Waves on deep water, II Lecture 14

Transcription

1 Waves on deep water, II Lecture 14 Main question: Are there stable wave patterns that propagate with permanent form (or nearly so) on deep water? Main approximate model: i" # A + $" % 2 A + &" ' 2 A + ( A 2 A = 0 Nonlinear Schrödinger equation (NLS)

2 Waves on deep water 1. The story so far A uniform train of periodic waves is unstable on deep water, according to NLS and to experiments.

3 Waves on deep water 1. The story so far A uniform train of periodic waves is unstable on deep water, according to NLS and to experiments The 1-D NLS equation is completely integrable.

4 Waves on deep water 1. The story so far A uniform train of periodic waves is unstable on deep water, according to NLS and to experiments The 1-D NLS equation is completely integrable. For focussing NLS in 1-D on ("#,#), arbitrary initial data evolve into a finite number of envelope solitons, plus a modulated wavetrain that disperses (so its amplitude decays) as " # $

5 Waves on deep water 1. The story so far A uniform train of periodic waves is unstable on deep water, according to NLS and to experiments The 1-D NLS equation is completely integrable! For focussing NLS in 1-D on ("#,#), arbitrary initial data evolve into a finite number of envelope solitons, plus a modulated wavetrain that disperses (so its amplitude decays) as " # $ Envelope solitons are stable in 1-D NLS. [For defocussing NLS, dark solitons are stable.]

6 Chapter 2: Waves on deep water Near recurrence of initial states a) Lake, Yuen, Rungaldier & Ferguson, 1977 proposed (correctly) that with periodic boundary conditions, focussing NLS should exhibit near recurrence of initial states, just as KdV does.

7 Waves on deep water b) What is near recurrence of initial states? Example from linearized equations on deep water, with periodic boundary conditions: N "(x,t) = # a m cos{mx $% m t + & m }, % 2 m = gm. m=1

8 Waves on deep water b) What is near recurrence of initial states? Example from linearized equations on deep water, with periodic boundary conditions: "(x,t) = N # a m cos{mx $% m t + & m }, % 2 m = gm. m=1 Frequencies are not rationally related: " m = " 1 m η(x,t) is not periodic in time, but for finite N the solution returns close to its initial state, over and over again

9 Experimental evidence of recurrence in deep water Lake et al, 1977 Initial frequency: ω = 3.6 Hz λ = 12 cm [First physical observation of FPU recurrence?]

10 Q: Stable wave patterns on deep water? A#1. NLS in 1-D with periodic b.c.: A uniform train of oscillatory plane waves is unstable But a continuous wave train exhibits near recurrence of initial states.

11 Q: Stable wave patterns on deep water? A#1. NLS in 1-D with periodic b.c.: A uniform train of oscillatory plane waves is unstable But a continuous wave train exhibits near recurrence of initial states. A#2. NLS in 1-D with localized initial data: Envelope solitons are stable (Envelope solitons have played an important role in communication through optical fibers)

12 Q: What about a 2-D free surface? (so a 3-D fluid flow) 1. 2-D NLS: σ = +1 for envelope solitons σ = -1 for dark solitons i" # A + " $ 2 A + %" & 2 A + 2' A 2 A = 0

13 Q: What about a 2-D free surface? (so a 3-D fluid flow) 1. 2-D NLS: σ = +1 for envelope solitons σ = -1 for dark solitons i" # A + " $ 2 A + %" & 2 A + 2' A 2 A = 0 2. Zakharov & Rubenchik, 1974 σ = +1: for either sign of β, envelope solitons are unstable to 2-D perturbations σ = -1: for either sign of β, dark solitons are unstable to 2-D perturbations The unstable perturbations have long transverse wavelengths (Problem in water waves, but not necessarily in optical fibers)

14 Recall experiment by Hammack on envelope soliton (a) 6 m from wavemaker (b) 30 m from wavemaker

15 Hammack repeated the experiment, using the same wavemaker, in a wider tank

16 Q: Stable patterns that propagate with (nearly) permanent form on 2-D surface in deep water? A. The story continues - stay tuned

