Transverse dynamics of gravity-capillary periodic water waves

Transcription

1 of gravity-capillary periodic water waves Mariana Haragus LMB, Université de Franche-Comté, France IMA Workshop Dynamical Systems in Studies of Partial Differential Equations September 24-28, 2012

2 Water waves Water-wave problem Two-dimensional waves

3 Water waves Water-wave problem Two-dimensional waves

4 Water waves Water-wave problem Two-dimensional waves

5 Water-wave problem Water waves Water-wave problem Two-dimensional waves

6 Water-wave problem Two-dimensional waves Water-wave problem gravity-capillary water waves three-dimensional inviscid fluid layer constant density ρ gravity and surface tension irrotational flow

7 Water-wave problem Two-dimensional waves Water-wave problem y z x gravity-capillary water waves three-dimensional inviscid fluid layer constant density ρ gravity and surface tension irrotational flow

8 Water-wave problem Water waves Water-wave problem Two-dimensional waves y = h + η(x,z,t) (free surface) y x z y = 0 (flat bottom) Domain D η = {(x,y,z) : x,z R, y (0,h + η(x,z,t))} depth at rest h

9 Water-wave problem Water waves Water-wave problem Two-dimensional waves y = h + η(x,z,t) (free surface) y x z y = 0 (flat bottom) Domain D η = {(x,y,z) : x,z R, y (0,h + η(x,z,t))} depth at rest h

10 Euler equations Water waves Water-wave problem Two-dimensional waves Laplace s equation φ xx + φ yy + φ zz = 0 in D η boundary conditions φ y = 0 on y = 0 η t = φ y η x φ x η z φ z on y = h + η φ t = 1 2 (φ2 x + φ2 y + φ2 z ) gη + σ ρ K on y = h + η

11 Euler equations Water waves Water-wave problem Two-dimensional waves Laplace s equation φ xx + φ yy + φ zz = 0 in D η boundary conditions φ y = 0 on y = 0 η t = φ y η x φ x η z φ z on y = h + η φ t = 1 2 (φ2 x + φ2 y + φ2 z ) gη + σ ρ K on y = h + η velocity potential φ; [ free surface ] [ h + η ] η mean curvature K = x η + z parameters ρ, g, σ, h 1+η 2 x +η 2 z 1+η 2 x x +ηz 2 z

12 Water-wave problem Two-dimensional waves Euler equations moving coordinate system, speed c dimensionless variables characteristic length h characteristic velocity c

13 Euler equations Water waves Water-wave problem Two-dimensional waves moving coordinate system, speed c dimensionless variables characteristic length h characteristic velocity c parameters inverse square of the Froude number α = gh Weber number β = σ ρhc 2 c 2

14 Euler equations Water waves Water-wave problem Two-dimensional waves φ xx + φ yy + φ zz = 0 for 0 < y < 1 + η φ y = 0 on y = 0 φ y = η t + η x + η x φ x + η z φ z on y = 1 + η ( ) φ t + φ x + 1 φ 2 2 x + φ2 y + φ2 z + αη βk = 0 on y = 1 + η η x K = 1 + ηx 2 + η2 z x η z ηx 2 + η2 z z

15 Water-wave problem Two-dimensional waves Euler equations very rich dynamics difficulties variable domain (free surface) nonlinear boundary conditions symmetries, Hamiltonian structure many particular solutions

16 Two-dimensional traveling waves Water-wave problem Two-dimensional waves periodic wave solitary waves generalized solitary waves solitary waves [Nekrasov, Levi-Civita, Struik, Lavrentiev, Friedrichs & Hyers,... Amick, Kirchgässner, Iooss, Buffoni, Groves, Toland, Lombardi, Sun,...]

17 Three-dimensional traveling waves Water-wave problem Two-dimensional waves [Groves, Mielke, Craig, Nicholls, H., Kirchgässner, Deng, Sun, Sandstede, Iooss, Plotnikov, Wahlen,...]

