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

Transcript

1 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 URL:

2 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

3 Motivation: NLS The nonlinear Schr o&& dinger (NLS) equation r r i t ( x, t) = + V( x) + ψ ψ ψ β ψ ψ r d x ( R ) t : time & : spatial coordinate ψ ( xt r, ) V( x r ) : complex-valued wave function : real-valued external potential (0 < 1) : scaled Planck constant β ( = 0, ± 1) : interaction constant =0: linear; =1: repulsive interaction = -1: attractive interaction

4 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 N( ψ ): = ψ = ψ ( x, t) d x ψ ( x,0) d x = ψ ( x) d x: = N( ψ ) ( = 1), 0 0 d d d R R R r r β r 4 r = d R E( ψ ): ψ ( x,) t V() x ψ (,) x t ψ ( x,) t d x E( ψ )

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

6 For ground & excited states For special initial data: A ( x r ) = φ ( x r )& S ( x r ) = 0 ψ ( x r, t) = φ ( x r ) e i μ t/ 0 0 Time-independent NLS:nonlinear eigenvalue problem r r r r μ φ ( x) = φ + V( x) φ + β φ φ, φ : = φ ( x) d x = 1 d Eigenvalue (or chemical potential) r r r 4 r μ : = μ( φ ): = φ ( x) + V( x) φ ( x) + β φ ( x) d x d Eigenfunctions are R Orthogonal in linear case & Superposition is valid for dynamics!! Not orthogonal in nonlinear case!!!! No superposition for dynamics!!! R

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

8 For ground state: Box Potential in 1D The potential: 0, 0 x 1, V( x) = d = 1, β = 1, otherwise. The nonlinear eigenvalue problem (Bao,Lim,Zhang,Bull. Inst. Math., 05 ) μ φ ( x) = ( φ ) ( x) + φ ( x) φ ( x), 0< x< 1, 1 (0) = (1) = 0 with ( x) dx= 1 φ φ φ 0 Leading order approximation, i.e. drop the diffusion term μ φ ( x) = φ ( x) φ ( x), 0 < x< 1, φ ( x) = μ TF TF TF TF TF TF g g g g g g g g g g g g Boundary condition is NOT satisfied, i.e. Boundary layer near the boundary 1 TF φg ( x) dx 1 0 = TF TF TF 1 φ (x) φ ( x) = 1, μ μ = 1, E E = φ 0< 1 TF TF g (0) = φg (1) = 1 0

9 For ground state: Box Potential in 1D Matched asymptotic approximation Consider near x=0, rescale We get The inner solution x = X, φg( x) = μg Φ( x) μ 1 Φ X = Φ X +Φ X X < Φ = Φ X = 3 ( ) ( ) ( ), 0 ; (0) 0, lim ( ) 1 X μ Φ ( X ) = tanh( X), 0 X < φ ( ) tanh( g g x μg x), 0 x= o(1) Matched asymptotic approximation for ground state MA MA MA MA MA g g g g( x) g ( x) = g tanh( x) + tanh( (1 x)) tanh( ), 0 x 1 φ φ μ 1 MA MA TF 1 = φg ( x) dx μg μg = = μg , 0< 1. 0 g μ μ μ

10 For ground state: Box Potential in 1D E Approximate energy g 1 4 E = MA g Asymptotic ratios: Width of the boundary layer: Semiclassical limits E g g 1 lim =, 0 μ 0 O( ) 0 1 0< x < 1 1 φg φg( x) = Eg μg 1 0 x = 0,1

11 For excited states: Box Potential in 1D Matched asymptotic approximation for excited states [( j+ 1)/] MA [ j/] MA MA MA MA g l g l+ 1 g j x j x = j x + x Cj l= 0 j+ 1 l= 0 j+ 1 μ μ μ φ ( ) φ ( ) μ [ tanh( ( )) tanh( ( )) tanh( )] Approximate chemical potential & energy μ μ = 1+ ( j+ 1) 1 + ( j+ 1) + ( j+ 1), j MA j MA 1 4 ( 1) 1 ( 1) ( 1) Ej Ej = + j+ + j+ + j+, 3 Boundary & interior layers Semiclassical limits O( ) 0 0 ± 1 x l/( j+ 1) 1 φj φj( x) = Ej μj 1 0 x= l/( j+ 1) E < E < E < L< E < L μ < μ < μ < L< μ < L g 1 j g 1 j