17 Intermission: wave collapse in 2-d Zakharov & Synakh, 1976: Consider elliptic, focusing NLS in 2-D i" # A + " $ 2 A + " % 2 A + 2 A 2 A = 0 (same signs for all coefficients not gravity waves)

18 Intermission: wave collapse in 2-d Zakharov & Synakh, 1976: Consider elliptic, focussing NLS in 2-D i" # A + " 2 $ A + " 2 % A + 2 A 2 A = 0 (same signs for all coefficients not gravity waves) Conserved quantities (finite list): I 1 = [ A 2 ] "" d#d$, I 2 = [A" # A * $ A * " # A] %% d#d&, I 3 = [A" & A * $ A * " & A] %% d#d&, I 4 = H = [ "A 2 # A 4 ] $$ d%d&.

19 Intermission: wave collapse in 2-d i" # A + " $ 2 A + " % 2 A + 2 A 2 A = 0 Consider J(") = %% [(# 2 + $ 2 ) A 2 ] d#d$ If we interpret: A 2 (ξ, ζ, τ) as mass density, then I 1 = "" [ A 2 ] d#d$ is total mass, and J(τ) is moment of inertia. J(τ) 0.

20 Intermission: wave collapse in 2-d i" # A + " $ 2 A + " % 2 A + 2 A 2 A = 0 Consider Compute Find: J(") = %% [(# 2 + $ 2 ) A 2 ] d#d$ dj d" and d 2 J d" 2 d 2 J d" 2 = 8H = 8 [ #A 2 $ A 4 ] %% d&d'. If Η < 0, then J(τ) < 0 in finite time. (Bad!) This happens while I 1, I 2, I 3, H are conserved. [Wave collapse has been important in nonlinear optics.]

21 Back to the main story Q: Are there stable wave patterns that propagate with permanent form (or nearly so) on a 2-D free surface in deep water? More complication: Lake, Yuen, Rungaldier & Ferguson (1977) Recall near recurrence of initial states

22 Lake, Yuen, Rungaldier & Ferguson Frequency downshifting also seen in optics

23 Frequency downshifting different from recurrence Frequency downshifting does not occur in simulations based on NLS, in 1-D or 2-D It does not occur in simulations based on Dysthe s (1979) generalization of NLS It has been observed & studied in optics (Mollenauer, 1986; Gordon, 1986) My opinion: No satisfactory model of the process has been found

24 Q: Stable patterns that propagate with (nearly) permanent form on 2-D surface in deep water? 1990s Joe Hammack built a new tank to study 2-D wave patterns (so 3-D fluid flows) on deep water

25 Experimental evidence of apparently stable wave patterns in deep water - ( 3 Hz frequency 4 Hz 17.3 cm wavelength 9.8 cm

26 How to reconcile the experimental observations with Benjamin-Feir instability? Options Modulational instability afflicts 1-D plane waves, but not 2-D periodic patterns The Penn State tank is too short to observe the (relatively slow) growth of the instability Other (please specify)

27 More experimental results ( 3 Hz 2 Hz old water new water

28 Main results The modulational (or Benjamin-Feir) instability is valid for waves in deep water without dissipation

29 Main results The modulational (or Benjamin-Feir) instability is valid for waves in deep water without dissipation But any amount of damping (of the right kind) stabilizes the instability (according to NLS & exp s) This dichotomy (with vs. without damping) applies to both 1-D plane waves and to 2-D periodic surface patterns Segur, Henderson, Carter, Hammack, Li, Pheiff, Socha, 2005 Controversial

30 Stability vs. existence in full water-wave equations Recall: Craig & Nicholls (2000) prove that the full equations of (inviscid) water waves, with gravity and surface tension, admit solutions with 2-D, periodic surface patterns of permanent form on deep water. Iooss & Plotnikov (2008) prove the existence of such patterns for (some) pure gravity waves on deep water. Neither paper considers stability.