18 Solitary wave Water waves Water-wave problem Two-dimensional waves

19 Periodic waves Water waves Water-wave problem Two-dimensional waves

20 Water-wave problem Two-dimensional waves Questions Existence two- and three-dimensional waves Dynamics 2D stability 3D stability new solutions (bifurcations) (Numerical results; Model equations; Cauchy problem;...)

21 Water-wave problem Two-dimensional waves Dynamics of solitary waves capillary-gravity waves β > 1 3 2D stability 3D instability (linear and nonlinear) bifurcations : dimension-breaking [H. & Scheel; Mielke; Groves, H. & Sun; Pego & Sun; Rousset & Tzvetkov]

22 Water-wave problem Two-dimensional waves Dynamics of solitary waves capillary-gravity waves β > 1 3 2D stability 3D instability (linear and nonlinear) bifurcations : dimension-breaking [H. & Scheel; Mielke; Groves, H. & Sun; Pego & Sun; Rousset & Tzvetkov] capillary-gravity waves 0 < β < 1 3 2D stability 3D instability (linear) bifurcations : dimension-breaking [Buffoni; Groves & Wahlen; Groves, Wahlen & Sun]

23 Water-wave problem Two-dimensional waves Dynamics of solitary waves capillary-gravity waves β > 1 3 2D stability 3D instability (linear and nonlinear) bifurcations : dimension-breaking [H. & Scheel; Mielke; Groves, H. & Sun; Pego & Sun; Rousset & Tzvetkov] capillary-gravity waves 0 < β < 1 3 2D stability 3D instability (linear) bifurcations : dimension-breaking [Buffoni; Groves & Wahlen; Groves, Wahlen & Sun] gravity waves β = 0 2D stability [Pego & Sun]

24 Dynamics of periodic waves Water-wave problem Two-dimensional waves gravity waves β = 0 Benjamin-Feir instability [Bridges & Mielke]

25 Water-wave problem Two-dimensional waves Dynamics of periodic waves gravity waves β = 0 Benjamin-Feir instability [Bridges & Mielke] gravity-capillary waves β > 1 3 3D instability (linear) bifurcations : dimension-breaking (see [Groves, H. & Sun, 2001, 2002])

26 Water-wave problem Two-dimensional waves Predictions : model equations gravity-capillary waves β > 1 3 2D stability : Korteweg-de Vries equation [Angulo, Bona & Scialom; Bottman & Deconinck; Deconinck & Kapitula] 3D instability : Kadomtsev-Petviashvili-I equation [H.; Johnson & Zumbrun; Hakkaev, Stanislavova & Stefanov]

27 Euler equations Water waves Spatial dynamics 2D periodic waves φ xx + φ yy + φ zz = 0 for 0 < y < 1 + η φ y = 0 on y = 0 φ y = η t + η x + η x φ x + η z φ z on y = 1 + η ( ) φ t + φ x + 1 φ 2 2 x + φ2 y + φ2 z + αη βk = 0 on y = 1 + η η x K = 1 + ηx 2 + η2 z x η z ηx 2 + η2 z z parameters : β > 1 3, α 1

28 Spatial dynamics 2D periodic waves Questions 3D instability bifurcations : new solutions

29 Spatial dynamics 2D periodic waves Spatial dynamics : Hamiltonian formulation time-like variable z [Kirchgässner, 1982] fixed domain R (0,1) : variable y = y/(1+η)

