Transverse dynamics of gravity-capillary periodic water waves...Transverse dynamics Water-wave...

Water wavesThe hydrodynamic problem

Transverse dynamics

Transverse dynamics of gravity-capillary

periodic water waves

Mariana Haragus

LMB, Universite de Franche-Comte, France

IMA Workshop

Dynamical Systems in Studies of Partial Differential Equations

September 24-28, 2012

Transverse dynamics of periodic water waves


Transverse dynamics

Water-wave problemTwo-dimensional waves

Water waves



Transverse dynamics


Water-wave problem



Transverse dynamics


Water-wave problem

gravity-capillary water waves

three-dimensional inviscid fluid layer

constant density ρ

gravity and surface tension

irrotational flow



Transverse dynamics


Water-wave problem

y

xz

gravity-capillary water waves

three-dimensional inviscid fluid layer

constant density ρ

gravity and surface tension

irrotational flow



Transverse dynamics


Water-wave problem

x

y

zy = 0 (flat bottom)

y = h + η(x, z, t)

(free surface)

Domain

Dη = {(x, y, z) : x, z ∈ R, y ∈ (0, h + η(x, z, t))}

depth at rest h



Transverse dynamics


Euler equations

Laplace’s equation

φxx + φyy + φzz = 0 in Dη

boundary conditions

φy = 0 on y = 0

ηt = φy − ηxφx − ηzφz on y = h + η

φt = −1

2(φ2

x + φ2y + φ2

z) − gη +σ

ρK on y = h + η



Transverse dynamics


Euler equations

Laplace’s equation

φxx + φyy + φzz = 0 in Dη

boundary conditions

φy = 0 on y = 0

ηt = φy − ηxφx − ηzφz on y = h + η

φt = −1

2(φ2

x + φ2y + φ2

z) − gη +σ

ρK on y = h + η

velocity potential φ ; free surface h + η

mean curvature K =

[

ηx√1+η2

x+η2z

]

x

+

[

ηz√1+η2

x+η2z

]

zparameters ρ, g, σ, h



Transverse dynamics


Euler equations

moving coordinate system, speed −c

dimensionless variables

characteristic length h

characteristic velocity c



Transverse dynamics


Euler equations

moving coordinate system, speed −c

dimensionless variables

characteristic length h

characteristic velocity c

parameters

inverse square of the Froude number α =gh

c2

Weber number β =σ

ρhc2



Transverse dynamics


Euler equations

φxx + φyy + φzz = 0 for 0 < y < 1 + η

φy = 0 on y = 0

φy = ηt + ηx + ηxφx + ηzφz on y = 1 + η

φt + φx +12

(

φ2x + φ2

y + φ2z

)

+ αη − βK = 0 on y = 1 + η

K =

ηx√

1 + η2x + η2

z

x

+

ηz√

1 + η2x + η2

z

z



Transverse dynamics


Euler equations

very rich dynamics

difficulties

variable domain (free surface)

nonlinear boundary conditions

symmetries, Hamiltonian structure

many particular solutions



Transverse dynamics


Two-dimensional traveling waves

periodic wave solitary waves

generalized solitary waves solitary waves

[Nekrasov, Levi-Civita, Struik, Lavrentiev, Friedrichs & Hyers, . . .

Amick, Kirchgassner, Iooss, Buffoni, Groves, Toland, Lombardi, Sun, . . . ]



Transverse dynamics


Three-dimensional traveling waves

[Groves, Mielke, Craig, Nicholls, H., Kirchgassner, Deng, Sun, Sandstede,

Iooss, Plotnikov, Wahlen, . . . ]



Transverse dynamics


Solitary wave



Transverse dynamics


Periodic waves



Transverse dynamics


Questions

Existence two- and three-dimensional waves

Dynamics

2D stability

3D stability

new solutions (bifurcations)

(Numerical results ; Model equations ; Cauchy problem ; . . .)



