Necklace solitary waves on bounded domains

Transcription

1 Necklace solitary waves on bounded domains with Dima Shpigelman Gadi Fibich Tel Aviv University 1

2 Physical setup r=(x,y) laser beam z z=0 Kerr medium Intense laser beam that propagates in a transparent medium (air, water, glass, ) Direction of propagation is z 2

3 NLS in nonlinear optics i,, 2 z z x y xy, 0, diffraction Kerr nonlinearity laser beam r=(x,y) z Competition between diffraction Δψ = ψ xx + ψ yy and focusing nonlinearity z = t (evolution variable) z 0, x, y x, y 0 z=0 Kerr medium Initial value problem in z 3

4 Optical collapse Experiments in the 1960s showed that intense laser beams become narrower with propagation (self-focusing) Sufficiently powerful beams undergo optical collapse Kelley (1965) : Solutions of 2D cubic NLS t,, 2 0, 0,,, i t x y x y x y can become singular (blowup) in finite time (distance) T c 0 4

5 NEW 5

6 Solitary waves NLS solutions of the form solitary it x x e R ( ), ( x, y), where R R R 3 ( x) 0 Dependence on µ through the dilation symmetry R( x) R( x), R : R 1 Power (L2 norm) of R µ is independent of µ: 2 2 P( Rµ ) P( R), P( Rµ ) : Rµ R 2 µ dxdy 2 6

7 Solitary waves cont. R R R x y 3 ( x) 0, x (, ) Infinite number of radial solutions R (n) (r) Of most interest is the ground state R (1) (Townes profile) Positive, monotonically decreasing Has exactly the critical power for collapse Single peak at the origin R (1) number of peaks P cr R (1) 2 2 7

8 Stability of solitary waves 2 i t, x, y 0, (0, x, y) ( x, y) t 0 Lemma (1) Let 0 cr ( x) If c 1, then blows up in finite time If 0< c 1, then scatters as t Ground-state solitary waves have a dual instability 8

9 Challenge How can we ``overcome the dual instability and propagate localized laser beams in a Kerr medium over long distances? Important for applications 9

10 Necklace beams Soljacic, Sears, Segev (1998): Use necklace beams e.g., consider the input beam csech( r r )cos( m) 0 max Consists of 2m identical beams (``pearls ), located at equal distances along a circle of radius r max Adjacent input beams have opposite signs (phases) Hence, Ψ 0 =0 on the rays between the beams ( n 1/ 2) n ' n1,, 2m m By anti-symmetry, Ψ=0 on these rays for t>0. These rays thus act as reflecting boundaries Beams do not interact Beams repel each other 10

11 Necklace beams Soljacic, Sears, Segev (1998): Use necklace beams e.g., consider the input beam csech( r r )cos( m) 0 max Consists of 2m identical beams (``pearls ), located at equal distances along a circle of radius r max Adjacent input beams have opposite signs (phases) Hence, Ψ 0 =0 on the rays separating the beams By uniqueness, Ψ=0 on these rays for t>0 11

12 Necklace beams Soljacic, Sears, Segev (1998): Use necklace beams e.g., consider the input beam csech( r r )cos( m) 0 max Consists of 2m identical beams (``pearls ), located at equal distances along a circle of radius r max Adjacent input beams have opposite signs (phases) Hence, Ψ 0 =0 on the rays separating the beams By uniqueness, Ψ=0 on these rays for t>0 These rays thus act as reflecting boundaries Beams do not interact Necklace structure preserved for t>0 12

13 Necklace beams Soljacic, Sears, Segev (1998): Use necklace beams e.g., consider the input beam csech( r r )cos( m) 0 max Consists of 2m identical beams (``pearls ), located at equal distances along a circle of radius r max Adjacent input beams have opposite signs (phases) Hence, Ψ 0 =0 on the rays separating the beams By uniqueness, Ψ=0 on these rays for t>0 These rays thus act as reflecting boundaries Beams do not interact Necklace structure preserved for t>0 Beams with opposite phases repel each other 13

14 Necklace beams csech( r r )cos( m) 0 max Idea: Let each beam (``pearl ) have power below P cr Nonlinearity weaker than diffraction, so beam expands Out-of-phase adjacent beams repel each other Slows down the expansion of each beam 14

15 Simulation 15

16 Simulation 16

17 Simulation Necklace expansion much slower than for a single beam Ultimately, necklace either scatters as t, or collapses in finite time (distance), simultaneously at 2m points 17