30 Spatial dynamics 2D periodic waves Spatial dynamics : Hamiltonian formulation time-like variable z [Kirchgässner, 1982] fixed domain R (0,1) : variable y = y/(1+η) Hamiltonian H(η, ω, φ, ξ) [Groves, H. & Sun, 2001] T H(η,ω,φ,ξ) = { 12 } 1 αη2 + β (β 2 W 2 ) 1/2 (1 + η 2x yφ yξ )1/2 dx dt W = ω + T R η dy T 1 { + (η t + η x)yφ y (1 + η)(φ t + φ x) 1 + η T 0 R 2 space X s,δ, s (0,1/2), δ > 1/2 ( φ x yηxφy 1 + η ) 2 + ξ2 φ 2 } y dx dy dt 2(1 + η) X s,δ = H s+1 δ (0,L) Hδ s (0,L) Hs+1 δ ((0,L) (0,1)) Hδ s ((0,L) (0,1)) { H s δ (0,L) = u = u m(x)e imπt/t u m H s (0,L), u 2 s,δ = } + m m Z m Z(1 2 ) 2δ u m 2 s

31 Hamiltonian system Water waves Spatial dynamics 2D periodic waves Hamilton s equations u = (η,ω,φ,ξ) u z = Du t + F(u)

32 Hamiltonian system Water waves Spatial dynamics 2D periodic waves Hamilton s equations u = (η,ω,φ,ξ) u z = Du t + F(u) Du = (0,φ y=1,0,0), F(u) = (f 1 (u),f 2 (u),f 3 (u),f 4 (u)) ( ) 1 + η 2 1/2 1 f 1 (u) = W x yφ yξ β 2 W 2, W = ω η dy f 2 (u) = { 1 ξ 2 φ 2 y 0 2(1 + η) ( ) β 2 W 2 1/2 + αη η x + W 1 + ηx 2 x (1 + η) 2 f 3 (u) = ξ 1 + η + yφyw ( ) 1 + η 2 1/2 x 1 + η β 2 W 2 ( φ x + yφyηx )( φ x yφyηx ) ( + [yφ y φ x yφyηx )] 1 + η 1 + η 1 + η x ( ) 1 + η 2 1/2 1 x β 2 W 2 yφ yξdy + φ y=1 x 0 f 4 (u) = φyy 1 + η [ [ ( ] (1 + η)φ x yη xφ y x + yη x φ x yφyηx )] + (yξ)yw 1 + η y 1 + η } dy ( ) 1 + η 2 1/2 x β 2 W 2

33 Hamiltonian system Water waves Spatial dynamics 2D periodic waves Hamilton s equations u = (η,ω,φ,ξ) u z = Du t + F(u) Du = (0,φ y=1,0,0), F(u) = (f 1 (u),f 2 (u),f 3 (u),f 4 (u)) boundary conditions φ y = b(u) t + g(u) on y = 0,1 b(u) = yη, g(u) = y(1+η)(1+φ x )η x yη 2 x Φ y +yξw ( ) 1+η 2 1/2 x β 2 W 2

34 Spatial dynamics 2D periodic waves 2D periodic waves parameters α = 1 + ε, β > 1/3 [Kirchgässner, 1988] family of 2D periodic waves, ε small η (x) = εη KdV (ε 1/2 x) + O(ε 2 ) φ (x,y) = ε 1/2 φ KdV (ε 1/2 x) + O(ε 3/2 ) η KdV solution of KdV : ( β 1 ) η = η η2 φ KdV = η KdV

35 Periodic solutions of KdV Spatial dynamics 2D periodic waves ( β 1 ) η = η η2 family of periodic waves η KdV (X) = P a (k a X), a I R P a even function, 2π periodic

36 2D periodic waves Water waves Spatial dynamics 2D periodic waves scaling x = k a x, η = ε η, φ = ε 1/2 φ, ω = ε ω, ξ = ε 1/2 ξ Hamilton s equations u = ( η, ω, φ, ξ) boundary conditions u z = D ε u t + F ε (u) φ y = b ε (u) t + g ε (u) on y = 0,1

