Weizhu Bao. Modeling, Analysis & Simulation for Quantized Vortices in Superfluidity and Superconductivity

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1 Modeling, Analysis & Simulation for Quantized Vortices in Superfluidity and Superconductivity Weizhu Bao Department of Mathematics & Center for Computational Science and Engineering National University of Singapore URL:

2 Vortex: From macroscale to microscale Cloud vortex Saturn hexagon at the north pole of the planet Saturn

3 Vortex: From macroscale to microscale Vortex galaxy vortex in cosmos

4 Vortex: From macroscale to microscale Tornado Vortex in air

5 Vortex: From macroscale to microscale Vortex in water generated by a boat

6 Vortex: From macroscale to microscale Vortex in water generated by an airplane

7 Vortex: From macroscale to microscale Magnetic vortex vortex in plasma

8 Vortex: From macroscale to microscale Vortex in plant

9 Vortex: From macroscale to microscale Quantized Vortex in liquid Helium 3

10 Vortex: From macroscale to microscale Quantized Vortex in a type-ii superconductor

11 Vortex: From macroscale to microscale Quantized Vortex in Bose-Einstein condensation (BEC)

12 Outline Vortex vs quantized vortex Mathematical models Gross-Pitaevekii equation (GPE) / nonlinear Schrodinger equation Ginzburg-Landau equation (GLE); nonlinear wave equation,. Numerical methods & results for vortex nucleation Numerical methods & results for dynamics Central vortex states & interactions Conclusions

13 Vortex In fluid dynamics, a vortex is a region, in a fluid medium, in which the flow is mostly rotating on an axis line, the vortical flow that occurs either on a straight-axis or a curved-axis. Vortices Form in stirred fluids: liquid, gas & plasma Evidenced in: smoke rings; whirlpool of the wake of a boat or paddle or an airplane; winds surrounding a tropical cyclone (hurricane); a dust devil, a tornado It is a major component of turbulence flow

14 Models for vortex in fluid dynamics Physical quantities: Velocity, pressure, vorticity, streamfunction Mathematical models Euler equations Navier-Stokes equations Shallow water equations,.. Classifications Viscous vs inviscid Compressible vs incompressible Vortex sheet in D vs vortex patch in 3D

15 Quantized vortex In physics, a quantized (quantum) vortex is a topological (particle-like) defect exhibited in superfluids & superconductors Zero of the complex scalar field localized phase singularities with integer topological charge: iφ ψ ( x 0 ) 0, ψ ( x) = ρ e, d(argψ ) = dφ = π n 0 = Key of superfluidity: ability to support dissipationless flow Γ Γ

16 Quantized vortex In different physical system Type-II superconductors: Ginzburg-Landau equations (GLE) Liquid helium: Two-fluid model Gross-Pitaevskii equation (GPE) Bose-Einstein condensation (BEC): Nonlinear Schroedinger equation (NLSE) or GPE Nonlinear optics & propagation of laser beams Nonlinear Schroedinger equation (NLSE) Nonlinear wave equation (NLWE)

17 Vortex in superconductivity Superconductivity: a phenomenon of exactly zero electrical resistance and expulsion of magnetic fields occurring in certain materials when cooled below a characteristic critical temperature. Heike Kamerlingh Onnes, 1911 Phase transition Type I single critical field Type II -- two critical fields, between which it allows partial penetration of the magnetic field. High-temperature superconductivity A superconductor that is capable of supporting vortex lattices is called a type-ii superconductor

18 Nobel Prizes for superconductivity H. K. Onnes (1913), "for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium" J. Bardeen, L. N. Cooper& J. R. Schrieffer (197), "for their jointly developed theory of superconductivity, usually called the BCS-theory" L. Esaki, I. Giaever, and B. D. Josephson (1973), "for their experimental discoveries regarding tunneling phenomena in semiconductors and superconductors, respectively," and "for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects" G. Bednorz and K. A. Müller (1987), "for their important break-through in the discovery of superconductivity in ceramic materials" A. A. Abrikosov, V. L. Ginzburg & A. J. Leggett (003), "for pioneering contributions to the theory of superconductors and superfluids"

19 Models for superconductivity Microscopic theory --- BCS theory By Bardeen, Cooper & Schrieffer, Macroscopic theory --- Ginzburg-Landau theory A phenomenal model & be formally derived from BCS theory Complex order parameter field Total energy density β 4 1 ( ) F = Fn + αψ + ψ + i ea ψ + m B = A--magnetic field & A---magnetic vector potential F n ψ --free energy in the normal phase & m-- effectiv mass B µ 0

