Waves on deep water, I Lecture 13

Transcription

1 Waves on deep water, I Lecture 13 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 (I,II) This lecture (13): A. Sketch derivation of NLS for waves on deep water B. Earlier work: Waves with 1-D surface patterns on deep water - existence and stability Next lecture (14): C. More recent work: Stability of waves with 2-D surface patterns Effect of small damping

3 A. Sketch derivation of NLS NLS describes the slow evolution of a train (or packet) of dispersive waves: of small or moderate amplitude travelling in nearly the same direction with nearly the same frequency. shallow water Linearized dispersion curve c p dispersive waves kh

4 Sketch derivation of NLS 1. Simplest case: Gravity waves only (no surface tension) Deep water ( ) kh " #

5 Sketch derivation of NLS 1. Simplest case: Gravity waves only (no surface tension) Deep water ( kh " #) 2. Small parameters (ε) Nearly monochromatic: Small amplitude: k 0 " = O(#)! k = (k 0,0) + O("), # = gk 0 + O("),

6 Sketch derivation of NLS 1. Simplest case: Gravity waves only (no surface tension) Deep water ( kh " #) 2. Small parameters (ε) Nearly monochromatic: Small amplitude: k 0 " = O(#)! k = (k 0,0) + O("), # = gk 0 + O("), 3. Look for: " = k 0 x #$(k 0 )t, "(x, y,t;#) = #[A(#x,#y,#t,# 2 t)e i$ + A * e %i$ ] +# 2 [stuff 2 ] + # 3 [stuff 3 ] + O(# 4 )

7 Sketch derivation of NLS 4. Insert formal expansions for η(x,y,t;ε), φ(x,y,z,t;ε) into full equations. Solve, order by order. 5. Algebra is fearsome. (Use maple or )

8 Sketch derivation of NLS 4. Insert formal expansions for η(x,y,t;ε), φ(x,y,z,t;ε) into full equations. Solve, order by order. 5. Algebra is fearsome. (Use maple or ) 6. Find: At O(ε): ω 2 = gk (linearized dispersion relation) At O(ε 2 ): expansion becomes disordered unless "A "(#t) + c g "A "(#x) = 0, (c g = group velocity) (Wave envelope travels at group velocity)

9 Sketch derivation of NLS 7. Recall " = #[A(#x,#y,#t,# 2 t)e i$ + A * e %i$ ] + O(# 2 ) Define: " = (#x) $ c g (#t), % = #y, & = # 2 t. 8. At O(ε 3 ), expansion becomes disordered (again) unless A(ξ,ζ,τ) satisfies i" # A + $" % 2 A + &" ' 2 A + ( A 2 A = 0 Nonlinear Schrödinger equation in 2-D (α, β, γ) are real numbers, defined by problem

10 B. History of NLS i" # A + $" % 2 A + &" ' 2 A + ( A 2 A = 0 This equation (or something equivalent) was derived by: Zakharov, 1968 water waves Ostrovsky, 1967 optics (& generalizations) Benjamin & Feir, 1967 water waves (nearby) Benney & Newell, 1967 general Whitham, 1967 his formulation was used by: Lighthill, 1965 had the basic idea Stokes, 1847 water waves (no spatial dependence) Zakharov & Ostrovsky, 2008 historical review

11 Q: Waves of permanent form on deep water? 1. Stokes (1847): Consider a spatially uniform train of plane waves "(x,t;#) = #[A(# 2 t)e i$ + A * e %i$ ] + O(# 2 ) i" # A + $ A 2 A = 0, $ = %4k 0 2

12 Q: Waves of permanent form on deep water? 1. Stokes (1847): Consider a spatially uniform train of plane waves "(x,t;#) = #[A(# 2 t)e i$ + A * e %i$ ] + O(# 2 ) i" # A + $ A 2 A = 0, $ = %4k 0 2 A(") = (A 0 e i# )e i$ A 0 2 "

13 Q: Waves of permanent form on deep water? 1. Stokes (1847): Consider a spatially uniform train of plane waves "(x,t;#) = #[A(# 2 t)e i$ + A * e %i$ ] + O(# 2 ) i" # A + $ A 2 A = 0, $ = %4k 0 2 A(") = (A 0 e i# )e i$ A 0 2 " "(x,t;#) = 2# A 0 cos{k 0 x $%(k 0 )t $ (2#k 0 A 0 ) 2 t} + O(# 2 ) Stokes nonlinear correction to frequency

14 Q: Do waves of permanent form exist on deep water? * Stokes (1847) found a nonlinear correction for water waves of permanent form, with finite amplitude. He did not prove that waves of permanent form exist.