37 2D periodic waves Water waves Spatial dynamics 2D periodic waves scaling x = k a x, η = ε η, φ = ε 1/2 φ, ω = ε ω, ξ = ε 1/2 ξ Hamilton s equations u = ( η, ω, φ, ξ) boundary conditions u z = D ε u t + F ε (u) φ y = b ε (u) t + g ε (u) on y = 0,1 equilibria (F ε (u a ) = 0) Q a = u a = (η a,0,φ a,0) = (P a,0,q a,0) + O(ε) x P a(ζ)dζ 0

38 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Questions 3D instability bifurcations : new solutions analysis of the linearized problem

39 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Linear operator linearized system boundary conditions u z = D ε u t + DF ε (u a )u φ y =Db ε (u a )u t + Dg ε (u a )u on y = 0,1

40 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Linear operator linearized system boundary conditions u z = D ε u t + DF ε (u a )u φ y =Db ε (u a )u t + Dg ε (u a )u on y = 0,1 linear operator L ε := DF ε (u a ) boundary conditions φ y = Dg ε (u a )u on y = 0,1 space of symmetric functions (x x) X s = H 1 e (0,2π) L2 e (0,2π) H1 o ((0,2π) (0,1)) L2 o ((0,2π) (0,1))

41 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Linear operator L ε = L 0 ε + L1 ε ω η L 0 ω β ε φ = εk 2 a βηxx + (1 + ǫ)η kaφx y=1, L 1 ε ξ ξ εk 2 aφxx φyy η ω φ ξ g 1 = g 2 G 1 G 2 g 1 = (1 + εk2 a η2 ax )1/2 β { 1 g 2 = 0 [ + G 1 = εηaξ G 2 = ( ω εη a εk 2 φayφy aφaxφx 1 (1 + εη a) 2 + εφ 2 0 ) yφ ayξ dy ω β εk 2 a yφayφx + εk2 a yφaxφy 2ε2 k 2 a y2 η axφ ayφ y 1 + εη a ε + εk 2 a βηxx εk2 a β [ ay η (1 + εη a) 3 ε 3 k 2 a y 2 ηax 2 φayφy (1 + εη a) 2 ε 3 k 2 a y 2 η axφ 2 ay ηx (1 + εη a) 2 + ε 3 k 2 a y 2 ηax 2 φ2 ay η (1 + εη a) 3 2 k 2 a y 2 φ 2 ay ηx + ε 3 k 2 a y 2 η axφ 2 ay η ] } 1 + εη a (1 + εη a) 2 dy x ] η x (1 + ε 3 k 2 a η2 ax )3/2 + (1 + ε 3 k 2 a ηax 2 )1/2 1 + εη a β(1 + εη a) ] [ εηaφ (1 + εη a) + εφaη (1 + η a) 2 ( ω + x εη a 1 0 ) yφ ayξdy yφ ay ε 2 k 2 a [ηaφx + φaxη yφayηx yηaxφy]x yy [ + ε 2 k 2 a yη axφ x + yφ axη x + ε2 y 2 ηax 2 φayη (1 + εη a) 2 εy 2 η 2 ax φ y 2εy ] 2 ηaxφayη x 1 + εη a 1 + εη a y

42 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Spectrum of L ε Theorem The linear operator L ε has the following properties (ε small) : pure point spectrum spec(l ε ); spec(l ε ) ir = { iεk ε,iεk ε } ; ±iεk ε are simple eigenvalues; resolvent estimate (L ε iλi) 1 c, λ λ λ

43 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof operator with compact resolvent spectral analysis pure point spectrum λ λ λ λ λ εl

44 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof STEP I : λ λ no eigenvalues L ε small relatively bounded perturbation of L 0 ε operator with constant coefficients L 0 ε a priori estimates

45 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof STEP II : λ λ reduction to a scalar operator B ε,l in L 2 o (0,2π) scaling λ = εl, ω = ε ω, ξ = ε ξ decomposition φ(x,y) = φ 1 (x)+φ 2 (x,y)

46 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof STEP II : λ λ reduction to a scalar operator B ε,l in L 2 o (0,2π) scaling λ = εl, ω = ε ω, ξ = ε ξ decomposition φ(x,y) = φ 1 (x)+φ 2 (x,y) B ε,l φ 1 = λ = εl eigenvalue iff B ε,l φ 1 = 0 ( β 1 ) k 4 a 3 φ 1xxxx k 2 a φ 1xx + l 2 (1 + ǫ)φ 1 3k 2 a (P aφ 1x ) x +...

