Quasipatterns in surface wave experiments

Transcription

1 Quasipatterns in surface wave experiments Alastair Rucklidge Department of Applied Mathematics University of Leeds, Leeds LS2 9JT, UK With support from EPSRC A.M. Rucklidge and W.J. Rucklidge, Convergence properties of the 8, 10 and 12 mode representations of quasipatterns Physica D 178 (2003) Isaac Newton Institute, 15 December 2005 p. 1/50

2 Pattern formation Patterns form in many different experiments: convection, Faraday waves, Taylor Couette, optical systems, reaction diffusion problems... Faraday (surface) wave experiment: Forcing: a 1 cos ω 1 t + a 2 cos ω 2 t, with ω 1 : ω 2 = m : n As the forcing is increased, the flat state becomes unstable and patterns form on the surface Isaac Newton Institute, 15 December 2005 p. 2/50

3 Experimental patterns Two frequency forcing, silicone oil Aspect ratio: up to 50 wavelengths The primary patterns are usually hexagons or squares: Arbell & Fineberg Phys. Rev. E (2002) Isaac Newton Institute, 15 December 2005 p. 3/50

4 Experimental patterns Various spatially periodic patterns are found with increased forcing: Arbell & Fineberg Phys. Rev. E (2002) Spatial periodicity allows the analysis of these transitions using standard techniques (cf. R, Silber and Fineberg (2003), Tse et al. Physica D 146 (2000) ) Isaac Newton Institute, 15 December 2005 p. 4/50

5 Experimental patterns The Faraday experiment appeals because observed patterns have such a high degree of symmetry The patterns have very clean Fourier spectra, even when the amplitude is high Arbell & Fineberg Phys. Rev. E (2002) Isaac Newton Institute, 15 December 2005 p. 5/50

6 Experimental quasipatterns Quasipatterns, like quasicrystals, have orientational but not translational order This is manifested in their Fourier spectra: 8, 10 or 12 peaks with the same wavenumber Kudrolli, Pier & Gollub Physica D (1998) Isaac Newton Institute, 15 December 2005 p. 6/50

7 Aside: Al Mn quasicrystal Shechtman et al. Phys. Rev. Lett. (1984) Isaac Newton Institute, 15 December 2005 p. 7/50

8 Experimental quasipatterns Quasipatterns and hexagons can coexist Kudrolli, Pier & Gollub Physica D (1998) Isaac Newton Institute, 15 December 2005 p. 8/50

9 Experimental quasipatterns Boundaries appear not to be important Edwards & Fauve J. Fluid Mech. (1994) Isaac Newton Institute, 15 December 2005 p. 9/50

10 Experimental quasipatterns 8, 10 and 12-fold quasipatterns have been observed in twoand three-frequency experiments Arbell & Fineberg Phys. Rev. E (2002) Isaac Newton Institute, 15 December 2005 p. 10/50

11 Experimental parameters Arbell & Fineberg Phys. Rev. E (2002) Isaac Newton Institute, 15 December 2005 p. 11/50

12 Pattern selection Forcing: f(t) = a m cos(mωt) + a n cos(nωt + φ) Waves can be harmonic (same period) or subharmonic (twice the period) as the forcing a Three-wave resonant interactions occur between excited, neutral and weakly damped waves k = k m k = k n k These resonances can stabilise or destabilise certain combinations of waves, and so favour or discourage certain angles from appearing in the pattern k y k m k m k n k x Cf. Silber, Topaz & Skeldon (2000), Topaz, Porter & Silber (2004) Isaac Newton Institute, 15 December 2005 p. 12/50

13 Pattern selection The selected angles are determined by k m and k n, which can be controlled experimentally π 2, π 3 2 cos ( ) θ 2 : squares and hexagons = k n k m other values: superlattice patterns π 4, π 5, π 6 : 8-, 10- and 12-fold quasipatterns We would like to predict which wavevectors are excited, and the amplitude and stability of the pattern Isaac Newton Institute, 15 December 2005 p. 13/50

14 Summary Many of these patterns look like they ought to be describable in terms of a relatively small number of modes There is a well developed theory for spatially periodic patterns But there are difficulties with quasipatterns: the problem of small divisors... Isaac Newton Institute, 15 December 2005 p. 14/50

15 A question of convergence To understand the problem with quasipatterns, we step back from Faraday waves, and consider a simpler pattern-forming PDE on the plane (for example, the Swift Hohenberg equation) U t = F ( U, U x, U ) y ; µ U(x, y, t) is a measure of the pattern, µ is a parameter, and (x, y) R 2 When µ increases through zero, the trivial (U = 0) solution loses stability to waves with k = 1 Waves in any direction are permitted, but choose a finite number (Q) of wavevectors: k 1 up to k Q Isaac Newton Institute, 15 December 2005 p. 15/50