15 Q: Do waves of permanent form exist on deep water? * Stokes (1847) found a nonlinear correction for water waves of permanent form, with finite amplitude. He did not prove that waves of permanent form exist. Nekrassov (1921) and Levi-Civita (1925) proved that such waves exist on deep water. Struik (1926) extended their result to water of any (constant) depth. Amick & Toland (1981a,b) obtained optimal results about existence of waves of permanent form in 2-D.

16 Q: Why don t we see uniform wave trains on deep water? Loch Ness Lake Superior

17 Are plane waves stable on deep water? Photos from Benjamin, 1967 (L = 2.3 m, h = 7.6 m, 60 m between photos) Benjamin & Feir: On the disintegration of wavetrains in deep water, Part 1, 1967

18 Stability of plane waves on deep water Zakharov (1968): NLS i" # A + $" % 2 A + &" ' 2 A + ( A 2 A = 0 For gravity waves on deep water, α < 0, β > 0, γ < 0 A(",#,$) = A 0 e i% A 0 2 $ Stokes wave, represents a uniform train of plane waves with finite amplitude in deep water Linearize NLS about a Stokes wave, and determine its linear stability in NLS

19 Linear stability of a Stokes wave NLS: i" # A + $" % 2 A + &" ' 2 A + ( A 2 A = 0 Linearize about a Stokes wave: A(",#,$) = e i% A 0 2 $ [ A 0 +µ & u(",#,$) + iµ & v(",#,$)] + O(µ 2 )

20 Linear stability of a Stokes wave NLS: i" # A + $" % 2 A + &" ' 2 A + ( A 2 A = 0 Linearize about a Stokes wave: A(",#,$) = e i% A 0 2 $ [ A 0 +µ & u(",#,$) + iµ & v(",#,$)] + O(µ 2 ) " # v = $" % 2 u + &" ' 2 u + 2( A 0 2 u, )" # u = $" % 2 v + &" ' 2 v.

21 Linear stability of a Stokes wave NLS: i" # A + $" % 2 A + &" ' 2 A + ( A 2 A = 0 Linearize about a Stokes wave: A(",#,$) = e i% A 0 2 $ [ A 0 +µ & u(",#,$) + iµ & v(",#,$)] + O(µ 2 ) " # v = $" % 2 u + &" ' 2 u + 2( A 0 2 u, )" # u = $" % 2 v + &" ' 2 v. Linear PDEs, constant coefficients Seek u = U " e im# +il$ +%& + (c.c.), v = V " e im# +il$ +%& + (c.c.).

22 Linear stability of a Stokes wave Algebraic equation determines linear stability: " 2 + (#m 2 + $l 2 )(#m 2 + $l 2 % 2& A 0 2 ) = 0 Re(Ω) > 0 linear instability

23 Linear stability of a Stokes wave Algebraic equation determines linear stability: " 2 + (#m 2 + $l 2 )(#m 2 + $l 2 % 2& A 0 2 ) = 0 Re(Ω) > 0 linear instability Unstable regions

24 Linear stability of a Stokes wave Result: In deep water, a uniform train of plane waves of finite amplitude is unstable! The most unstable mode has a growth rate: Ω max = γ Α 0 2. (Nonlinear instability in the sense that the growth rate depends on A 0. As A 0 " 0, # max " 0. )