20 Ginzburg-Landau equation (GLE) Time-independent GLE 1 αψ + β ψ ψ + ( i ea) ψ = 0 m e µ ; Re{ ψ * ψ} B = 0 j j = ( i ea) m GLE without external magnetic potential 1 ( ) ψt = ψ + 1 ψ ψ, x Ω, t > 0, ε = 0, x Ω ( ) ψ ψ ε GLE energy functional ( ) E 1 ( ) V ( x ψ ψ ) ψ dx = +, ε Ω A dissipative PDE --- complex heat-type (parabolic) equation

21 Vortex in superfluidity Superfluidity is a state of matter in which the matter behaves like a fluid with zero viscosity; where it appears to exhibit the ability to self-propel and travel in a way that defies the forces of gravity and surface tension. It was originally discovered in liquid helium in 1937 It is also found in astrophysics, high-energy physics, and theories of quantum gravity. The phenomenon is related to the Bose Einstein condensation (BEC), especially ultracold atomic gas.

22 Bose-Einstein condensation (BEC) Bose-Einstein condensation: Bosons at nano-kevin temperature many atoms occupy in one obit (at quantum mechanical ground state) `super-atom new matter of wave. i.e., the fifth matter of state Theoretical predication: Bose & Einstein Bose, Z. Phys., 6 (194) 8 Einstein, Sitz. Ber. Kgl. Preuss. Adad., Wiss. (194) 61 Experimental realization: JILA 1995 Anderson et al., Science, 69 (1995), 198: JILA Group; Rb Davis et al., Phys. Rev. Lett., 75 (1995), 3969: MIT Group; Rb Bradly et al., Phys. Rev. Lett., 75 (1995), 1687, Rice Group; Li

23 Experimental techniques for BEC Laser cooling Magnetic trapping Evaporative Cooling ($100k 300k)

24 Experimental Results of BEC JILA (95,Rb,5,000) ETH (0,Rb, 300,000)

25 BEC 001 Nobel prize in physics: C. Wiemann: U. Colorado; E. Cornell: NIST & W. Ketterle: MIT Mathematical models: Gross-Pitaevskii equation (mean field theory) Quantum Boltzmann master equation (kinetic) Mathematical analysis Existence, dynamical laws, soliton-like solution, damping effect, etc. Numerical Simulations Numerical methods Guiding and predicting outcome of new experiments

26 Possible applications of BEC Quantized vortex for studying superfluidity Square Vortex lattices in spinor BECs Giant vortices Vortex lattice dynamics Test quantum mechanics theory Bright atom laser: multi-component Quantum computing Quantum Transport, Atomic circuit,..

27 Mathematical models N-body problem (3N+1)-dim linear Schroedinger equation Mean field theory--zeor or `extremely low temperatures Gross-Pitaevskii equation (GPE): T < T = O(nK) (3+1)-dim nonlinear Schroedinger equation (NLSE) Quantum kinetic theory-- high temperatures High temperature: QBME (3+3+1)-dim Around critical temperature: QBME+GPE Below critical temperature: GPE c

28 Model for a BEC at zero temperature with N identical bosons N-body problem 3N+1 dim. (linear) Schrodinger equation i Ψ ( x, x,, x, t) = H Ψ ( x, x,, x, t), with t N 1 N N N 1 N N HN = j + V( xj) Vint( xj xk) j= 1 m + 1 j< k N Hartree ansatz Fermi interaction N Ψ ( x, x,, x,) t ψ ( x,), t x N 1 N j j j= 1 Dilute quantum gas -- two-body elastic interaction N N N N N 1 N 3N 4 as Vint( x π j x k) = gδ ( x j x k) with g = m E ( Ψ ) : = Ψ H Ψ dx dx NEψ ( )--energy per particle 3

29 Model for a BEC with N identical bosons Energy per particle mean field approximation (Lieb et al, 00 ) Ng 4 E( ψ) = ψ + V( x) ψ + ψ dx with ψ : = ψ( x, t) 3 m Dynamics (Gross, Pitaevskii 1961 ; Erdos, Schlein & Yau, Ann. Math. 010 ) δe( ψ) i tψ( x,) t = = V ( x) Ng ψ ψ, x δψ + + m 3 Proper non-dimensionalization & dimension reduction GPE/NLSE 1 d 4π Na i tψ( xt, ) = ψ + V( x ) ψ + β ψ ψ, x with β = x s s

30 Other applications of GPE/NLSE 1 i ( xt,) = + V( x) + tψ ψ ψ β ψ ψ For laser beam propagation In plasma physics: wave interaction between electrons and ions Zakharov system---nlse +wave equation,.. In quantum chemistry: chemical interaction based on the first principle Schrodinger-Poisson system In materials science: First principle computation, DFT, Semiconductor industry In nonlinear (quantum) optics In biology protein folding H + O HO In superfluids flow without friction, liquid 4 He

