Chaos in disordered nonlinear lattices

Transcription

1 Chaos in disordered nonlinear lattices Haris Skokos Physics Department, Aristotle University of Thessaloniki Thessaloniki, Greece URL: Work in collaboration with Ioannis Gkolias, George Voyatzis, Sergej Flach, Joshua Bodyfelt, Tanya Laptyeva, Dima Krimer, Stavros Komineas

2 Dynamical Systems: Outline The quartic Klein-Gordon (KG) disordered lattice The disordered nonlinear Schrödinger equation (DNLS) Numerical methods Different dynamical behaviors Single site excitations Block excitations Summary

3 Interplay of disorder and nonlinearity Waves in disordered media Anderson localization [Anderson, Phys. Rev. (1958)]. Experiments on BEC [Billy et al., Nature (008)]. Waves in nonlinear disordered media localization or delocalization? Theoretical and/or numerical studies [Shepelyansky, PRL (1993) Molina, PRB (1998) Pikovsky & Shepelyansky, PRL (008) Kopidakis et al. PRL (008)] Experiments: propagation of light in disordered 1d waveguide lattices [Lahini et al., PRL (008)]. 3

4 The Klein Gordon (KG) model p ε 1 H = + u + 1 u + u u N l l 4 K l l l+1 - l l=1 4 W with fixed boundary conditions u 0 =p 0 =u N+1 =p N+1 =0. Typically N= Parameters: W and the total energy E. ε l chosen uniformly from,. The discrete nonlinear Schrödinger (DNLS) equation N β 4 * * 1 H = ε ψ + ψ - ψ ψ + ψ ψ, ψ = q + ip D l l l l+1 l l+1 l l l l l=1 W W where ε l chosen uniformly from, and is the nonlinear parameter. 4

5 Linear case: Δ K = 1+ W ω ν, + W 4 (Δ D = W + 4) Scales, width of the squared frequency spectrum: Localization volume of an eigenstate: V~ N l=1 1 A 4 ν,l Average spacing of squared eigenfrequencies of NMs within the range of a localization volume: d K Δ V K Nonlinearity induced squared frequency shift of a single site oscillator δ = l 3E ε E The relation of the two scales dk ΔK with the nonlinear frequency shift δ l determines the packet evolution. l l (δ = β ψ ) l l 5

6 Distribution characterization We consider normalized energy distributions in normal mode (NM) space z ν Eν m m of the νth NM. 1 E with E = A + ωa ν ν ν ν N ν ν=1 Second moment: Participation number:, where A ν is the amplitude m = ν - ν z N with ν = νzν P= 1 N z ν=1 ν ν=1 measures the number of stronger excited modes in z ν. Single mode P=1, equipartition of energy P=N. 6

7 Lyapunov Exponents Roughly speaking, the Lyapunov exponents of a given orbit characterize the mean exponential rate of divergence of trajectories surrounding it [see e.g. Ch.S., LNP, (010)]. Consider an orbit in the N-dimensional phase space with initial condition x(0) and an initial deviation vector from it v(0). Then the mean exponential rate of divergence is: 1 σ(x(0), v(0)) = lim ln t t σ 1 =0 Regular motion σ 1 0 Chaotic motion v(t) v(0) 8 July 011

8 Computational methods Consider an N degree of freedom autonomous Hamiltonian systems of the form: As an example, we take the Hénon-Heiles system: Hamilton equations of motion: Variational equations:

9 Symplectic integration schemes If the Hamiltonian H can be split into two integrable parts as H=A+B, a symplectic scheme for integrating the equations of motion from time t to time t+τ consists of approximating the operator e τl H, i.e. the solution of Hamilton equations of motion, by j τl τ(l +L ) c τl d τl e = e e e H A B i A i B for appropriate values of constants c i, d i. So the dynamics over an integration time step τ is described by a series of successive acts of Hamiltonians A and B. We consider a particular symplectic integrator [Laskar & Robutel, Cel. Mech. Dyn. Astr. (001)]. i=1 (3-3) (3-3) τl A τ 3τ τ τ L 6 LB LA LB 3 6 SABA = e e e e e A

10 Tangent Map (TM) Method We use symplectic integration schemes for the integrating the equations of motion AND THE VARIATIONAL EQUATIONS. The Hénon-Heiles system can be split as: The system of the Hamilton equations of motion and the variational equations is split into two integrable systems which correspond to Hamiltonians A and B.