Transverse dynamics


Dynamics of solitary waves

capillary-gravity waves β > 13

2D stability

3D instability (linear and nonlinear)

bifurcations : dimension-breaking

[H. & Scheel ; Mielke ; Groves, H. & Sun ; Pego & Sun ; Rousset & Tzvetkov]



Transverse dynamics




2D stability




capillary-gravity waves 0 < β < 13

2D stability

3D instability (linear)


[Buffoni ; Groves & Wahlen ; Groves, Wahlen & Sun]



Transverse dynamics




2D stability




capillary-gravity waves 0 < β < 13

2D stability



[Buffoni ; Groves & Wahlen ; Groves, Wahlen & Sun]

gravity waves β = 02D stability [Pego & Sun]



Transverse dynamics


Dynamics of periodic waves

gravity waves β = 0

Benjamin-Feir instability [Bridges & Mielke]



Transverse dynamics


Dynamics of periodic waves

gravity waves β = 0

Benjamin-Feir instability [Bridges & Mielke]

gravity-capillary waves β > 13



(see [Groves, H. & Sun, 2001, 2002])



Transverse dynamics


Predictions : model equations

gravity-capillary waves β > 13

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]



Transverse dynamics

Spatial dynamics2D periodic waves

Euler equations

φxx + φyy + φzz = 0 for 0 < y < 1 + η

φy = 0 on y = 0

φy = ηt + ηx + ηxφx + ηzφz on y = 1 + η

φt + φx +12

(

φ2x + φ2

y + φ2z

)

+ αη − βK = 0 on y = 1 + η

K =

ηx√

1 + η2x + η2

z

x

+

ηz√

1 + η2x + η2

z

z

parameters : β >1

3, α ∼ 1



Transverse dynamics


Questions

Transverse dynamics

3D instability

bifurcations : new solutions



Transverse dynamics


Spatial dynamics : Hamiltonian formulation

time-like variable z [Kirchgassner, 1982]

fixed domain R × (0, 1) : variable y′ = y/(1 + η)



Transverse dynamics


Spatial dynamics : Hamiltonian formulation

time-like variable z [Kirchgassner, 1982]

fixed domain R × (0, 1) : variable y′ = y/(1 + η)

Hamiltonian H(η, ω, φ, ξ) [Groves, H. & Sun, 2001]

H(η, ω, φ, ξ) =

∫

T

−T

∫

R

{

−1

2αη

2+ β − (β

2− W

2)1/2

(1 + η2x )

1/2}

dx dt W = ω +

∫ 1

0

yφy ξ

1 + ηdy

+

∫

T

−T

∫ 1

0

∫

R

{

(ηt + ηx )yφy − (1 + η)(φt + φx ) −1 + η

2

(

φx −yηxφy

1 + η

)2

+ξ2 − φ2

y

2(1 + η)

}

dx dy dt

space Xs,δ, s ∈ (0, 1/2), δ > 1/2

Xs,δ = Hs+1δ (0, L) × H

sδ(0, L) × H

s+1δ ((0, L) × (0, 1)) × H

sδ((0, L) × (0, 1))

Hsδ(0, L) =

{

u =∑

m∈Z

um(x)eimπt/T

∣

∣

∣

∣

um ∈ Hs(0, L), ‖u‖

2s,δ =

∑

m∈Z

(1 + |m|2)2δ

‖um‖2s

}



Transverse dynamics


Hamiltonian system

Hamilton’s equations u = (η, ω, φ, ξ)

uz = Dut + F(u)



Transverse dynamics


Hamiltonian system


uz = Dut + F(u)

Du = (0, φ∣

∣

y=1, 0, 0), F(u) = (f1(u), f2(u), f3(u), f4(u))

f1(u) = W

(

1 + η2x

β2 − W 2

)1/2

, W = ω +

∫

1

0

yφy ξ

1 + ηdy

f2(u) =

∫

1

0

{

ξ2 − φ2y

2(1 + η)2+

1

2

(

φx +yφy ηx

1 + η

)(

φx −yφy ηx

1 + η

)

