Singularity Formation in Nonlinear Schrödinger Equations with Fourth-Order Dispersion

Transcription

1 Singularity Formation in Nonlinear Schrödinger Equations with Fourth-Order Dispersion Boaz Ilan, University of Colorado at Boulder Gadi Fibich (Tel Aviv) George Papanicolaou (Stanford) Steve Schochet (Tel Aviv) Shimshon Bar-Ad (Physics, Tel Aviv)

2 Outline Review of Nonlinear Schrödinger equation (NLS) theory Higher-order dispersion (HOD) effects in: 1. Numerical discretization effects of NLS singularity 2. Optical light bullets Analysis of NLS eqs. w/ HOD Applying results to: 1. Numerical discretization effects of NLS singularity 2. Optical light bullets Summary

3 Nonlinear Schrödinger equation (NLS) iψ t (t, x) + ψ + κ ψ 2σ ψ = 0, ψ(0, x) = ψ 0 (x) x = (x 1,..., x d ) = 2 x 1 x x d x d (2 nd -order dispersion) ψ 0 H 1 (R d ), ψ H 1 ψ ψ 2 2 Applications (σ = 1): Laser propagation in Kerr media (water/air/silica) Deep water waves w/ surface-tension Bose-Einstein Condensation...

4 Conserved quantities f p p = ψ p dx iψ t (t, x) + ψ + κ ψ 2σ ψ = 0 power (mass): N(ψ) ψ 2 2 = N(0) Hamiltonian: H(ψ) ψ 2 2 κ σ+1 ψ 2σ+2 2σ+2 = H(0) κ < 0: self-defocusing (same sign) κ > 0: self-focusing (opposite signs)

5 Singularity formation (blowup, collapse) Variance identity (Vlasov et al 75): V (t) x 2 ψ 2 dx V tt = 8H(0) 4κ σd 2 σ + 1 ψ 2σ+2 2σ+2 κ > 0, σd 2 = V tt 8H(0) H(0) < 0 = T <, V (T ) = 0 = finite-time singularity Singularity occurs in all norms: ψ H 1 ψ 2 ψ ψ 2σ+2 as t t c

6 Global existence (GE) Local existence (Ginibre & Velo 79; Kato 87): ψ H 1 formally bounded = global existence (GE) ψ H 1 ψ ψ 2 2 ψ 2 2 ψ = need a bound for ψ 2 Hamiltonian conservation = H = H(0) = ψ 2 2 = H(0) + κ σ + 1 ψ 2σ+2 2σ+2 κ < 0 (defocusing): ψ 2 2 H(0) = GE

7 Global existence (cont.) κ > 0 (focusing): Gagliardo-Nirenberg inequality ψ 2σ+2 2σ+2 C σ,d ψ 2σ+(2 σd) 2 ψ σd 2, ψ H 1 Combined with H = H(0) get ψ 2 2 H(0) + C σ,d σ + 1 ψ 2σ+(2 σd) 2 ψ σd 2 σd < 2 or σd = 2 & ψ < N c ( ) 1/σ σ + 1 = GE C σ,d σ NLS = 2/d - critical exponent N c - critical power

8 Summary 1. κ < 0 (defocusing NLS): GE 2. κ > 0 (focusing NLS): (a) σ < 2/d (sub-critical): GE (b) σ = 2/d (critical): i. H(0) < 0 sufficient for singularity formation ii. ψ N c necessary for singularity formation iii. collapse can be arrested by small perturbations (c) σ > 2/d (super-critical): can have singularity formation σ NLS = 2/d

9 Blowup simulations Kelley, PRL 65: (2+1)D NLS radially-symmetric iψ t (t, r) + ψ + ψ 2 ψ = 0, = 2 rr + 1 r r Specialized methods: 1. Dynamic rescaling (McLaughlin, Papanicolaou, Sulem, Sulem 86) 2. Galerkin finite-elements (Akrivis, Dougalis, Karakashian, McKinney 98) 3. Iterative grid redistribution (Ren, Wang 00)

10 10 15 ψ 0 = 4 exp( x 2 y 2 ) ψ t Can resolve huge gradients for radially-symmetric NLS But less useful/harder to implement for: NLS + perturbations No radial symmetry In such cases use standard finite-difference schemes