25 Stability of a uniform wave train, according to NLS i" # A + $" 2 % A + &" 2 ' A + ( A 2 A = 0 For other applications: αβ < 0, αγ > 0 unstable αβ < 0, αγ < 0 unstable αβ > 0, αγ > 0 unstable αβ > 0, αγ < 0 stable

26 Instability of a uniform train of water waves -Benjamin & Feir L = 2.3 m, T = 1.2 s

27 Instability of a uniform train of EM waves in optical fiber -Tai, Hasegawa & Tomita (1986) L = 1.3*10-6 m, T = 4*10-15 s

28 Q: Are there stable wave patterns of permanent form on deep water? A#1 (Zakharov & others, about 1967) A uniform wave train is not stable

29 Q: Are there stable wave patterns of permanent form on deep water? A#1 (Zakharov & others, about 1967) A uniform wave train is not stable A#2 (Zakharov & Shabat, 1972) Consider NLS in 1-D i" # A = " $ 2 A + 2% A 2 A σ = +1 focussing (gravity waves on deep water) σ = -1 defocussing

30 Focussing NLS, in 1-D i" # A = " $ 2 A + 2 A 2 A Look for travelling waves of permanent form. Special case: A(",#) = 2a $ e %i(2a )2# sec h{2a(" + " 0 )} How does the free surface look? "(x,t;#) = #[Ae i$ + A * e %i$ ] + O(# 2 ) " = (2#a)sec h{(2#a)(x % c g t)}cos{kx %[&(k) + (2#a) 2 ]t} wave packet with special shape

31 i" # A = " $ 2 A + 2 A 2 A "(x,t;#) = 2(2#a)sec h{(2#a)(x $ c g t)}cos{kx $[%(k) + (2#a) 2 ]t}

32 i" # A = " $ 2 A + 2 A 2 A "(x,t;#) = 2(2#a)sec h{(2#a)(x $ c g t)}cos{kx $[%(k) + (2#a) 2 ]t} Nice, but so what?

33 i" # A = " $ 2 A + 2% A 2 A Zakharov & Shabat (1972): NLS in 1-D is completely integrable, for either sign of σ! Just like KdV. Conservation laws i" # ( A 2 ) = " $ (A * " $ A % A" $ A * ), i" # (A * " $ A % A" $ A * ) = " $ (...), i" # ( " $ A 2 +& A 4 ) = " $ (...), :

34 i" # A = " $ 2 A + 2% A 2 A Zakharov & Shabat (1972): Scattering problem: Time-dependence: Compatibility: " # v 1 = $i%v 1 + Av 2, " # v 2 = $&A * v 1 + i%v 2, " # v 1 = [2i$ 2 + i% A 2 ]v 1 + [2A$ + i" & A]v 2, " # v 2 = ['2%A * $ + i%" & A * ]v 1 '[2i$ 2 + i% A 2 ]v 2, % v 1 ( % v 1 ( " # " $ ' * = " $ " # ' * " & ) & ) v 2 v 2 i" # A = " $ 2 A + 2% A 2 A

35 i" # A = " $ 2 A + 2% A 2 A Zakharov & Shabat (1972): For σ = +1 (focussing NLS): Any smooth initial data, A(ξ,0), with A d" < # evolves under focussing NLS into a N envelope solitons, which persist forever, plus an oscillatory wavetrain that decays in amplitude as " # $ Envelope solitons are stable, within NLS in 1-D. $

36 Stability of envelope solitons - experimental evidence a) 6 m from wavemaker b) 30 m from Wavemaker k = 4 m -1 - Hammack

37 Stability of envelope solitons - 3 experiments by Yuen & Lake (1975)

38 Tentative conclusions According to NLS in 1-D or 2-D, a uniform train of plane waves is unstable in deep water. According to focussing NLS in 1-D with initial data in L 1, envelope solitons are stable in deep water. Experimental evidence seems to support these conclusions

39 Tentative conclusions According to NLS in 1-D or 2-D, a uniform train of plane waves is unstable in deep water. According to focussing NLS in 1-D with initial data in L 1, envelope solitons are stable in deep water. Experimental evidence seems to support these conclusions But wait for Lecture 14