18 Experiments Grow et al. (07): First observation of a necklace beam in a Kerr medium input beam after propagating 30cm in glass 18

19 Recent application Jhajj et al (14): used a necklace beam configuration to set up a thermal waveguide in air 19

20 Comparison with vortex beams Necklace beams Vortex beams csech( r r )cos( m) 0 max c sech( r r )exp( im) 0 max 20

21 Necklace solitary waves? Proposition (Fibich and Shpigelman, 16): The NLS 2 i t, x, y 0, x, y t does not admit necklace solitary waves. In other words, there are no necklace solutions of 3 R( x, y) R 0, x, y R 21

22 Multi-peak solitary waves Theorem (Ao, Musso, Pacard, Wei, 2012 & 2015) There exist multi-peak, sign-changing solutions of 3 R( x, y) R 0, x, y R These solutions do not have the necklace form 22

23 Multi-peak solitary waves Theorem (Ao, Musso, Pacard, Wei, 2012 & 2015) There exist multi-peak, sign-changing solutions of 3 R( x, y) R 0, x, y R These solutions do not have the necklace form As with all solitary-wave solutions NLS, they are strongly-unstable it er( x) of the critical 23

24 Propagation in PR crystal Yang et al. (05): Propagation in a photorefractive (PR) crystal with optically-induced photonic lattice E i t t x y 1I ( x, y) 0,, 0, 2 I x y 2 2 (, ) : 0 sin ( x y) sin ( x y) I Showed theoretically and experimentally that necklace solitary waves exist, and can be stable Because of the need to induce a photonic lattice, not applicable for propagation in a Kerr medium 24

25 How to stabilize necklace beams in a Kerr medium? Idea: Confine beam to a hollow-core fiber To leading-order, fiber walls are perfect reflectors Therefore, propagation governed by the NLS on a bounded domain 2 i t, x, y 0, ( x, y) D, t 0, ( x, y) D Typically, D is the unit disk B 1 = {x 2 +y 2 1} 25

26 How to stabilize necklace beams in a Kerr medium? Idea: Confine beam to a hollow-core fiber To leading-order, fiber walls are perfect reflectors Propagation governed by the NLS on a bounded domain D R 2 Typically, D is the unit disk B 1 = {x 2 +y 2 1} 2 i t, x, y 0, ( x, y) D, t 0, ( x, y) D 26

27 NLS on a bounded domain 2 i t, x, y 0, ( x, y) D, t 0, ( x, y) D Admits solitary waves solitary x iµt e R µ ( ), where 3 R R R 0, ( x, y) D, R 0, ( x, y) D Studied by Fibich and Merle (01), Fukuizumi, Hadj Selem and Kikuchi (12), Noris, Tavares and Verzini (15) Mostly for radial solutions on the unit disk B 1 27

28 Radial solitary waves on the unit disk B 1 R R R 3 ( r) 0, 0 r 1, R 0, r 1 Unlike free-space, no dilation symmetry (1) ( ) (1) r ( r ) R R d P R d (1) ( ) 0, lin R R 0, 0 r 1, R 0, r 1 28

29 Linear stability - review Let ψ = e iμt [R μ + ε[u x + iv x ] e Ωt ] Then the linearized equation for the perturbation reads where 0 L + L 0 u v =Ω u v, L + = Δ μ + 3 R μ 2, L = Δ μ + R μ 2 Linear instability if Re(Ω)>0 Lemma (Vakhitov and Kolokolov): If R μ >0, then ψ = e iμt R μ is linearly stable if d and unstable if d P ( R ) 0 P ( R ) 0 d d VK condition 29

30 Orbital stability - review Lemma If R μ >0, then ψ = e iμt R μ and unstable if d P ( R ) 0 d is orbitally stable if d P ( R ) 0 d Weinstein, Grillakis, Shatah, Strauss 30

31 Orbital stability - review Lemma If R μ >0, then ψ = e iμt R μ and unstable if d P ( R ) 0 d is orbitally stable if d P ( R ) 0 d Weinstein, Grillakis, Shatah, Strauss On the unit disc, VK condition is satisfied: 31