11 Singularity formation w/ FD schemes? Fibich & Ilan 03: 1. How do finite-difference (FD) methods break down near NLS singularity? 2. 2 nd -order vs. higher-order FD discretizations? No theory! Compare conservation laws for hyperbolic eqs.: How do FD schemes break down near shocks 1 st -order discretization = +numerical diffusion 2 nd -order discretization = +numerical dispersion Extensive theory

12 Semi-discrete (2+1)D NLS iψ t (t, x, y) + ψ + ψ 2σ ψ = 0, = 2 xx + 2 yy 2 nd -order semi-discrete FD scheme (continuous in t) ψ δ 2 ψ n,k = ψn 1,k + ψ n,k 1 4ψ n,k + ψ n,k+1 + ψ n+1,k h 2 δ 2 ψ n,k = ψ + h2 12 (ψ xxxx + ψ yyyy ) + O(h 4 ) modified equation w/ fourth-order dispersion (4OD): iψ t (t, x, y) + ψ + ψ 2σ ψ + h2 12 (ψ xxxx + ψ yyyy ) = 0

13 Modified equations For 2 nd -order FD discretization get iψ t + ψ + ψ 2σ ψ + h2 12 (ψ xxxx + ψ yyyy ) = 0 For m th -order FD discretization get iψ t + ψ + ψ 2σ ψ +( 1) m ε ( 2m ) ψ x 2m + 2m ψ y 2m = 0, ε = ch 2m 2 NLS w/ small, mildly-anisotropic HOD

14 Optical light bullets Silberberg (1990): Under the combined effect of diffraction, anomalous dispersion, and nonlinear refraction, an optical pulse can collapse[...] in time and space[...] yield pulses with extremely high optical fields. Potential applications: optical switches high-energy physics... Remains a holy grail in NL Optics

15 Ultrashort pulses in a pure Kerr medium NL Maxwell s equations for ultrashort pulses in Kerr medium iψ z (z, x, t) + ψ γ 2 ψ tt + ψ 2 ψ +γ }{{} 4 ψ tttt = 0 same sign Bulk medium: x = (x, y), = 2 xx + 2 yy Planar waveguide: x = x, = 2 xx Anomalous 2 nd -order dispersion: γ 2 < 0 = same sign = spatio-temporal self-focusing Small 4 th -order dispersion (4OD), γ 4 1, becomes important for ultrashort pulses

16 After rescaling & renaming of variables get Bulk media (3+1)D: iψ t (t, x, y, z)+ ψ + ψ 2 ψ +εψ xxxx = 0, = 2 xx + 2 yy + 2 zz Planar waveguides (2+1)D: iψ t (t, x, y) + ψ + ψ 2 ψ + εψ xxxx = 0, = 2 xx + 2 yy NLS w/ small, strongly-anisotropic 4OD x - physical time ε = 0 = spatio-temporal collapse in bulk medium & planar waveguides (σ σnls = 2/d for d = 2, 3)

17 Key questions Effects of: 1. mildly-anisotropic high-order dispersion (HOD) 2. strongly-anisotropic fourth-order dispersion (4OD) on singularity formation and global-existence in NLS Application to: 1. FD discretization effects of NLS eqs. 2. Optical light-bullets

18 Isotropic 4OD Mixed-dispersion NLS iψ t (x, t) + ψ + ψ 2σ ψ + ε 2 ψ = 0, = 2 x 1 x x d x d Simpler problem: Biharmonic NLS (B-NLS): iψ t (x, t) + ɛ 2 ψ + ψ 2σ ψ = 0, ɛ = ±1 Stability of solitons (Karpman 91-99, Karpman and Shagalov 00) Local existence (Ben-Artzi, Koch, and Saut 00) Almost no theory!