47 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking... β 1 1 ω = (1 + ǫ 3 ηx 2 (η + ikη) )1/2 1 + ǫη yφ y ξdy, 0 ξ = (1 + ǫη )(Φ + ikφ) ǫyφ y (η + ikη) (1 + ǫ) ǫ 2 η 1 ǫ 2 Φx y=1 1 ǫ βηxx ikβ(hǫ 1 + ikη) = hǫ 2 B ǫ 0 = B ǫ (η,φ) = ǫη x + B ǫ 0 + Bǫ 1, ǫη Φ y 1 + ǫη + ǫφy η (1 + ǫη ) 2, y=1 B ǫ 1 = ǫ 2 η x Φx + ǫ2 Φ x ηx + ǫ4 ηx 2 Φ y η (1 + ǫη ) 2 ǫ 3 η 2 x Φy 1 + ǫη 1 ǫ Φxx 1 ǫ 2 Φyy ik(hǫ 1 + ikφ) = Hǫ 2, h ǫ 2 = ω g ǫ 2, ˆΦ yy + q 2ˆΦ = ǫ 2 (Ĥ2 ǫ + ikĥǫ 1 ), 0 < y < 1 H ǫ 2 = ξ G ǫ ˆΦ y = 0, y = 0 2 h1 ǫ = ω β ikη ǫµ ˆΦ 2ˆΦ 3 iµ(ĥ y 1 + ǫ + βq 2 = ǫ 2 ǫ + ikβĥǫ 1 ) 1 + ǫ + βq 2 + ˆB ǫ 0 + ˆB ǫ 1, y = = β(1 + ǫη yφ y ) [ ǫyφ y (ikη + η ) + (1 + ǫη )(ikφ + Φ )]dy 0 ( ) cosh qy (1 + ǫ + βq 2 )coshq(1 ζ) + (ǫµ 2 /q) 1 + (1 + ǫ 3 ηx 2 1 η ikη + )1/2 (1 + ǫ 3 ηx 2, coshq q 2 (1 + ǫ + βq 2 )qtanhq ǫ )1/2 G(y,ζ) = H ǫ 1 = ξ ikφ cosh qζ (1 + ǫ + βq 2 )coshq(1 y) + (ǫµ 2 /q) = (1 + ǫη )Φ + ikǫη Φ ǫyφ y (η coshq q + ikη). 2 (1 + ǫ + βq 2 )qtanhq