32 Solitary waves on the unit disk - stability THM (Fibich and Merle, 2001) The ground-state solitary waves it on the unit disk are orbitally stable: e R (1) () r (1) i solitary If 0 R, then inf 02 ( t) e ( t) Unlike in free-space, where they are strongly unstable ``Reflecting boundary has a stabilizing effect But, (1) 0 ( ) P R P cr Goal: Find stable solitary waves with P>P cr Important for applications Idea: use necklace solitary waves! 32

33 Solitary waves on the unit disk - stability THM (Fibich and Merle, 2001) The ground-state solitary waves it on the unit disk are orbitally stable: e R (1) () r (1) i solitary If 0 R, then inf 02 ( t) e ( t) Unlike in free-space, where they are strongly unstable ``Reflecting boundary has a stabilizing effect But, (1) 0 ( ) P R P cr Goal: Find stable solitary waves with P>P cr Important for applications Idea: use necklace solitary waves! 33

34 Types of necklace solitary waves 3 R R R 0, ( x, y) D, R 0, ( x, y) D 34

35 Types of necklace solitary waves 3 R R R 0, ( x, y) D, R 0, ( x, y) D circle 35

36 Types of necklace solitary waves 3 R R R 0, ( x, y) D, R 0, ( x, y) D circle rectangle 36

37 Types of necklace solitary waves 3 R R R 0, ( x, y) D, R 0, ( x, y) D New solitary waves circle rectangle annulus 37

38 Rectangular necklace solitary wave Compute a positive/ground-state/single-pearl solitary wave on a square Construct rectangular necklace using 3*2 single pearls 38

39 Single pearl on a square 3 R R R 0, 1 x, y 1, R 0, x, y 1 R 0, 1 xy, 1, Compute using non-spectral renormalization method 39

40 Single pearl on a square 3 R R R 0, 1 x, y 1, R 0, x, y 1 R 0, 1 xy, 1, Compute using non-spectral renormalization method 40

41 Single pearl on a square 3 R R R 0, 1 x, y 1, R R 0, x, y 1 0, 1 xy, 1, R R 0, 1 x, y 1, R 0, x, y 1 41

42 Single pearl on a square 3 R R R 0, 1 x, y 1, R R 0, x, y 1 0, 1 xy, 1, weakly nl regime R cr lin 42

43 Single pearl on a square 3 R R R 0, 1 x, y 1, R R 0, x, y 1 0, 1 xy, 1, R strongly nl regime R free-space free-space R ( x) weakly nl regime R cr lin 43

44 Single pearl on a square 3 R R R 0, 1 x, y 1, R R 0, x, y 1 0, 1 xy, 1, d dμ P R (1) μ > 0 44

45 From single pearl to a rectangular necklace 45

46 From single pearl to a rectangular necklace 46

47 From single pearl to a rectangular necklace R µ =0 on interfaces between pearls Lemma : R (32) is a necklace solution of 0, 3 3, 2 2, 3 R R R x y with 3 2 pearls R 0, x 3, y 2 47

48 Can also construct necklace necklace solitary waves on non-rectangular domains 48

49 Circular necklace solitary wave 3 R R R 0, x 1, R 0, x 1 Compute a positive/ground-state/single-pearl on a sector of a circle strongly nl regime d dμ P R μ > 0 weakly nl regime 49

50 Circular necklace solitary wave 3 R R R 0, x 1, R 0, x 1 Compute a positive/ground-state/single-pearl on a sector of a circle strongly nl regime d dμ P R (1) μ > 0 weakly nl regime 50

51 From single pearl to a circular necklace 51

52 From single pearl to a circular necklace R µ =0 on interfaces between pearls 52

53 Annular necklace solitary wave 3 R R R 0, r min x r R 0, x rmin, r Compute a positive/ground-state/single-pearl on a sector of an annulus max max Strongly nl regime d dμ P R μ > 0 weakly nl regime 53

54 Annular necklace solitary wave 3 R R R 0, r min x r R 0, x rmin, r Compute a positive/ground-state/single-pearl on a sector of an annulus max max Strongly nl regime d dμ P R μ > 0 weakly nl regime 54

55 From single pearl to an annular necklace 55

56 From single pearl to an annular necklace R µ =0 on interfaces between pearls 56

57 Stability Single-pearl solitary waves Necklace solitary waves 57

58 Single-pearl stability Perturbed single-pearl solutions ψ 0 = 1.05R μ 1 t=0 t=10 Numerical observation: All single-pearl solitary waves are stable. Indeed, they satisfy the VK condition d P R 1 dμ μ > 0 58

