r r r ⊗ 0 0 J J ∂ + ∇• + ∇ ρ + ρ ∇ = 0 0 0 ( ) ( ) ( ) 0 J P V ρ t 0 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