16 A question of convergence Close to the bifurcation point (µ small), suppose that we can write the solution as U(x, y, t) = ɛ Q j=1 A j (t) exp(ik j x) + O(ɛ 2 ) ɛ 1 is a small parameter measuring the distance above onset, and A j (t) is the amplitude of the mode with wavevector k j This is implicitly a series solution for the pattern Question: does this series converge? Isaac Newton Institute, 15 December 2005 p. 16/50

17 A question of convergence For spatially periodic patterns, the problem is posed on a periodic domain, not the plane A Lyapunov Schmidt or centre manifold reduction can be used to prove the existence of solutions Applying standard perturbation theory results in amplitude equations: A j = f j (A 1,..., A Q ; µ) + O(ɛ) The same procedure can be used formally for quasipatterns, in spite of the question of convergence Isaac Newton Institute, 15 December 2005 p. 17/50

18 A question of convergence The temptation is to truncate, not worry about the higher order terms and obtain: Ȧ j = µa j + Q/2 k=1 β jk A k 2 A j + resonant terms From this, predictions about amplitudes and stability of quasipatterns can be made Isaac Newton Institute, 15 December 2005 p. 18/50

19 A question of convergence It has been known since Poincaré that allowing quasiperiodicity can (and will) cause standard perturbation theory to fail, owing to the presence of small divisors In spite of this, many authors have written down amplitude equations for quasipatterns, and used these to calculate the amplitude and stability of quasipatterns The question of convergence was resolved for Hamiltonian and other problems by Kolmogorov, Arnol d and Moser (KAM) in the 1950 s and 60 s The same methods are not directly applicable here Isaac Newton Institute, 15 December 2005 p. 19/50

20 Perturbation theory How does perturbation theory fail? We will address this in the context of the Swift Hohenberg equation, but exactly the same issues arise in any treatment of quasipatterns in the Faraday wave problem or other problems We will not address here the mechanism for selecting quasipatterns in the Faraday wave experiment (an important and interesting question) Isaac Newton Institute, 15 December 2005 p. 20/50

21 Perturbation theory Write the governing PDE as U t = L(U) + N µ (U) where L is the linearised PDE at the bifurcation point, and N µ has the parameter dependence and nonlinear terms Write a series solution for U: U = ɛu 1 + ɛ 2 U 2 + ɛ 3 U Relate µ to ɛ: µ = ɛ (or µ = ɛ 2 ) Scale the time evolution: d/dt = ɛd/dt (or ɛ 2 ) Solve order by order Isaac Newton Institute, 15 December 2005 p. 21/50

22 Leading order At O(ɛ), we have L(U 1 ) = 0 At the bifurcation point, the solution is made up of waves with k = 1 and any orientation U 1 = Q j=1 A j (T ) exp(ik j x) The Q vectors k 1,..., k Q lie on the unit circle These vectors are usually chosen to be evenly spaced around the unit circle, but this isn t required at leading order Isaac Newton Institute, 15 December 2005 p. 22/50

23 Next order At O(ɛ 2 ), we have L(U 2 ) = N (U 1 ) Time dependence (du 1 /dt ) and parameter dependence are included in N All combinations of wavevectors are generated by the nonlinear terms: (k j1 + k j2 ) j 2 j 1 All waves with k = 1 are in the kernel of L, so L(exp(ik x)) = 0 whenever k = k = 1 More generally, L(exp(ik x)) (k 2 1) 2 exp(ik x) whenever k is close to 1, since the growth-rate curve typically has a quadratic maximum Isaac Newton Institute, 15 December 2005 p. 23/50

24 Next order Divide the nonlinearly generated waves into two sets k = 1: coefficients of these modes must be zero This solvability condition yields equations for A j k 1: for these modes, L can be inverted to give U 2 of the form U 2 = C j1j2a j1a j2 (k 2 1) 2 exp(ik x) j 1 j 2 where k = k j1 + k j2 in the sum, with k 1, and C j1 j 2 are constants The form of the denominator is only approximate, though it is exact for certain model PDEs Isaac Newton Institute, 15 December 2005 p. 24/50

25 Higher order At N th order, we divide the nonlinear terms in the same way k = 1: these provide corrections to the evolution equations for A j k 1: for these modes, L can be inverted for U N : U N = j 1 j N C j1...jn A j1... A jn (k 2 1) 2 exp(ik x) where k = k j1 + + k jn in the sum, with k 1 Question: how close can k get to 1, as N increases? Isaac Newton Institute, 15 December 2005 p. 25/50

26 Small divisors If k gets close to 1, the denominator becomes small, resulting in a U N that is much larger than we had intended If the pattern is spatially periodic, k is bounded away from 1, since all combinations of wavevectors lie on a lattice If the pattern is quasiperiodic, then k comes arbitrarily close to 1 as N becomes large In this case, the assumption we made in writing U can break down Is ɛ N U N small compared to ɛ? Isaac Newton Institute, 15 December 2005 p. 26/50