59 Necklace stability Necklace is stable under perturbations that preserve its antisymmetries, since then it inherits the single-pearl stability e.g. ψ 0 = 1.05R μ n To excite the instability numerically, use perturbations that break the anti-symmetries, such as ψ 0 = [1 + noise x, y ]R μ n Multiply bottom-left pearl of R μ n by

60 Necklace stability Since P R ( n) np R (1), necklace satisfies the VK condition d P R n dμ μ > 0 This does not imply stability, however, since the VK condition implies stability only for positive solitary waves Go back to the original linearized eigenvalue problem: Let ψ = e iμt [R μ n + ε[u x + iv x ] e Ωt ] Then where 0 L + L 0 u v =Ω u v, L + = Δ μ + 3 R μ n 2, L = Δ μ + R μ n 2 Instability if Re(Ω)>0 60

61 Circular necklace with 4 pearls Stable for Unstable for lin cr Instability not related to collapse! ``Less stable'' than radial solitary waves cr µ (4) P R cr 0.24P cr 61

62 Circular necklace with 4 pearls Stable for Unstable for lin cr cr cr Necklace becomes unstable when the power of each pearl is 0.08P necklace (4) Pth : P R 0.24 P cr Necklace instability not related to collapse! Necklace ``less stable'' than radial solitary waves µ cr 62

63 Circular necklace with 4 pearls Stable for Unstable for lin cr cr cr Necklace becomes unstable when the power of each pearl is 0.08P necklace (4) Pth : P R 0.24 P cr Necklace instability not related to collapse! Necklace ``less stable'' than radial solitary waves µ cr 63

64 Circular necklace with 4 pearls Stable for Unstable for lin cr cr cr Necklace becomes unstable when the power of each pearl is 0.08P P R cr necklace (4) th : P? Necklace instability not related to collapse! Necklace ``less stable'' than radial solitary waves µ 64

65 Circular necklace with 4 pearls Stable for Unstable for lin cr cr cr Necklace becomes unstable when the power of each pearl is 0.08P necklace (4) Pth : P R 0.24 P cr Necklace instability not related to collapse! Necklace ``less stable'' than radial solitary waves µ cr 65

66 Dynamics Multiply bottom-left pearl of R μ 4 by 1.05 cr 1 cr 1 66

67 Dynamics Multiply bottom-right pearl of R μ 4 by 1.05 cr 1 cr 1 Instability but no collapse! 67

68 Stable necklace propagation above P cr? Saw that (4) P R cr Idea 1: take more pearls 0.24P cr 68

69 Stable necklace propagation above P cr? Saw that (4) P R cr Idea 1: take more pearls 0.24P cr (6) (8) P R 0.23 Pcr, P R 0.2 P cr cr Threshold power for necklace instability decreases with number of pearls cr 69

70 Stable necklace propagation above P cr? Saw that (4) P R cr Idea 1: take more pearls 0.24P cr (6) (8) P R 0.23 Pcr, P R 0.2 P cr cr Threshold power for necklace instability decreases with number of pearls cr 70

71 Stable necklace propagation above P cr? Saw that (4) P R cr Idea 1: take more pearls 0.24P cr (6) (8) P R 0.23 Pcr, P R 0.2 P cr cr Threshold power for necklace instability decreases with number of pearls Idea 2 : Use rectangular domain cr 71

72 Stable necklace propagation above P cr? Saw that (4) P R cr Idea 1: take more pearls 0.24P cr (6) (8) P R 0.23 Pcr, P R 0.2 P cr cr Threshold power for necklace instability decreases with number of pearls Idea 2 : Use rectangular domain P R cr (2*2) 0.55P Surprising, but not good enough cr cr 72

73 Stable necklace propagation above P cr? Saw that (4) P R cr Idea 1: take more pearls 0.24P cr (6) (8) P R 0.23 Pcr, P R 0.2 P cr cr Threshold power for necklace instability decreases with number of pearls Idea 2 : Use rectangular domain P R cr (2*2) 0.55P Surprising, but not good enough cr cr 73

74 Instability dynamics To raise the threshold power for necklace stability, need to understand the instability dynamics linearized eigenvalue problem: Instability if Re(Ω)>0 Yes/No approach 0 L + L 0 u v =Ω u v Fibich, Ilan, Sivan, Weinstein (06-08): Deduce instability dynamics from the unstable eigenfunctions 74