19 Biharmonic NLS (B-NLS) iψ t (x, t) + ɛ 2 ψ + ψ 2σ ψ = 0, ɛ = ±1 Conserved quantities: ψ 2 2 ψ (power/mass) H(t) = ɛ ψ σ + 1 ψ 2σ+2 2σ+2 H(0) (Hamiltonian) ɛ > 0: self-defocusing (same sign) ɛ < 0: self-focusing (opposite signs)

20 Global existence in B-NLS Local existence theory (Ben-Artzi, Koch, & Saut 00): ψ H 2 formally bounded = GE ψ H 2 ψ ψ ψ 2 2 need only to bound ψ 2 Fibich, Ilan, Papanicolaou 02 H = H(0) = ɛ ψ 2 2 = H(0) + 1 σ + 1 ψ 2σ+2 2σ+2 ɛ = 1 (defocusing): ψ 2 2 H(0) = GE

21 ɛ = 1 (focusing): Gagliardo-Nirenberg inequality ψ 2σ+2 2σ+2 B σ,d ψ 2σ+(2 σd)/2 2 ψ σd/2 2, ψ H 2 Combined with H = H(0) get ψ 2 2 H(0) + B σ,d σ + 1 ψ 2σ+(2 σd)/2 2 ψ σd/2 2 σd < 4 or σd = 4 & ψ < N B c ( ) 1/σ σ + 1 = GE B σ,d N B c - critical power, σ B NLS = 4/d - critical exponent

22 Global existence in B-NLS iψ t (x, t) + ɛ 2 ψ + ψ 2σ ψ = 0, ɛ = ±1 Theorem 1 (Fibich, Ilan, Papanicolaou 02) Sufficient conditions for GE: 1. defocusing: ɛ > 0 2. sub-critical focusing: ɛ < 0 and σd < 4 3. critical: ɛ < 0, σd = 4, and ψ < N B c := ( σ+1 B σ,d ) 1/σ σ B NLS = 4/d Similar to GE-theory for NLS (σ NLS = 2/d)

23 Singularity formation in B-NLS Can observe blowup numerically (ɛ < 0 & σd 4) But no proof Open problems: Variance identity for B-NLS? Special symmetries of B-NLS? Related questions: Optimal constant in inequality = N B c =? Ground-state of B-NLS: ψ(x, t) = R B (x) e it 2 R B R B + R 8/d+1 B = 0 In NLS theory (Weinstein 83): N c = R 2 2

24 Ground-states & optimal constants N B c = ( ) 1/σ σ + 1, B σ,d 1 B σ,d = f σd/2 inf 0 f H 2 2 f 2+2σ σd/2 2 f 2σ+2 2σ+2 Lemma 1 If infimum is attained, then Nc B R B is the ground-state of = R B 2 2, where 2 R B R B + R 8/d+1 B = 0 R B (r) ce r/ 2 cos(r/ 2) as r Ground-state is neither positive nor monotonic!

25 Calculation of waveguide solns d = 1, σ = 4 (critical B-NLS) R (4) B R B + RB 9 = 0 (0) = R B (0) = 0, R B( ) = R B ( ) = 0 R B R B (r) 1.15 R B (0)<0 R B (0)> Critical power for collapse: N B c = R B r

26 Asymptotic blowup profile Collapsing core converges to a modulated ground-state ψ L 1/2 (t)r B (r/l(t)), L(t) = c/ ψ(0, 0, t) t t c 0 R(0) A R(0) B L 1/2 ψ R B (r) z=0 z r/l R B (r) z=0 z r/l Figure 1: (A) ψ 0 = c (1 + r 4 ) 1, (B) ψ 0 = c e r2

27 Mixed-dispersion eqs. Theorem 2 (Fibich, Ilan, Papanicolaou 02) Sufficient conditions for GE in iψ t (t, x) + ψ + ψ 2σ ψ + ε 2 ψ = 0 and in the mildly-anisotropic eqs. iψ t (t, x) + ψ + ψ 2σ ψ + ε d i=1 ψ xi x i x i x i = 0 are same as for B-NLS. In particular: ε < 0, σd = 4, ψ < N B c ( σ+1 B σ,d ) 1/σ

28 Corollary 1 (Fibich & Ilan 02) GE of solns of 2 nd and m th -order modified eqs. for FD schemes iψ t + ψ + ψ 2σ ψ + h2 12 (ψ xxxx + ψ yyyy ) = 0 (2 nd ) iψ t + ψ + ψ 2σ ψ + ( 1) m ε ( 2m ) ψ x 2m + 2m ψ y 2m Solns of semi-discrete NLS do not blow-up Dynamics of the numerical solns? = 0, (m th )