31 Mathematical model for BEC mean field theory The Gross-Pitaevskii equation (GPE/NLSE) 1 i ( xt,) = + V( x) + t ψ ψ ψ β ψ ψ d t : time & x ( ) : spatial coordinate (d=1,,3) ψ (,) xt : complex-valued wave function V( x ) : real-valued external potential β : dimensionless interaction constant =0: linear; >0(<0): repulsive (attractive) interaction β = 0 :Schrodinger equation (E. Schrodinger 195 ) :GPE (E.P. Gross 1961 ; L.P. Pitaevskii 1961 ) β 0 E. Schrodinger

32 Dispersive Time symmetric: Conservation laws Time transverse (gauge) invariant Mass conservation d Energy conservation 1 i ( xt,) = + V( x) + tψ ψ ψ β ψ ψ t t & take conjugate unchanged!! V( x) V( x) + α ψ ψe iαt ρ = ψ --unchanged!! N(): t = N( ψ(,)) t = ψ( x,) t dx ψ( x,0) dx= 1, t 0 1 β 4 E( t) : = E( ψ(, t)) = ψ + V( x) ψ + ψ dx E(0), t 0 d d

33 Stationary states Stationary states (ground & excited states) it ψ( xt,) = φ( x) e µ Nonlinear eigenvalue problems: Find 1 µφ( x) = φ( x) + V( x) φ( x) + β φ( x) φ( x), x with φ : = (x) dx= 1 Time-independent NLSE or GPE: Eigenfunctions are φ d 1 i xt = + V x + tψ(,) ψ ( ) ψ β ψ ψ ( µφ, ) s.t. Orthogonal in linear case & Superposition is valid for dynamics!! Not orthogonal in nonlinear case!!!! No superposition for dynamics!!! d

34 Ground states The eigenvalue is also called as chemical potential β 4 µ : = µφ ( ) = E( φ) + φ(x) dx d With energy 1 β φ = φ + φ + φ d 4 E( ) [ ( x) V( x) ( x) ( x) ] dx Ground states -- nonconvex minimization problem { } E( φ ) = min E( φ) S = φ φ = 1, E( φ) < g φ S Euler-Lagrange equation nonlinear eigenvalue problem

35 Existence & uniqueness Theorem (Lieb, etc, PRA, 0 ; Bao & Cai, KRM, 13 ) If potential is confining d V( x) 0 for x & lim V( x) = x There exists a ground state if one of the following holds ( i) d = 3& β 0; ( ii) d = & β > C ; ( iii) d = 1& β The ground state can be chosen as nonnegative Nonnegative ground state is unique if β 0 The nonnegative ground state is strictly positive if There is no ground stats if one of the following holds b φ,.. ieφ = φ g g g V( x) L ( i)' d = 3& β < 0; ( ii)' d = & β Cb C b = inf 1 0 f H ( ) f f L ( ) L ( ) 4 f 4 L ( ) loc i 0 e θ

36 Key Techniques in Proof Positivity & semi-lower continuous E( φ) E( φ ) = E( ρ), φ S with ρ = φ The energy E( ρ): = E( ρ) is bounded below if conditons (i) or (ii) or (iii) and strictly convex if Confinement potential implies decay at far field The set Using convex minimization theorem Non-existence result S= ρ ρ( x) dx=1 & E ( ρ) < is convex in ρ d ε 0 E( φ ε ) 1 x d φε ( x) = exp, x with ε 0 d /4 ( πε ) ε β 0

37 Linear case Ground state approximation 1 µφ( x) = φ( x) + V( x) φ( x) + β φ( x) φ( x), x β = 0 d Weakly interaction regime β 1 Strongly repulsive interaction regime β 1

38 Computing ground states Idea: Steepest decent method + Projection 1 δ E( ϕ) 1 = = < δϕ φ 0 φ ϕ t 1 φ φˆ 1 tϕ( xt,) ϕ V( x) ϕ β ϕ ϕ, tn t tn+ 1 ϕ( x, ) n+ 1 ( xt, n+ 1) =, n= 0,1,, ϕ( x, ) n+ 1 t ϕ( x,0) = ϕ (x) with ϕ ( x) = The first equation can be viewed as choosing in NLSE For linear case: (Bao & Q. Du, SIAM Sci. Comput., 03 ) E0( φ(., tn+ 1) ) E0( φ(., tn) ) E0( φ(., 0) ) For nonlinear case with small time step, CNGF φ g t = E( ˆ φ1) < E( φ0) E( ˆ φ1) < E( φ1) E( φ ) < E( φ ) i τ 1 0??