75 Instability dynamics

76 Instability dynamics

77 Instability dynamics e.f. pert

78 Instability dynamics e.f. pert

79 Instability dynamics

80 Instability dynamics

81 Instability dynamics

82 Instability dynamics

83 Stable necklace propagation above P cr? Saw that (4) P R cr Idea 1: take more pearls 0.24P cr (6) (8) P R 0.23 Pcr, P R 0.2 P cr cr Threshold power for necklace instability decreases with number of pearls Idea 2 : Use rectangular domain P R cr (2*2) 0.55P Surprising, but not good enough Idea 3: Use annular domain, since hole reduces interactions between pearls 83 cr cr

84 Annular necklace with 4 pearls 84

85 Annular necklace with 4 pearls Stable for lin (4) P R 0.24Pcr cr Hole does not help cr 85

86 Annular necklace with 4 pearls Stable for lin (4) P R 0.24Pcr cr Hole does not help cr Second stability regime! (4) 3.1P cr P R 3.7Pcr Hole does help! 86

87 Dynamics ψ 0 = [ noise x, y ]R μ

88 Dynamics ψ 0 = [ noise x, y ]R μ

89 Dynamics ψ 0 = [ noise x, y ]R μ

90 Dynamics ψ 0 = [ noise x, y ]R μ 4 10 stable solitary wave with P>3P cr 30 90

91 Positive solitary waves on the annulus 3 R( x) R R 0, r min x r max Can look for radial solutions R 0, x rmin, rmax R 0, r x r Are these the ground-state (minimal power) solutions? Compute ground-state solutions numerically using the nonspectral renormalization method min R r r 3 R( r) R 0, min r R 0, r rmin, rmax R 0, rmin r r max max max 91

92 Ground states on the annulus 3 R( x) R R 0, r min x r max R 0, x rmin, rmax R 0, r x r min max Radial for lin cr Non-radial for Symmetry breaking Related to variational characterization of the ground state 2 U P 2 cr inf HU ( ) 92

93 R equation on the annulus 3 R( x) R R 0, r min x r max R 0, x rmin, rmax First example of 1. Non-uniqueness of positive solutions of the R equation 2. A positive solution of the R equation which is not a groundstate Previously observed for the inhomogeneous NLS (Kirr et al., 08) 93

94 Stability on the annulus Radial, cr 0.3 stable Radial, cr 0.3 unstable Non-radial, cr 0.3 stable 94

95 Computation of necklace solitary waves 3 R R R 0, ( x, y) D, R 0, ( x, y) D R 0, ( xy, ) D Nonlinear elliptic problem Multidimensional cannot use radial symmetry Non-positive solution with necklace structure Bounded domain Plan: Compute a single pearl (positive) Extend into a necklace using the anti-symmetries 95

96 Computation of a single pearl 3 R R R 0, ( x, y) D, R 0, ( x, y) D R Nonlinear elliptic problem Multidimensional Non-positive solution Bounded domain 0, ( xy, ) D When D = R 2, can use Petviashvili s spectral renormalization method Cannot apply on a bounded domain Introduce a non-spectral version 96

97 Non-spectral renormalization method Rewrite equation for R µ as LR 3 R, L : Solve using fixed point iterations 1 3 k1 k, k 0,1,... R L R L discretized using finite differences Problem: Iterations diverge to zero or to infinity To prevent divergence, renormalize at each stage R k c k R k 3 ck k 1 R k 1 L R, k 0,1,... Determine c k from the requirement that c k R k satisfies an integral relation, such as R, R R, L R

98 1D NLS on bounded domain Fukuizumi, Hadj Selem, and Kikuchi (12) studied multi-peak solitary waves of 1D NLS On B 1 : 2D radial analog are R (n) (r) 2D non-radial analog are necklace solutions 2 i t, x 0, 1 x 1, t xx 0, x 1 Theory in 1D and 2D cases is very similar 98

99 Summary New type of solitary waves of the 2D NLS on bounded domains Stable at low powers Threshold for necklace instability < critical power for collapse Instability unrelated to collapse Second stability region for necklace solitary waves on the annulus Stable solitary waves with P>P cr Symmetry breaking of the ground state on the annulus Non-spectral renormalization method for computing solitary waves on bounded domains G. Fibich and D. Shpigelman Positive and necklace solitary waves on bounded domains Physica D 315: 13-32,