27 Small divisors Q = 12 evenly spaced vectors, and N = 11 Isaac Newton Institute, 15 December 2005 p. 27/50

28 Small divisors Q = 12, N = 7 N = 11 N = 15 The nonlinearly generated wavevectors are denser and approach the unit circle as N increases This quasilattice is dense in the limit N Isaac Newton Institute, 15 December 2005 p. 28/50

29 Small divisors Q = 12, N = 7 N = 11 N = 15 The vectors that are closest to the unit circle are vertically above (1, 0) Isaac Newton Institute, 15 December 2005 p. 29/50

30 Small divisors Q = 8 Q = 10 Q = 12 Q = 14 Q = 8, 10 and 12 are qualitatively similar Q = 14 is much more dense, for the same N Isaac Newton Institute, 15 December 2005 p. 30/50

31 Small divisors Given Q (the number of vectors around the circle), and N (the number of vectors in the sum), how close can k get to 1? Take Q = 12, for example, and write k = n 1 k n 6 k 6, where n n 6 = N. Then k 2 = n n 2 6+n 1 n 3 + n 6 n 2 + 3(n 1 n 2 + n 6 n 1 ) 1 So we have a rational approximation to 3: 3 1 (n n2 6 + n 1n 3 + n 6 n 2 ) n 1 n 2 + n 6 n 1 = p l q l p l q l should be a continued fraction approximation to 3: p l q l = 1 1, 2 1, 5 3, 7 4, 19 11, 26 15, 71 41,... for k to be close to 1 Isaac Newton Institute, 15 December 2005 p. 31/50

32 Small divisors The combination k = p l (0, 1) + (q l 1) ( 1 2, 3 2 ) + (q l + 1) ( 1 2, ) 3 2 k = (1, p l 3q l ) has N = p l + 2q l, and k 2 1 (p l 3q l ) 2 1 q 2 l 1 N 2 using the properties of continued fraction representations of quadratic irrationals Isaac Newton Institute, 15 December 2005 p. 32/50

33 Small divisors The closest any combination of N vectors can come to the unit circle satisfies K 1 N 2 < k 2 1 < K 2 N 2 for constants K 1 and K 2, apart from combinations that give k = 1, for Q = 8, 10 and 12 This result is independent of which PDE we are solving Rucklidge & Rucklidge Physica D (2003) Isaac Newton Institute, 15 December 2005 p. 33/50

34 Small divisors k - 1 k - 1 k - 1 k (a) N (b) N (c) N (d) N Closest distance to the unit circle, Q = 8, 10, 12, 14 N up to 10 5 Distances scale as N 2, N 2, N 2 and (apparently) N 4 Isaac Newton Institute, 15 December 2005 p. 34/50

35 Perturbation theory We now have a bound (a Diophantine condition) on the smallness of the small divisors But this is not enough to save the perturbation theory Recall: U N = C j1...jn A j1... A jn (k 2 1) 2 exp(ik x) j 1 j N U N = O(N 4 ) U N 1 = O(N 4 ) O((N 1) 4 ) U 1 = O(N N ) U 1 so the series for U contains terms of the order of ɛ N U N ɛ N N N and fails to converge The series could still be asymptotic but to what? Isaac Newton Institute, 15 December 2005 p. 35/50

36 Swift Hohenberg equation To illustrate this failure of convergence, consider the model PDE: U t = µu (1 + 2 ) 2 U U 3 This has an instability to k = 1 modes when µ = 0 Assume an equal-amplitude steady quasipattern solution: 12 U(x, y) = A (N) (µ) exp(ik j x) +... j=1 Carry out perturbation theory to order N up to 33 Look at how the amplitude of the quasipattern depends on µ and N Isaac Newton Institute, 15 December 2005 p. 36/50

37 Swift Hohenberg equation 0.8 (a) 0.50 (b) 0.6 A (N) µ (c) µ (d) 0.2 A (N) µ µ Q = 6: the series converges for µ < 0.65 Q = 8, 10 and 12, the series diverge Isaac Newton Institute, 15 December 2005 p. 37/50

38 Swift Hohenberg equation 0.2 N = 31 A (N) 0.1 N = 29 N = 17 N = µ Q = 12: the series diverges at a value of µ that gets smaller as N gets larger For fixed N, and µ small enough, we have A (N) µ At best, the series is asymptotic Isaac Newton Institute, 15 December 2005 p. 38/50

39 Swift Hohenberg equation A n A n+2 /A n Q = order n (b) (a) Q = 12 Q = 8 Q = 6 Q = 4 Q = 2 Q = 12 Q = 10 Q = 8 Q = 6 Q = order n The radius of convergence is related to a bound on the ratio of coefficients as N For quasipatterns, this ratio increases in steps as new vectors are generated, ever closer to k = 1 Isaac Newton Institute, 15 December 2005 p. 39/50