29 Dynamics of modified equation iψ t + ψ + ψ 2σ ψ + ( 1) m ε ( 2m ) ψ x 2m + 2m ψ y 2m = 0, ε > 0 Initially HOD accelerates focusing H = ψ 2 2 }{{} defocusing 1 σ + 1 ψ 2σ+2 }{{} focusing ε ( ψ x m ψ y m 2 ) 2 }{{} weakly focusing but collapse is arrested by HOD (defocusing case) H 1 σ + 1 ψ 2σ+2 }{{} defocusing ε ( ψ x m ψ y m 2 2 ) }{{} defocusing

30 FD discretizations of critical NLS (σ = 1, d = 2, m = 2, 4) 70 A 70 B ψ t t 70 C 70 D ψ t t discretizations initially accelerate focusing no blowup, multiple focusing-defocusing cycles 1 st -order modified eqs. valid only in the first cycle

31 Strong anisotropy iψ t (t, x) + ψ + ψ 2σ ψ + ε k i=1 ψ xi x i x i x i = 0, (0 k d) H = ψ k σ + 1 ψ 2σ+2 ε Theorem 3 (Fibich Ilan, Schochet 03) i=1 ψ xi x i 2 2 critical exponent = σ (d, k) = 2 d k/2 Proof: use anisotropic Gagliardo-Nirenberg inequality

32 σ (d, k) = σ (d, 0) = σ NLS = 2/d 2 d k/2 σ NLS < σ (d, k) < σ B NLS, σ (d, d) = σ B NLS = 4/d 0 < k < d Interpolates between critical exponents of NLS & B-NLS

33 Arrest of spatiotemporal collapse Bulk medium (3+1)D iψ t (t, x, y, z) + ψ + ψ 2σ ψ + εψ xxxx = 0, ε < 0 σ = 1 (Kerr) & σ (d = 3, k = 1) = 4/5 = super-critical Negative 4OD cannot arrest collapse Planar waveguide (2+1)D iψ t (t, x, y) + ψ + ψ 2σ ψ + εψ xxxx = 0, ε < 0 σ = 1 & σ (d = 2, k = 1) = 4/3 = sub-critical Negative 4OD can arrest collapse = optical light bullets?

34 Robustness of light-bullets Fibich & Ilan 04: Bullets robust to 3OD (w/ physical parameters) & noise Bullets undergo almost-elastic collisions Conclusion: bullets possible in planar waveguides Explanation of experiments: Cheskis et al. 03

35 Positive anisotropic 4OD iψ t (t, x, y) + ψ + ψ 2σ ψ + εψ xxxx = 0, ε > 0 H = ψ 2 2 ε ψ xx 2 }{{} 2 1 σ + 1 ψ 2σ+2 opposite signs Open problem: opposite signs = no GE theory Related open problem: GE theory for iψ t + ψ xx ψ yy + ψ 2 ψ = 0

36 σ = 1, d = 2, ɛ = 0.01, N(0) = 1.25N c single focusing-defocusing cycle strong radiation Open problem: GE theory?

37 Summary Global existence theory for Biharmonic & mixed-dispersion NLS Breakdown of FD schemes for NLS: role of HOD effects NLS w/ negative anisotropic 4OD - critical exponent Planar waveguides: GE = optical bullets (robust) Bulk media: singularity formation (numerical) NLS w/ higher-order dispersion beginning of theory, lots of open problems: Proof of singularity formation Existence of ground-states GE theory w/ opposite-signs of dispersion...

38 References 1. G. Fibich, B. Ilan and G.C. Papanicolaou, Self focusing with fourth-order dispersion, SIAM Journal on Applied Mathematics, G. Fibich and B. Ilan, Discretization effects in the nonlinear Schrödinger equation, Applied Numerical Mathematics, G. Fibich, B. Ilan and S. Schochet, Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion, Nonlinearity, G. Fibich and B. Ilan, Optical light bullets in a pure Kerr medium, Optics Letters, 2004