31 Reconsider stability of plane waves in 1-D i(" t A + c g " x A) + #[$" x 2 A + % A 2 A ] = 0 [" = t # x c g, X = $ x c g ] i" X A + #" $ 2 A + % A 2 A = 0

32 Reconsider stability of plane waves in 1-D, with damping i(" t A + c g " x A) + #[$" x 2 A + % A 2 A +i"a] = 0 [" = t # x c g, X = $ x c g ] " # 0 i" X A + #" $ 2 A + % A 2 A +i"a = 0 [A(", X) = e #$X A (", X)] i" X A + #" $ 2 A + % & e '2(X A 2 A = 0

33 NLS in 1-D, cont d i" X A + #" $ 2 A + % & e '2(X A 2 A = 0 Hamiltonian equation, but dh dx " 0 H = i % [" # $ A 2 & ' 2 e&2(x A 4 ]d$ Conjugate variables: A, A*

34 i" X A + #" $ 2 A + % & e '2(X A 2 A = 0, cont d Uniform (in ξ) wave train: 1# A = A 0 exp{i" A 0 2 e#2$x ( 2$ Perturb: )} 1$ A (",X) = exp{i# A 0 2 e$2%x ( 2% algebra.. )}[ A 0 +µ(u + iv)]} + O(µ 2 ) d 2 ˆ u dx 2 + ["m 2 ("m 2 # 2$ % e #2&X A 0 2 )] % ˆ u = 0

35 a d 2 ˆ u dx 2 + ["m 2 ("m 2 # 2$ % e #2&X A 0 2 )] % ˆ u = 0 Hasegawa &Kodama (1995)

36 d 2 u ˆ dx + ["m 2 ("m 2 # 2$ % e #2&X A )] % u ˆ = 0, cont d

37 d 2 u ˆ dx + ["m 2 ("m 2 # 2$ % e #2&X A )] % u ˆ = 0 There is a growing mode if, cont d ["m 2 ("m 2 # 2$ % e #2&X A 0 2 )] < 0

38 d 2 u ˆ dx + ["m 2 ("m 2 # 2$ % e #2&X A )] % u ˆ = 0 There is a growing mode if, cont d ["m 2 ("m 2 # 2$ % e #2&X A 0 2 )] < 0

39 d 2 u ˆ dx + ["m 2 ("m 2 # 2$ % e #2&X A )] % u ˆ = 0 There is a growing mode if, cont d ["m 2 ("m 2 # 2$ % e #2&X A 0 2 )] < 0 For any δ > 0, growth stops eventually No mode grows forever Total growth is bounded

40 What is linearized stability? (Lyapunov) A uniform wave train solution is linearly stable if for every ε > 0 there is a Δ(ε) > 0 such that if a perturbation (u,v) satisfies # [u 2 (",0) + v 2 (",0)]d" < $(%) at X = 0, then necessarily # [u 2 (",X) + v 2 (", X)]d" < $ for all X > 0.

41 1-D NLS with damping, conclusion d 2 ˆ u dx 2 + ["m 2 ("m 2 # 2$ % e #2&X A 0 2 )] % ˆ u = 0 There is a universal bound, B: the total growth of any Fourier mode cannot exceed B To demonstrate stability, choose Δ(ε) so that "(#) < 1 B 2 $ # Nonlinear stability is similar, but more complicated

42 Experimental verification of theory (old) 1-D tank at Penn State

43 Experimental wave records X 1 X 8

44 Amplitudes of seeded sidebands (damping factored out of data) damped NLS theory Benjamin-Feir growth rate experimental data

45 Q: Are there stable wave patterns that propagate with permanent form (or nearly so) on deep water? A: YES, in the presence of (weak) damping Apparently NO, with no damping

46 Q: Stable wave patterns that propagate with nearly permanent form on deep water? A: YES, in the presence of (weak) damping Apparently NO, with no damping Q: Is this the final chapter of this story? A: Almost certainly not. Downshifting is still unexplained. Its physical importance is largely unexplored. More surprises?

47 Amplitudes of unseeded sidebands (damping factored out of data) damped NLS theory experimental data

48 Numerical simulations of full water wave equations, plus damping Wu, Liu & Yue 2006