40 Summary so far For Q = 8, 10 and 12, combinations of N vectors approach the unit circle no faster than O(N 2 ) Standard perturbation theory does not converge fast enough (if at all) to provide a reliable method of computing properties of quasipatterns, even though there is a Diophantine condition There is not yet an existence proof of quasipatterns And yet... Isaac Newton Institute, 15 December 2005 p. 40/50

41 And yet... Kudrolli, Pier & Gollub Physica D (1998) Isaac Newton Institute, 15 December 2005 p. 41/50

42 A way forward This is work in progress with Ian Melbourne Focus on the existence problem; worry about stability later We want to find a steady quasipattern solution of the PDE: F(U; µ) = 0, taking the Swift Hohenberg equation as an example Let P be a projection operator onto the Q modes on the unit circle (k = 1) Perform a Lyapunov Schmidt construction: Q U(x, y) = A exp(ik j x) + Φ(x, y) j=1 Isaac Newton Institute, 15 December 2005 p. 42/50

43 A way forward Project the PDE onto modes with k 1: Q 0 = µφ (1 + 2 ) 2 Φ (I P) A exp(ik j x) + Φ j=1 This is the hard part: solve this equation implicitly for Ψ as a function of A and µ This requires the Hard Implicit Function Theorem (Cf. de la Llave (2001) Proc. Symp. Pure Math ) Substitute Ψ into the PDE projected on to k = 1: 0 = µa (... )A Unfortunately, one of the hypotheses of the theorem is not satisfied, so this approach may not work without considerably more effort Isaac Newton Institute, 15 December 2005 p. 43/50 3

44 Another way forward Use a method of Craig and Wayne (1993 Comm. Pure Appl. Math ) developed for constructing periodic solutions of nonlinear wave equations If this works, the method will provide quasipatterns as a function of µ, but only for an open full measure set of parameter values, not a complete branch The method relies on having well behaved small divisors: as N increases, the vectors coming close to k = 1 appear in a regular fashion This condition is satisfied for Q = 8, 10 and 12 This condition is probably not satisfied for Q = 14 Work in progress... Isaac Newton Institute, 15 December 2005 p. 44/50

45 Numerical quasipatterns Solve F(U; µ) = 0 numerically using a Newton-like method: U 1 = Q µ exp(ik j x) j=1 U 2 = U 1 J 1 F(U 1 ; µ) U 3 = U 2 J 1 F(U 2 ; µ) U 4 = U 3 J 1 F(U 3 ; µ)... At each step, nonlinear interactions generate more modes In practice, the Jacobian J can only be estimated By the fourth step, with Q = 12, there are O(10 6 ) modes Isaac Newton Institute, 15 December 2005 p. 45/50

46 Numerical quasipatterns Truncate in two ways: Retain only modes with k 5 Retain modes only up to N = 1, 3, 9 or 27 Symmetry implies that many modes have the same amplitude Q = 8: N = 1 8 modes 1 amplitude N = 3 64 modes 5 amplitudes N = modes 28 amplitudes N = modes 195 amplitudes Q = 12: N = 1 12 modes 1 amplitude N = modes 11 amplitudes N = modes 62 amplitudes N = modes 460 amplitudes Isaac Newton Institute, 15 December 2005 p. 46/50

47 Numerical quasipatterns Q = 8, µ = 0.01, modes closest to k = 1 N k = 1 k = 1.47 k = 1.16 k = Q = 8, µ = 0.1 N k = 1 k = 1.47 k = 1.16 k = Q = 8, µ = 0.2: method does not converge Isaac Newton Institute, 15 December 2005 p. 47/50

48 Numerical quasipatterns (a) Q=8 µ=0.01 (b) Q=8 µ= If the method is to have a hope of converging, the amplitudes must decay exponentially with N This appears to be the case, even for µ = 0.1 where the perturbation method diverged Isaac Newton Institute, 15 December 2005 p. 48/50

49 Numerical quasipatterns N = 27 N = 3 N = A (N) µ Newton s method converges where perturbation theory fails using the same modes Isaac Newton Institute, 15 December 2005 p. 49/50

50 Conclusions For Q = 8, 10 and 12, combinations of N vectors approach the unit circle no faster than O(N 2 ) Standard perturbation theory should be treated with caution There is hope for an existence proof for quasipatterns Questions: Do the 8-, 10- and 12-mode truncated amplitude equations have predictive power for stability, in spite of the convergence problems? Why do experiments produce Q = 2, 4, 6, 8, 10 and 12, but apparently not 14? Spatial chaos? One dimensional methods (Iooss and Los 1990) do not apply here Isaac Newton Institute, 15 December 2005 p. 50/50