48 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking... ˆΦ 1 = { 1 + ǫ 1 1 ǫ 2 (k 2 (1 + ǫ) + µ 2 + (β 1/3)µ 4 ) ǫ 2 (ˆξ iµĝ2,2 ǫ + ikĥǫ 1 )dζ ǫq2 ˆp 2 ǫ dζ 0 0 } ǫ3 iµ(ĥ2 ǫ + ikβĥǫ 1 ) 1 + ǫ + βq 2 + ǫ µ 2ˆp 2 ǫ ζ=1 1 + ǫ + βq 2, ˆΦ = G 1 (ˆξ iµĝ ǫ 2,2 + ikĥǫ 1 )dζ G 1ζ Ĝ ǫ 2,1 dζ + (ǫk 2 + µ 2 )G 1ˆp ǫ 2 dζ + ǫˆpǫ ( G 1 ζ=1 ǫiµ(ĥǫ 2 + ikβĥǫ 1 ) + µ 2 ǫ ) ζ=1, 1 + ǫ + βq ǫ + βq ˆΦ = Gǫ 2 (ˆξ ıµĝ ǫ 2,2 + ıkĥǫ 1 )dζ G ζ ǫ 2 Ĝ ǫ 2,1 dζ + ǫ3 ıµg ζ=1 (ĥ2 ǫ + ıkβĥǫ 1 ) ǫ + βq 2 + ǫq 2 Gˆp ǫ 2 dζ + ǫˆpǫ 2 ǫ2 µ ǫq 2 Gˆp ǫ 2 dζ + ǫˆpǫ 2 ǫ2 µ 2 G ζ=1ˆp 2 ǫ ζ= ǫ + βq = Gǫˆp 2ζζ ǫ dζ ǫg ζ=1ˆp 2ζ ǫ ζ=1 = Gǫ 2 (Ĝ2,0 ǫ ) ζζdζ G ζ=1ˆb 0 ǫ, ˆΦ 1 + ˆΦ 2 = Gǫ 2 (ˆξ ıµĝ ǫ 2,2 + ıkĥǫ 1 )dζ G ζ ǫ 2 Ĝ ǫ 2,1 dζ ǫ3 ıµg ζ=1 (ĥ2 ǫ + ıkβĥǫ 1 ) 1 ǫq 2 Gˆp ǫ 2 dζ + ǫˆpǫ 2 ǫ2 µ 2 G ζ=1ˆp 2 ǫ ζ=1 1 + ǫ + βq ǫ + βq 2,

49 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking... ıµĥ ǫ 2 = [ ıµˆω + ıµf 1 { 1 ǫ 2 ǫφ x Φx Φ y Φy 0 (1 + ǫη ) 2 + ǫφ 2 y η (1 + ǫη ) 3 ǫ 3 y 2 η 2 x Φ y Φy (1 + ǫη ) 2 ǫ 3 y 2 η x Φ 2 y ηx (1 + ǫη ) 2 + ǫ 4 y (1 [ { + µ2 1 ǫ F yφ y Φx + yφ x Φy 2ǫy2 ηx ΦyΦ y ǫη ǫy 2 Φ 2 y ηx 1 + ǫη + ǫ 2 y 2 η x Φ 2 y η } ] [ (1 + ǫη ) 2 dy βµ2 ǫ F (1 + [ F 1 ǫıµĥ2 ǫ ] [ ] [ 1 + ǫ + βq 2 = F ǫ + βq 2 F[(Φ 1x Φ 1x) x] + F 1 µ 2 [ 1 ] ] 1 + ǫ + βq 2 F yφ x Φ 2ydy 0 [ [ ( + {F ǫ + βq 2 F Φ 2x Φ 1x + Φ x Φ Φ y 2x 2 0 ǫ(1 + ǫη ) 2 + Φy (1 + ǫη ) 3 ǫ2 y 2 ηx 2 y Φy (1 + ǫη ) 2 ǫ 2 y 2 η x Φ 2 y ηx (1 + ǫη ) 2 + ǫ 3 y 2 η x Φ 2 y η ) ] (1 + ǫη ) 3 dy [ ( ıµ ǫ + βq 2 F yφ y Φx 2ǫy2 ηx Φ y Φy ǫη ǫy 2 Φ 2 y ηx 1 + ǫη + ǫ 2 y 2 η x Φ 2 y η ) ] (1 + ǫη ) 2 dy [ ]]} [ βıµ ǫ + βq 2 F η x (1 + ǫ 3 ηx 2 ηx + F 1 ǫıµˆω ] )3/2 1 + ǫ + βq x [ ] [ = F ǫ + βq 2 F[(Φ 1x Φ 1x) x] + F 1 µ 2 [ 1 ] ] 1 + ǫ + βq 2 F yφ x Φ 2ydy 0 + (L(ǫΦ 1x,Φ 2x,Φ 2y,ǫ 2 η,ǫ 4 η x)) x + ǫ 1/2 (L(ǫΦ x,ǫ 2 Φ 2y,ǫ 4 η,ǫ 3 η x)) x + ǫ 1/2 L(ω ),