39 Normalized gradient glow Idea: letting time step go to 0 (Bao & Q. Du, SIAM Sci. Comput., 03 ) 1 µφ ( (., t)) tφ xt = φ V x φ β φ φ+ φ t φ(., t) φ( x,0) = φ ( x) with φ ( x) = 1. (, ) ( ), 0, 0 0 Mass conservation & energy diminishing Numerical discretizations ϕ(., t) = ϕ0 = 1, d E( ϕ(., t)) 0, t 0 dt BEFD: Energy diminishing & monotone (Bao & Q. Du, SIAM Sci. Comput., 03 ) TSSP: Spectral accurate with splitting error (Bao & Q. Du, SIAM Sci. Comput., 03 ) BESP: Spectral accuracy in space & stable (Bao, I. Chern & F. Lim, JCP, 06 )

40 Ground states in 1D & 3D

41 Vortex nucleation in BEC With an external perturber 1 i ( xt,) = + [ V( x) + W( xt,)] + tψ ψ ψ β ψ ψ BEC with an angular momentum rotation

42 Rotating BEC GPE / NLSE with an angular momentum rotation 1 d i tψ( xt,) = [ + V( x) Ω Lz + β ψ ] ψ, x, t> 0 L : = xp yp = i( x y ) i, L = x P, P = i z y x y x θ Mass conservation N() t = ψ ( x,) t dx Energy conservation 1 β 4 EΩ( ψ): = ψ + V( x) ψ Ω ψlzψ + ψ dx d R d

43 Ground states Ground states Seiringer, CMP, 0 ; Bao,Wang & Markowich, CMS, 05 ;.. min E ( φ) φ S Ω Existence & uniqueness Exists a ground state when β 0 & Ω min { γ x, γ y} Uniqueness when Ω <Ω c ( β ) Ω Ω Quantized vortices appear when c ( β ) Phase transition & bifurcation in energy diagram Numerical methods --- GFDN & BEFD or BEFP

44 Ground states with different Ω

45 Vortex nucleation in BEC

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48 In D: point vortex Central vortex states 1 µφ ( x) = [ + V( x) Ω L β z + φ ] φ φ (, xy) = f () re imθ m In 3D: vortex line vortex ring φ ( xyz,, ) = f ( rze, ) imθ m

49 Dynamics of vortex in BEC Time-dependent NLSE / GPE 1 i tψ xt = ψ + V xψ Ω Lψ + β ψ ψ x t> ψ( x,0) = ψ ( x) d (,) ( ) z,, 0 0 Well-posedness & dynamical laws Well-posedness & finite time blow-up Dynamical laws Center-of-mass An exact solution under special initial data Numerical methods and applications

50 Well-posedenss Theorem (T. Cazenave, 03 ; C. Sulem & P.L. Sulem, 99 ) Assumptions d d α d (i) V( x) C ( ), V( x) 0, x & D V( x) L ( ) α X= u H u = u + u + V x u x dx< 1 d (ii) ψ 0 ( ) ( ) ( ) X L L d Local existence, i.e. T (0, ], s. t. the problem has a unique solution ψ C([0, T ), X) max Global existence, i.e. T = + if max d = 1 or d = with β C / ψ or d = 3 & β 0 b 0 d L ( ) max

51 Finite time blowup Theorem (T. Cazenave, 03 ; C. Sulem & P.L. Sulem, 99 ) Assumptions d There exists finite time blowup, i.e. T max < + if one of the following holds (i) E( ψ ) < 0 Proof: d β <0 & V( x) d + x V( x) 0, x with d =,3 0 X with finite variance V(0):= x 0( x) dx ψ δ ψ < 0 (ii) E( ψ ) = 0 & Im ψ ( x) ( x ψ ( x)) dx< (iii) E( ψ ) > 0 & Im ψ ( x) ( x ψ ( x)) dx< E( ψ ) d xψ δv(): t = x ψ( x,) t dx δv() t d E( ψ0), t 0, d =,3 d (i) Variance identity: ' δ ( t) de( ψ ) t + δ (0) t+ δ (0) 0 < t* < & δ ( t*) = 0!! (ii) Uncertainty principle: d d V 0 V V V 1 = ψ xψ ψ = δ () t ψ L L L V L L