+

[

yφy

(

φx −yφy ηx

1 + η

)]

x

}

dy

+ αη −

ηx

(

β2 − W 2

1 + η2x

)1/2

x

+W

(1 + η)2

(

1 + η2x

β2 − W 2

)1/2 ∫ 1

0yφy ξ dy + φx

∣

∣

y=1

f3(u) =ξ

1 + η+

yφyW

1 + η

(

1 + η2x

β2 − W 2

)1/2

f4(u) = −φyy

1 + η−[

(1 + η)φx − yηxφy

]

x+

[

yηx

(

φx −yφy ηx

1 + η

)]

y

+(yξ)yW

1 + η

(

1 + η2x

β2 − W 2

)1/2



Transverse dynamics


Hamiltonian system


uz = Dut + F(u)

Du = (0, φ∣

∣

y=1, 0, 0), F(u) = (f1(u), f2(u), f3(u), f4(u))

boundary conditions

φy = b(u)t + g(u) on y = 0, 1

b(u) = yη, g(u) = y(1 + η)(1+Φx)ηx − yη2xΦy + yξW

(

1 + η2xβ2 −W 2

)1/2



Transverse dynamics


2D periodic waves

parameters α = 1 + ε, β > 1/3 [Kirchgassner, 1988]

family of 2D periodic waves, ε small

η⋆(x) = ε ηKdV(ε1/2x) + O(ε2)

φ⋆(x, y) = ε1/2φKdV(ε1/2x) + O(ε3/2)

ηKdV solution of KdV :

(

β −1

3

)

η′′ = η +3

2η2

φ′

KdV = ηKdV



Transverse dynamics


Periodic solutions of KdV

(

β −1

3

)

η′′ = η +3

2η2

family of periodic waves

ηKdV(X) = Pa(kaX), a ∈ I ⊂ R

Pa even function, 2π–periodic



Transverse dynamics


2D periodic waves

scaling

x = kax, η = εη, φ = ε1/2φ, ω = εω, ξ = ε1/2ξ


uz = Dεut + Fε(u)

boundary conditions

φy = bε(u)t + gε(u) on y = 0, 1



Transverse dynamics


2D periodic waves

scaling

x = kax, η = εη, φ = ε1/2φ, ω = εω, ξ = ε1/2ξ


uz = Dεut + Fε(u)

boundary conditions

φy = bε(u)t + gε(u) on y = 0, 1

equilibria (Fε(ua) = 0) Qa =

∫

x

0Pa(ζ)dζ

ua = (ηa, 0, φa, 0) = (Pa, 0,Qa, 0) + O(ε)



Transverse dynamics

Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking

Questions

Transverse dynamics

3D instability

bifurcations : new solutions

– analysis of the linearized problem –



Transverse dynamics


Linear operator

linearized system

uz = Dεut + DFε(ua)u

boundary conditions

φy =Dbε(ua)ut + Dgε(ua)u on y = 0, 1



Transverse dynamics


Linear operator

linearized system


boundary conditions


linear operator Lε := DFε(ua)

boundary conditions

φy = Dgε(ua)u on y = 0, 1

space of symmetric functions (x → −x)

Xs = H1e(0, 2π)× L2

e(0, 2π)× H1o((0, 2π)× (0, 1))× L2

o((0, 2π)× (0, 1))



Transverse dynamics


Linear operator

Lε = L0ε + L1

ε L0ε

η

ω

φ

ξ

=

ω

β

−εk2aβηxx + (1 + ǫ)η − kaφx |y=1

ξ

−εk2aφxx − φyy

, L1ε

η

ω

φ

ξ

=

g1

g2

G1

G2

g1 =(1 + εk2a η

2ax )

1/2

β

(

ω +1

1 + εηa

∫ 1

0yφay ξ dy

)