50 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking... = [ F 1 µ 2 [ 1 ] ] [ 1 + ǫ + βq 2 F yφ x Φ 2ydy = F 1 µ 2 [ ǫ + βq 2 F Φ 1x Φ 2 y=1 Φ 1x Φ 2dy + yφ 2x Φ 2y 0 0 [ [F 1 µ 1/2 1 + ǫ + βq 2 µ1/2 F[Φ 1x Φ y=1 ] 1 + ǫ + βq 2 (Φ 1x Φ µ 1 ]] 2) xdy ǫ + βq 2 yφ 2x Φ 2ydy 0 x = ǫ 1/4 (L(Φ 2 )) x + (L(Φ 2,Φ 2x,ǫ 1/2 Φ 2y )) x, [ F 1 ǫıµĥ2 ǫ ] [ ] 1 + ǫ + βq 2 = F ǫ + βq 2 F[(Φ 1x Φ 1x) x] + ǫ 1/4 (L(Φ 2 )) x + ǫ 1/2 (L(ǫΦ x,ǫ 2 Φ 2y,ǫ 4 η,ǫ 3 η x) x + (L(ǫΦ 1x,Φ 2,Φ 2x,Φ 2y,ǫ 2 η,ǫ 4 η x) x + H. [ F 1 ǫıµ.ıkĥ 1 ǫ ] [ ] 1 + ǫ + βq 2 = (L(Φ 2,ǫ 2 η)) x + ǫ 2 k 2 (L(Φ 1 )) x + H, F 1 µ ǫ 2ˆp 2 ζ=1 1 + ǫ + βq 2 = ǫ 1/4 (L(Φ 2,ǫη)) 1 ] F [(ǫk µ 2 ) ˆp ǫ 2 dζ = k 2 L(ǫΦ 2,ǫ 2 η) + (L(Φ 2,Φ 2x,ǫη,ǫη x)) x 0 1 (ξ (G ǫ 2,2 )x + ıkhǫ 1 )dζ = (η Φ 1x ) x + (Φ 1x η)x + (L(Φ 2x,Φ 2y,ǫη,ǫη x)) x + 0 [ ] (β 1/3)Φ 1xxxx Φ 1xx + k 2 (1 + ǫ)φ 1 = (η Φ 1x ) x + (Φ 1x η)x + 1 F ǫ + βq 2 F[(Φ 1x Φ 1x) x] + (L(ǫ 1/2 Φ 1x [,ǫ 1/4 Φ 2,Φ] 2x,Φ 2y,ǫ 3/4 η,ǫη x)) x + k 2 [L(ǫΦ 1,ǫΦ 2,ǫ 2 η) + ǫ 2 L(Φ 1 ) x] + H, η = F 1 ıµˆφ ǫ + βq 2 + L(ǫΦ 1x,ǫ 3/4 Φ 2,Φ 2x,Φ 2y,ǫ 3 η,ǫ 7/2 η x) + k 2 ǫ 3 L(Φ 1 ) + H.

51 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof STEP II a : εl λ λ no eigenvalues B ε,l small relatively bounded perturbation of ( C ε,l = β 1 ) k 4 a 3 φ 1xxxx k 2 a φ 1xx + l 2 (1 + ǫ)φ 1 B ε,l selfadjoint operator with constant coefficients a priori estimates

52 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof STEP II b : λ εl two simple eigenvalues ±iεκ ε B ε,l small relatively bounded perturbation of B 0,l ( B 0,l = k 2 a xa x + l 2 A = β 1 ) k 2 a 3 xx 1 3P a

53 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof STEP II b : λ εl two simple eigenvalues ±iεκ ε B ε,l small relatively bounded perturbation of B 0,l ( B 0,l = k 2 a xa x + l 2 A = β 1 ) k 2 a 3 xx 1 3P a spectrum of A is known (KdV!) x A x : one simple negative eigenvalue ω 2 a perturbation arguments...