52 ψ Dynamics with no potential V( x) 0, x Momentum conservation Dispersion relation Soliton solutions in 1D: Bright soliton β < decaying to zero at far-field Dark (or gray) soliton β > 0 -- nonzero &oscillatory at far-field d J( t) : = Im ψ ψ dx J(0) t 0 d i( k x ωt) 1 ψ( x,) t = Ae ω = k + βa 1 a ψ x t = a x vt x0 e x t β 1 ( ( ) 0 ) (, ) sech( ( )) i vx v a t + θ,, 0 [ ] i( kx ( k + a + ( v k ) ) t+ θ0 ) ( x, t) = a tanh( a( x vt x )) + i( v k) e, x, t 0 β 0 1

53 ψ Dynamics with harmonic potential Harmonic potential Center-of-mass: = d x t x t t An analytical solution if γ x x d = 1 1 V( x) = γ x x + γ y y d = γ x x + γ y y + γ z z d = 3 xc() t xψ ( x,) t dx c( ) + diag( γ x, γ y, γ z) c( ) = 0, > 0 each component is periodic!! ψ ( x ) = φ ( x x ) 0 s 0 (,) (,) i s t ( ()) iw x µ xt = e φ t s x xc t e, xc(0) = x0 & wxt (,) = 0 ρ = ψ = φ ( xt, ) : ( xt, ) s( x xc( t)) -- moves like a particle!! 1 µφ s s( x) = φs + V( x) φs + β φs φs

54 Semiclassical limits 0< ε 1 ψ : = ψ ε ε iε tψ (,) xt = ψ + V( x) ψ + β ψ ψ is0 ( x)/ ( x,0): ( x) ( x) e ε ε ψ = ψ = ρ ε ε ε ε ε ε ε ε 0 0 WKB analysis -- Gregor Wentzel, Hans Kramers & Leon Brillouin, 196 Formally assume ε ψ = ρ = = ρ ε ε is / ε ε ε ε ε ε e, v S, J v Geometrical Optics: Transport + Hamilton-Jacobi ε ε ε ρ + ( ρ S ) = 0, t ε ts + S + V x + = ρ ε 1 ε ε 1 ε d ( ) βρ ρ ε

55 From QM to fluid dynamics Quantum Hydrodynamics (QHD): Euler +3 rd dispersion ε ε ε t ρ + ( ρ v ) = 0 P( ρ) = βρ / ε ε ε J J ε ε ε ε ε t ( J ) + ( ) + P( ρ ) + ρ V = ( ρ ln ρ ) ε ρ 4 Formal Limits --- Euler equations for fluids t ρ + ( ρ v ) = 0 P( ρ) = βρ / 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 NLSE before caustics

56 Time-splitting spectral method (TSSP) For [ t,, apply time-splitting technique (Bao, Jaksch&Markowich, JCP, 03 ) n t n + 1] Step 1: Discretize by spectral method & integrate in phase space exactly 1 i tψ(,) xt = ψ Step : solve the nonlinear ODE analytically i tψ(,) xt = V( x) ψ(,) xt + β ψ(,) xt ψ(,) xt = = i ψ xt = V xψ xt + β ψ xt ψ xt t 1 i xt = + V x + t( ψ( xt, ) ) 0 ψ( xt, ) ψ( xt, n) (, ) ( ) (, ) (, n) (, ) i( t tn)[ V( x) + βψ ( xt, n) ] ψ( xt, ) = e ψ( xt, n) t ψ(,) ψ ( ) ψ β ψ ψ Use nd order Strang splitting (or 4 th order time-splitting)

57 Properties of TSSP Explicit & computational cost per time step: Time symmetric: yes Time transverse invariant: yes Mass conservation: yes Stability: yes Dispersion relation without potential: yes Accuracy Spatial: spectral order; Temporal: nd or 4 th order OM ( ln M) Best resolution in strong interaction regime: β 1

58 Numerical methods for rotating BEC Use polar coordinates (B., Q. Du & Y. Zhang, SIAP 06 ) Time-splitting + ADI technique (B. & H. Wang, JCP, 06 ) Generalized Laguerre-Hermite functions (B., H. Li & J. Shen, SISC, 09 ) TSSP via Langrage coordinate (B., D. Marharens, Q. Tang & Y. Zhang, SISC, 13 ). 1 i tψ( xt,) = [ + V( x) Ω Lz + β ψ ] ψ

59 Numerical methods for rotating BEC Numerical Method 1: (Bao, Q. Du & Y. Zhang, SIAM, Appl. Math. 06 ) Ideas Time-splitting Use polar coordinates: angular momentum becomes constant coefficient Fourier spectral method in transverse direction + FD or FE in radial direction Crank-Nicolson in time Features Time reversible; Time transverse invariant Mass Conservation in discretized level Implicit in 1D & efficient to solve; Accurate & unconditionally stable 1 iψt( xt,) = ψ Ω Lzψ, Lz : = ix ( y y x) = i θ iψ xt = V xψ xt + β ψ xt ψ xt ψ xt = ψ xt t(, ) d( ) (, ) d (, ) (, ) (, ) (, n)