−ω

β

g2 =

∫

1

0

{

εk2aφaxφx −

φayφy

(1 + εηa)2+

εφ2ayη

(1 + εηa)3−

ε3k2a y2η2

axφayφy

(1 + εηa)2−

ε3k2a y2ηaxφ

2ayηx

(1 + εηa)2+

ε3k2a y2η2

axφ2ayη

(1 + εηa)3

+

[

εk2a yφayφx + εk

2a yφaxφy −

2ε2k2a y2ηaxφayφy

1 + εηa−

ε2k2a y2φ2

ayηx

1 + εηa+

ε3k2a y2ηaxφ

2ayη

(1 + εηa)2

]

x

}

dy

+ εk2aβηxx − εk

2aβ

[

ηx

(1 + ε3k2aη2ax )

3/2

]

x

G1 = −εηaξ

1 + εηa+

(1 + ε3k2aη2ax )

1/2

β(1 + εηa)

(

ω +1

1 + εηa

∫

1

0yφay ξ dy

)

yφay

G2 =

[

εηaφ

(1 + εηa)+

εφaη

(1 + ηa)2

]

yy

− ε2k2a [ηaφx + φaxη − yφay ηx − yηaxφy ]x

+ ε2k2a

[

yηaxφx + yφaxηx +ε2y2η2

axφayη

(1 + εηa)2−

εy2η2axφy

1 + εηa−

2εy2ηaxφayηx

1 + εηa

]

y



Transverse dynamics


Spectrum of Lε

Theorem

The linear operator Lε has the following properties (ε small) :

pure point spectrum spec(Lε) ;

spec(Lε) ∩ iR = {−iεkε, iεkε} ;

±iεkε are simple eigenvalues ;

resolvent estimate

‖(Lε − iλI)−1‖ ≤c

|λ|, ∀ |λ| ≥ λ⋆



Transverse dynamics


Proof

operator with compact resolvent

−→ pure point spectrum

spectral analysis

|λ| ≥ λ∗

|λ| ≤ λ∗

|λ| ≤ εℓ∗



Transverse dynamics


Proof

STEP I : |λ| ≥ λ∗

no eigenvalues

Lε small relatively bounded perturbation of L0ε

L0ε operator with constant coefficients

a priori estimates



Transverse dynamics


Proof

STEP II : |λ| ≤ λ∗

reduction to a scalar operator Bε,ℓ in L2o(0, 2π)

scaling λ = εℓ, ω = εω, ξ = εξ

decomposition φ(x, y) = φ1(x) + φ2(x, y)



Transverse dynamics


Proof

STEP II : |λ| ≤ λ∗

reduction to a scalar operator Bε,ℓ in L2o(0, 2π)

scaling λ = εℓ, ω = εω, ξ = εξ

decomposition φ(x, y) = φ1(x) + φ2(x, y)

λ = εℓ eigenvalue iff Bε,ℓφ1 = 0

Bε,ℓφ1 =

(

β −1

3

)

k4aφ1xxxx − k2aφ1xx + ℓ2(1 + ǫ)φ1 − 3k2a(Paφ1x)x + . . . . . .



Transverse dynamics


. . . . . . . . .

ω =β

(1 + ǫ3η⋆2x )1/2

(η†+ ikη) −

1

1 + ǫη⋆

∫ 1

0yΦ

⋆y ξdy,

ξ = (1 + ǫη⋆)(Φ

†+ ikΦ) − ǫyΦ

⋆y (η

†+ ikη)

(1 + ǫ)

ǫ2η −

1

ǫ2Φx |y=1 −

1

ǫβηxx − ikβ(h

ǫ1 + ikη) = h

ǫ2

−1

ǫΦxx −

1

ǫ2Φyy − ik(H

ǫ1 + ikΦ) = H

ǫ2 ,

hǫ2 = ω

†− g

ǫ2 ,

Hǫ2 = ξ

†− G

ǫ2

hǫ1 =

ω

β− ikη

= −1

β(1 + ǫη⋆)

∫

1

0yΦ

⋆y [−ǫyΦ

⋆y (ikη + η

†) + (1 + ǫη

⋆)(ikΦ + Φ

†)]dy

+

(

1

(1 + ǫ3η⋆2x )1/2

− 1

)

ikη +η†

(1 + ǫ3η⋆2x )1/2

,

Hǫ1 = ξ − ikΦ

= (1 + ǫη⋆)Φ

†+ ikǫη

⋆Φ − ǫyΦ

⋆y (η

†+ ikη).