54 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Spectrum of L ε Theorem The linear operator L ε has the following properties (ε small) : pure point spectrum spec(l ε ); spec(l ε ) ir = { iεk ε,iεk ε } ; ±iεk ε are simple eigenvalues; resolvent estimate (L ε iλi) 1 c, λ λ λ

55 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Transverse linear instability linearized system boundary conditions u z = D ε u t + DF ε (u a )u φ y =Db ε (u a )u t + Dg ε (u a )u on y = 0,1

56 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Transverse linear instability linearized system boundary conditions u z = D ε u t + DF ε (u a )u φ y =Db ε (u a )u t + Dg ε (u a )u on y = 0,1 Definition The periodic wave u a is linearly unstable if the linearized system possesses a solution u(t,x,y,z) = e λt v λ (x,y,z) with λ C, Reλ > 0, v λ bounded function.

57 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Transverse linear instability bounded solutions of boundary conditions v z = λd ε v + DF ε (u a )v φ y =λdb ε (u a )v + Dg ε (u a )v on y = 0,1

58 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Transverse linear instability bounded solutions of boundary conditions v z = λd ε v + DF ε (u a )v φ y =λdb ε (u a )v + Dg ε (u a )v on y = 0,1 Theorem For any λ R sufficiently small, there exists a solution v λ which is 2π periodic in x and periodic in z. The periodic wave u a is linearly unstable with respect to 3D periodic perturbations.

59 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof bounded solutions of boundary conditions v z = λd εv + DF ε(u a)v φ y =λdb ε(u a)v + Dg ε(u a)v on y = 0, 1 the linear operator L ε,λ := λd ε + DF ε (u a ) possesses two purely imaginary eigenvalues

60 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof bounded solutions of boundary conditions v z = λd εv + DF ε(u a)v φ y =λdb ε(u a)v + Dg ε(u a)v on y = 0, 1 the linear operator L ε,λ := λd ε + DF ε (u a ) possesses two purely imaginary eigenvalues for small and real λ, L ε,λ is a small relatively bounded perturbation of L ε ; L ε possesses two simple eigenvalues ±iεκ ε ; reversibility z z; boundary conditions...

61 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Transverse linear instability Theorem For λ R sufficiently small, there exists a solution v λ, 2π periodic in x and periodic in z. The periodic wave u a is linearly unstable with respect to 3D periodic perturbations.

62 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking 3D solutions Hamiltonian system boundary conditions u z = F ε (u) φ y = g ε (u) on y = 0,1

63 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking 3D solutions Hamiltonian system boundary conditions u z = F ε (u) φ y = g ε (u) on y = 0,1 family of equilibria (F ε (u a ) = 0) Q a = u a = (η a,0,φ a,0) = (P a,0,q a,0) + O(ε) x P a(ζ)dζ 0 3D solutions : u = u a + v

64 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Dimension-breaking Theorem A family of 3D doubly periodic waves u a,b (x,y,z), b small, emerges from the 2D periodic wave u a (x,y) in a dimension-breaking bifurcation : u a,b (x,y,z) = u a (x,y)+o( b ); u a,b and u a have the same period in x; u a,b is periodic in z with period 2π/κ, κ = εκ ε + O( b 2 ).

65 Spectral analysis Transverse linear instability Bifurcations : dimension-breaking Proof Lyapunov center theorem Hamiltonian formulation spectrum of L ε : spec(l ε ) ir = { iεκ ε,iεκ ε } boundary conditions...

66 2D periodic water waves β > 1 3 transverse linear instability dimension-breaking Questions transverse nonlinear instability other periods in the direction of propagation parameter β < 1 3 2D stability (spectral, linear, nonlinear)...

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68 Q.E.D.