60 Numerical methods for rotating BEC Numerical Method : (Bao & H. Wang, J. Comput. Phys. 06 ) Ideas Time-splitting ADI technique: Equation in each direction become constant coefficient Fourier spectral method Features Time reversible; Time transverse invariant Mass Conservation in discretized level Explicit & unconditionally stable; Spectrally accurate in space 1 1 iψt( xt,) = xxψ iωy xψ, iψt( xt,) = yyψ + iωx yψ iψ xt V xψ xt β ψ xt ψ xt ψ xt ψ xt t(, ) = d( ) (, ) + d (, ) (, ) (, ) = (, n)

61 Numerical methods for rotating BEC Numerical Method 3: (B., H. Li & J. Shen, 09 ) Ideas Time-splitting Use polar coordinates: angular momentum becomes constant coefficient Use the generalized Laguerre-Fourier Hermite functions as basis Features Time reversible; Time transverse invariant Mass Conservation in discretized level Explicit; spectral accurate & unconditionally stable 1 iψt( xt,) = ψ + Vho( x) ψ( xt,) ΩLzψ, iψ xt V x V x ψ xt β ψ xt ψ xt t(,) = [ d ( ) ho( )] (,) + d (,) (,)

62 Method 4 Bao, Du & Zhang, SIAP, 05 ; Bao & Cai, KRM, 13 ; 1 i tψ( xt,) = [ + V( x) Ω Lz + β ψ ] ψ Numerical methods A new formulation A rotating Lagrange coordinate: 1 x = At () x & φ( xt,): = ψ( xt,) = ψ( Atxt (),) cos( Ωt) sin( Ωt) 0 cos( Ωt) sin( Ωt) At ( ) = for d ; At ( ) sin( t) cos( t) 0 for d 3 sin( t) cos( t) = = Ω Ω = Ω Ω GPE in rotating Lagrange coordinates 1 d i t φ( xt,) = [ + V( At () x ) + β φ ] φ, x, t> 0 Analysis & numerical methods -- Bao & Wang, JCP6 ; Bao, Li & Shen, SISC, 09 ; Bao, Marahrens, Tang & Zhang, 13,.

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70 Models for central vortex states Ginzburg-Landau equation (GLE):Superconductivity, nonlinear heat flow, etc. 1 ( ) ψt = ψ + V( x) ψ ψ, x R, t > 0, ε Gross-Pitaevskii equation (GPE): nonlinear optics, BEC, superfluidity 1 ( ) iψt = ψ + V( x ) ψ ψ, x R, t > 0, ε Nonlinear wave equation (NLWE): wave motion 1 ( ) ψtt = ψ + V( x) ψ ψ, x R, t > 0, ε Here ψ : complex-valued wave function or order parameter, ε > 0: dimensionless constant, V( x) : real-valued external potential

71 Model property `Free Energy or Lyapunov functional: E 1 ( ) ( ) V ( x ψ ψ ) ψ dx, ψ δe( ψ) i = t δψ * E( ψ ) E( ψ 0), t 0 = + ε R Dispersive system (GPE): Energy conservation: Density conservation: Admits particle like solutions: solitons, kinks & vortices Dissipative system (GLE): Energy diminishing, etc. ( ψ ( x, t) ψ 0( x) ) dx 0, t 0 R ψ δe( ψ) = t δψ *

72 Two kinds of vortices `Bright-tail vortex: GLE, GPE & NLWE (`order parameter ) im V( x) 1, when x, ψ( x, t) 1, x, ( eg.. ψ e θ ), Time independent case (Neu, 90 ): Vortex solutions: 1 n f ( ) n r + fn ( r) r r f (0) = 0, n Asymptotic results f ψ ( x ) = φ ( x) = f ( r) e f n n ( r) + (1 f n f n ( ) = 1. ε = 1, V ( x ) 1 n ( r)) f n inθ n n + ar + O( r ), r) 1 n / r + O(1/ r ( n 4 ( r) = 0, 0 < r <, r 0, ), r.

73 `Bright-tail vortex

74 Two kinds of vortices `Dark-tail vortex: BEC (wave function) x i, x R, t 0, dx 1 ψt = ψ + ψ + β ψ ψ > ψ = ψ = n n ψ(,) xt = e φ ( x) = e f() r e Center vortex states: iµ t iµ t inθ n n d dfn() r n r 3 n fn() r = r + + f () (), 0, n r + fn r < r< µ β dr dr r n(0) = 0, n( ) = 0, π n ( ) = 1 0 f f f r r dr Asymptotic results f n () r n n + ar + O r r n r br e ( ), 0,, r.