12 Extension & numerical computation Extension High dimension Nonzero external potential Numerical method & results Normalized gradient flow Backward Euler finite difference method

13 For dynamics: Formal limits WKB analysis r r i tψ ( x, t) = ψ + V( x) ψ + β ψ ψ r r r is0 ( x)/ ( x,0): 0( x) 0( x) e r ψ = ψ = ρ Formally assume is / r r r ψ = ρ e, v = S, J = ρ v Geometrical Optics: Transport + Hamilton-Jacobi ρ + ( ρ S ) = 0, t 1 r 1 d ( ) β ρ ρ ts + S + V x + = Δ ρ

14 For dynamics: Formal limits Quantum Hydrodynamics (QHD): Euler +3 rd dispersion r t ρ + ( ρ v ) = 0 P( ρ) = β ρ / r r r J J t ( J ) + ( ) + P( ρ ) + ρ V = ( ρ Δln ρ ) ρ 4 Formal Limits 0 0 r0 t ρ + ( ρ v ) = 0 P( ρ) = β ρ / r r r J J 0 0 t ( J ) + ( ) + P( ρ ) + ρ V = 0 0 ρ 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

15 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 to 0.1 in typical BEC setups Provide benchmark results for other approaches Hints for analysis after caustics and/or with vacuum

16 Time reversible NLS and its properties r r i tψ ( x, t) = ψ + V( x) ψ + β ψ ψ r r r is0 ( x)/ ( x,0) ( x) A ( x) e r ψ = ψ = 0 0 Time transverse invariant (gauge invariant) r r itα/ V( x) V( x) + α ψ ψ e ψ unchanged Mass (wave energy) conservation r r r r r r Nψ(): t = ψ ( xt,) dx Nψ(0): = ψ ( x,0) dx= ψ0 ( x) dx, t 0 d d d Energy ( or Hamiltonian) conservation β ψ r r r 4 r E (): t = ψ (,) xt + V() xψ (,) xt + ψ (,) xt dx Eψ (0), t 0 d Dispersion relation without external potential i( kx ωt) ψ ( x, t) = ae (plane wave solution) ω = k + β a

17 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 ψ = A e is / (solution has wavelength of O( ))

18 Time-splitting spectral method (TSSP) For, apply time-splitting technique [ tn, t n + 1] Step 1: Discretize by spectral method & integrate in phase space exactly i r tψ ( x, t) = ψ Step : solve the nonlinear ODE analytically r r r r r i tψ x t = V x ψ x t + β ψ x t ψ x t r r r = = r r r r r i ψ x t = V x ψ x t + β ψ x t ψ x t t (, ) ( ) (, ) (, ) (, ) t( ψ ( xt, ) ) 0 ψ ( xt, ) ψ ( xt, n) (, ) ( ) (, ) (, n) (, ) r r r i( t t )[ V( x) + βψ ( x, t ) ]/ n n ψ ( xt, ) = e ψ ( xt, n ) Use nd order Strang splitting (or 4 th order time-splitting)

19 An algorithm in 1D for NLS Choose time step: ; set b a Choose mesh size h x ; set The algorithm (10 lines code in Matlab!!!) (Bao,Jin,Markowich, JCP,0 ) ψ ψ (1) j n ψ j with = e k n j ik[ V( x ) + βψ ]/ 1 j = Δ t tn n k, n 0,1, M = = L =Δ = x = a+ j h& ψ n ψ( x, t ) () j ψ j j j n M / 1 () i k μ / (1) i μl ( xj a) l ˆ j = e ψ j e, j = 0,1, L, M 1 M l= M / + 1 ik[ V( xj) + βψ ]/ () = e ψ j M 1 (1) (1) l ( j ) ˆ iμ x a l l j e b a j= 0 n j lπ M M M μ =, ψ = ψ, l =, + 1, L, 1 ( )