11 Tangent Map (TM) Method So any symplectic integration scheme used for solving the Hamilton equations of motion, which involves the act of Hamiltonians A and B, can be extended in order to integrate simultaneously the variational equations [Ch. S. & Gerlach, PRE (010) Gerlach & Ch.S., Discr. Cont. Dyn. Sys. (011), Gerlach, Eggl, Ch.S., IJBC (01)].

12 Symplectic Integrator SABA C The integrator has only small positive steps and its error is of order O(τ ). In the case where A is quadratic in the momenta and B depends only on the positions the method can be improved by introducing a corrector C, having a small negative step: 3 c -τ L A,B,B with - 3 c =. 4 (3-3) (3-3) τl A τ 3τ τ τ L 6 LB LA LB 3 6 SABA = e e e e e e Thus the full integrator scheme becomes: SABAC = C (SABA ) C and its error is of order O(τ 4 ). A 1

13 Symplectic Integrator SABA C for the KG system We apply the SABAC integrator scheme to the KG Hamiltonian by using the splitting: A= N l=1 p l ε 1 1 N l 4 B = u l + u l + ul+1 - ul l=1 4 W with a corrector term which corresponds to the Hamiltonian function: N 1 C A,B,B = u ε u u + u - u l=1 W l ( l l ) ( l -1 l+1 l ). 13

14 Different Dynamical Regimes Three expected evolution regimes [Flach, Chem. Phys (010) - Ch.S., Flach, PRE (010) - Laptyeva et al., EPL (010) - Bodyfelt et al., PRE(011)]: Weak Chaos Regime: δ<d, m ~t 1/3 Frequency shift is less than the average spacing of interacting modes. NMs are weakly interacting with each other. [Molina PRB (1998) Pikovsky, Shepelyansky, PRL (008)]. Intermediate Strong Chaos Regime: d<δ<δ, m ~t 1/ m ~t 1/3 Almost all NMs in the packet are resonantly interacting. Wave packets initially spread faster and eventually enter the weak chaos regime. Selftrapping Regime: δ>δ Frequency shift exceeds the spectrum width. Frequencies of excited NMs are tuned out of resonances with the nonexcited ones, leading to selftrapping, while a small part of the wave packet subdiffuses [Kopidakis et al., PRL (008)]. 14

15 Single site excitations DNLS W=4, β= 0.1, 1, 4.5 KG W = 4, E = 0.05, 0.4, 1.5 slope 1/3 slope 1/3 slope 1/6 slope 1/6 No strong chaos regime In weak chaos regime we averaged the measured exponent α (m ~t α ) over 0 realizations: α=0.33±0.05 (KG) α=0.33±0.0 (DLNS) [Flach, Krimer, Ch. S., PRL (009) Ch. S., Krimer, Komineas, Flach, PRE (009) Ch. S., Flach, PRE (010)] 15

16 Single site excitations (E=0.4) Lyapunov exponents 16

17 KG: Weak Chaos (E=0.4) 17

18 KG: Selftrapping (E=1.5) 18

19 Single site excitations: Different spreading regimes KG W = 4, E = 0.4, 1.5 Average over 0 realizations 19

20 Crossover from strong to weak chaos We consider compact initial wave packets of width L=V [Laptyeva, Bodyfelt, Krimer, Ch. S., Flach, EPL (010) Bodyfelt, Laptyeva, Ch. S., Krimer, Flach, PRE (011)] Time evolution DNLS KG 0

21 Crossover from strong to weak chaos DNLS β= 0.04, 0.7, 3.6 KG E/L= 0.01, 0., 0.75 W=4, L=1 Average over 1000 realizations! (log t) d log m dlog t α=1/ α=1/3 1

22 Block excitations (E/L=0.) Lyapunov exponents

23 KG: Weak Chaos (E/L=0.01) 3

24 KG: Strong Chaos (E/L=0.) 4

25 KG: Selftrapping (E/L=0.75) 5

26 Block excitations: Different spreading regimes W = 4, E/L = 0.01, 0., 0.75 Average over 0 realizations 6

27 Summary We predicted theoretically and verified numerically the existence of three different dynamical behaviors: Weak Chaos Regime: δ<d, m ~t 1/3 Intermediate Strong Chaos Regime: d<δ<δ, m ~t 1/ m ~t 1/3 Selftrapping Regime: δ>δ Generality of results: a) Two different models: KD and DNLS, b) Predictions made for DNLS are verified for both models. Our results suggest that Anderson localization is eventually destroyed by the slightest amount of nonlinearity, since spreading does not show any sign of slowing down. Questions under investigation: What is the actual chaotic nature of spreading? What is the final fate of the wave packet? 7