75 `Dark-tail vortex

76 Stability of vortices Data chosen h ε = 1, V ( x, t) 1 ( + W ( x, t)), ψ 0 ( x) = φ ( x) ( + n noise) Results (Bao,Du & Zhang, Eur. J. Appl. Math., 07 ) For GLE or NLWE with initial data perturbed n=1: velocity density N=3: velocity density For NLSE with external potential perturbed Results n=1: velocity density N=3: velocity density

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85 Summary of Stability For GLE or NLWE: under perturbation either in initial data or in external potential n =1: dynamically stable, n >1: dynamically unstable. Splitting pattern depends on perturbation, when t >>1, it becomes n well-separated vortices with index 1 or 1. For NLSE or GPE: under perturbation in initial data dynamically stable: Angular momentum expectation is conserved!! under perturbation in external potential n =1: dynamically stable, n >1: dynamically unstable. Vortex centers can NOT move out of core size.

86 Vortex interaction For `bright-tail vortex: ψ φ φ N N ( x) = m ( x x ) (, ) j j = m x x j j y yj j= 1 j= 1 For `dark-tail vortex 0 Well-separate N ψ ( xy, ) = φ ( x x, y y) / j= 1 Overlapped n j j j

87 Reduced dynamic laws For `bright-tail vortex Take ansatz N ψ( xt, ) = φ ( x x( t)) + high order terms j= 1 N m j = φ ( x x ( t), y y ( t))+high order terms j j= 1 Plug into GLE or GPE or NLWE Using matched asymptotic techniques Prove rigorously j m j j

88 Reduced dynamic laws For `bright-tail vortex For GLE For GPE For NLWE () N d xj t xj() t xl() t κvj( t): = κ = mj ml, t 0 dt l= 1, l j xj() t xl() t x x j N 0 j(0) = j, 1. d x () t J( x () t x ()) t vj( t) : = = m t 0, () () N j j l l dt l= 1, l j xj t xl t xj(0) = xj 1 j N; J = 1 0 d x () N j t xj() t xl() t κ = m, 0 j ml t dt l= 1, l j xj() t xl() t x = x x = v j N 0 ' 0 j(0) j, j(0) j, 1.

89 Existing mathematical results For GLE (or NLHE): Pairs of vortices with like (opposite) index undergo a replusive (attractive) interaction: Neu 90, Pismen & Rubinstein 91, W. E, 94, Bethual, etc. Energy concentrated at vortices in D & filaments in 3D: Lin 95-- Vortices are attracted by impurities: Chapman & Richardson 97, Jian 01 For NLSE: Vortices behave like point vortices in ideal fluid: Neu 90 Obeys classical Kirchhoff law for fluid point vortices: Lin & Xin 98 Equations for vortex dynamics & radiation: Ovchinnikov& Sigal 98--; Jerrard, Bethual, etc.

90 Conservation laws Mass center N 1 xt (): = x () t N j = 1 Lemma The mass center of the N vortices for GLE is conserved xt ( ) : = x( t) x(0) : = x(0) = x N N N N N N 0 j j j j= 1 j= 1 j= 1 Lemma The mass center of the N vortices for NLWE is conserved if initial velocity is zero and it moves linearly if initial velocity is nonzero j

91 Conservation laws Signed mass center N 1 xt (): = m x () t N j = 1 j j Lemma The signed mass center of the N vortices for NLSE is conserved x ( t) : = m x ( t) x(0) : = m x (0) = m x N N N N N N 0 j j j j j j j= 1 j= 1 j= 1

92 Analytical Solutions N ( ) like vortices on a circle (Bao, Du & Zhang, SIAP, 07 ) 0 jπ jπ xj = a cos, sin, mj = m0 =± 1, 1 j N N N Analytical solutions for GLE figure ( N 1) jπ jπ xj ( t) = a + t cos, sin κ N N Analytical solutions for NLSE figure next jπ N 1 jπ N 1 xj ( t) = a cos + t, sin t + N a N a

93 N= N=3 back N=4

94 N= N=3 back

95 Analytical Solutions N ( 3) like vortices on a circle and its center(bao,du&zhang, SIAP, 07 ) 0 0 jπ jπ xn = 0; xj = a cos, sin, 1 j N 1 N 1 N 1 Analytical solutions for GLE figure N jπ jπ xn() t = 0; xj() t = a + t cos, sin, 1 j N 1 κ N 1 N 1 Analytical solutions for NLSE figure next jπ N jπ N xn() t = 0; xj() t = a cos + t, sin t + N 1 a N 1 a