20 Properties of the method Explicit & computational cost per time step: Time reversible: yes n+ n ψ ψ + Time transverse invariant: yes Mass conservation: yes M 1 M 1 n n 0 = j = 0 = l 0 j = L Stability: yes OM ( ln M) n n 1 1 & j j scheme unchanged!! V x V x j M e n n inkα/ n ( ) ( ) + α (0 ) ψ j ψ j ψ j unchanged!!! ψ : h ψ ψ ψ : h ψ ( x ), n 0,1, for any h& k l l j= 0 j= 0

21 Properties of the method Dispersion relation without potential: yes ψ 0 j ikxj n i( kxj ω tn) ψ j = ae (0 j M) = ae (0 j M& n 0) = k + a M > k with ω β if Exact for plane wave solution Energy conservation (Bao, Jin & Markowich, JCP, 0 ): cannot prove analytically Conserved very well in computation

22 Properties of the method Accuracy Spatial: spectral order Temporal: nd or 4 th order Resolution in semiclassical regime (Bao, Jin & Markowich, JCP, 0 ) Linear case: β = 0 h= O( ) & k independent of Weakly nonlinear case: Strongly repulsive case: β = O( ) h= O( ) & k independent of 0 < β = O(1) h= O( ) & k = O( ) Error estimate: not available yet!!

23 Numerical Results Example 1. Linear Schrodinger equation 5( x 0.5) 1 5( x 0.5) 5( x 0.5) β = 0, V( x) = 10, A0( x) = e, S0( x) = ln[ e + e ] 5 Density at t=0.54 Current at t=0.54

24 = = =

25 Crank-Nicolson finite difference method Crank-Nicolson spectral method

26 Numerical Results Example. NLS with defocusing nonlinearity 1 x x < 1 x x β = 1, V( x) = 0, A0( x) =, S0( x) = ln[ e + e ] 0 otherwise t 0 ρ ( x, s ) d s

27 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!!

28 Numerical Results Example. NLS with focusing nonlinearity (Bao,Jin,Markowich, SISC,03 ) β =, V( x) = 0, A ( x) = e, S ( x) = 0 x 0 0 t 0 ρ ( x, s ) d s

29 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!!

30 Initial data with vacuum Vacuum at a point Vacuum at a region

31 x (1 x) 0 < x< 1 β = 1, V( x) = 0, A0( x) =, S0( x) = 0 0 otherwise

32

33 x < xe x 0 β = 1, V( x) = 0, A0( x) = 0 0 x 1, S0( x) = 0 -( x-1) ( x-1) e x> 1

34 Observations For fixed : the location of vacuum moves and interact 0 When : compare the motion of vacuum with Euler system, compressible Navier-Stokes equations (with Z.P. Xin & H.L. Li,ongoing)

35 Difficulties in rotating frame GPE in a rotational frame r r i ψ ( x, t) = [ + V( x) Ω L z + ψ ] ψ t Angular momentum rotation r r r r L : = xp yp = i( x y ) i, L= x P, P= i z y x y x Formal WKB analysis r ( ) ˆ 0, ˆ t ρ + ρ v +Ω Lz ρ = Lz: = ( x y y x) θ r r r J J r r ( ) ( ) ( ) ˆ t J + + P ρ + ρ Vd +Ω LzJ +Ω AJ = ( ρ Δln ρ ) ρ 4 P r r 0 1 ρ = ρ = ρ v = 1 0 ( ) /, J, A Analysis & critical threshhold: E. Tadmor, H.L Liu, C. Sparber,. θ

36 Difficulties in rotational frame ψ Efficient computation for rotating GPE Polar (D) & 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 (, ) ( ) 0 ( x y )/ is ( x, y)/ (, ) (, ) is ( x, t)/, 0 xy x iye e xt A xte r + = + ψ = A = a + ib Grenier s approach (Grenier, 98 ; Carles, CMP, 07 ) r r 1 r ) r i r ta + S A + A ΔS Ω LzA = ΔA 1 r r ) ts + S + Vd( x) + A Ω LzS = 0 r r r r

37 Difficulties in system Couple GPE system r r i tψ x t = ψ + V x ψ + β ψ + δ φ ψ + λφ r r i tφ ( x, t) = φ + V( x) φ + [ δ ψ + β φ ] φ + λψ (, ) ( ) [ 1 ] 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 )

38 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