96 N=3 back N=4

97 N=3 back N=4

98 Analytical Solutions Two opposite vortices (Bao, Du & Zhang, SIAP, 07 ) 0 0 x1 = x = a( cos ( θ0), sin ( θ0) ), m1 = m = 1 Analytical solutions for GLE figure x1() t = x() t = a t cos θ0, sinθ0 κ ( ( ) ( )) Analytical solutions for NLSE figure next t x t x ( ( θ0) ( θ0) ) j a 0 j( ) = j + sin, cos, = 1,

99 back

100 back

101 Analytical Solutions N ( 3) opposite vortices on a circle and its center 0 0 jπ jπ xn = 0 ( ); xj = a cos, sin ( + ), 1 j N 1 N 1 N 1 Analytical solutions for GLE figure ( N 4) jπ jπ xn() t = 0; xj() t = a + t cos, sin, 1 j N 1 κ N 1 N 1 Analytical solutions for NLSE figure next jπ N 4 jπ N 4 xn() t = 0; xj() t = a cos + t, sin t + N 1 a N 1 a

102 N=3 N=4 back N=5

103 N=3 N=4 back N=5

104 Numerical difficulties for `bright-tail vortex Numerical difficulties: Highly oscillatory nature in the transverse direction: resolution Quadratic decay rate in radial direction: large domain For NLSE, more difficulties: Time reversible Time transverse invariant Dispersive equation Keep conservation laws in discretized level Radiation

105 Numerical methods For `bright-tail vortex (Bao,Du & Zhang, Eur. J. Appl. Math., 07 ) Apply a time-splitting technique: decouple nonlinearity Adapt polar coordinate: resolve solution in transverse better Use Fourier pesudo-spectral discretization in transverse Use nd or 4 th order FEM or FDM in radial direction For `dark-tail vortex Apply a time-splitting technique: decouple nonlinearity Use Fourier pseudo-spectral discretization Use generalized Laguerre-Hermite pseudo-spectral method

106 Vortex dynamics & interaction Data chosen (Bao, Du & Zhang, SIAM, J. Appl. Math., 07 ) ε = 1, V ( x, t) Two vortices with like winding numbers N = For GLE: velocity density For GPE: velocity density Trajectory Summary 1, ψ N 0 ( x) = φn ( x x j j ), j= 1 m =, n1 = 1, n = 1, x1 = ( a,0), x = ( a,0) N j= 1 n j

107 back

108 back

109 back

110 back

111 GLE GPE back NLWE

112 Summary of interaction For GLE (Bao, Du & Zhang, SIAP, 07 ) Undergo a repulsive interaction Core size of the two vortices will well separated Energy decreasing: For NLSE (Bao, Du & Zhang, SIAP, 07 ) Behave like point vortices in ideal fluid The two vortex centers move almost along a circle Energy conservation & radiation For NLWE (Bao & Zhang, Physica D 07 ) Similar as GLE but with different speed E ψ ( x, t)) E( ψ ( x)), ( 1 t

113 Vortex dynamics & interaction Two vortices with opposite winding numbers N =, n = 1, n = 1, x1 = ( a,0), x ( 1 = a,0) For GLE: velocity density For NLSE: Case 1: velocity density Case : velocity density Trajectory Summary

114 back

115 back

116 back

117 back

118 back

119 back

120 GLE NLSE NLSE NLWE back

121 Summary of interaction For GLE (Bao, Du & Zhang, SIAP, 07 ) Undergo an attractive interaction & merge at finite time Energy decreasing: For GPE (Bao, Du & Zhang, SIAP, 07 ) depends on initial distance of the two centers Small: merge at finite time & generate shock wave Large: move almost paralleling & don t merge, solitary wave Energy conservation & radiation Sound wave generation E ( ψ ( x, t)) 0, For NLWE (Bao & Zhang, Physica D 07 ) Undergo an attractive interaction & merge at finite time t

122

123 Vortex-pair on bounded domain

124 Vortex-dipole on bounded domain

125 Interaction of 3 like vortices

126 Interaction of 3 opposite vortices

127 Interaction of a lattice

128 `dark-tail vortex-pair in BEC well-separate

129 `dark-tail vortex-dipole in BEC

130 `dark-tail vortex lattice in BEC

131 Conclusions Models for vortex in superfluidity & superconductivity Ginzburg-Landau equation (GLE) Gross-Pitaevskii equation (GPE)/ nonlinear Schrodinger equation (NLSE) Nonlinear wave equation Methods for vortex nucleation in superfluidity Gradient flow with normalization Backward Euler spectral (BESP) method Methods for vortex dynamics in superfluidity Time-splitting spectral (TSSP) method Central vortex stability and interaction