Bǫ(η,Φ) = −ǫηx + B

ǫ0 + B

ǫ1 ,

Bǫ0 =

ǫη⋆Φy

1 + ǫη⋆+

ǫΦ⋆y η

(1 + ǫη⋆)2

∣

∣

∣

∣

∣

y=1

,

Bǫ1 = ǫ

2η⋆x Φx + ǫ

2Φ⋆x ηx +

ǫ4η⋆2x Φ⋆

y η

(1 + ǫη⋆)2−

ǫ3η⋆2x Φy

1 + ǫη⋆−

−Φyy + q2Φ = ǫ2(Hǫ2 + ikHǫ

1 ), 0 < y < 1

Φy = 0, y = 0

Φy −ǫµ2Φ

1 + ǫ + βq2= −

ǫ3iµ(hǫ2 + ikβhǫ1 )

1 + ǫ + βq2+ B

ǫ0 + B

ǫ1 , y = 1

G(y, ζ) =

cosh qy

cosh q

(1 + ǫ + βq2) cosh q(1 − ζ) + (ǫµ2/q)

q2 − (1 + ǫ + βq2)q tanh q − ǫ

cosh qζ

cosh q

(1 + ǫ + βq2) cosh q(1 − y) + (ǫµ2/q)

q2 − (1 + ǫ + βq2)q tanh q −



Transverse dynamics


. . . . . . . . .

Φ1 =1 + ǫ

ǫ2(k2(1 + ǫ) + µ2 + (β − 1/3)µ4)×

{∫ 1

0ǫ2(ξ

†− iµG

ǫ2,2 + ikH

ǫ1 )dζ − ǫq

2∫ 1

0pǫ2 dζ

−ǫ3iµ(hǫ2 + ikβhǫ1 )

1 + ǫ + βq2+

ǫ2µ2 pǫ2 |ζ=1

1 + ǫ + βq2

}

,

Φ2 = −

∫

1

0G1(ξ

†− iµG

ǫ2,2 + ikH

ǫ1 )dζ −

∫

1

0G1ζ G

ǫ2,1dζ +

∫

1

0(ǫk

2+ µ

2)G1p

ǫ2 dζ + ǫp

ǫ2

− G1|ζ=1

(

−ǫiµ(hǫ2 + ikβhǫ1 )

1 + ǫ + βq2+

µ2pǫ2 |ζ=1

1 + ǫ + βq2

)

,

Φ = −

∫

1

0Gǫ

2(ξ

†− ıµG

ǫ2,2 + ıkH

ǫ1 )dζ −

∫

1

0Gζǫ

2Gǫ2,1dζ +

ǫ3 ıµG |ζ=1(hǫ2 + ıkβhǫ1 )

1 + ǫ + βq2+

∫

1

0ǫq

2Gp

ǫ2 dζ + ǫp

ǫ2 −

ǫ2µ2

1

∫

1

0ǫq

2Gp

ǫ2 dζ + ǫp

ǫ2 −

ǫ2µ2G |ζ=1 pǫ2 |ζ=1

1 + ǫ + βq2

=

∫ 1

0Gǫp

ǫ2ζζdζ − ǫG |ζ=1 p

ǫ2ζ |ζ=1 =

∫ 1

0Gǫ

2(G

ǫ2,0)ζζdζ − G |ζ=1B

ǫ0 ,

Φ1 + Φ2 = −

∫

1

0Gǫ

2(ξ

†− ıµG

ǫ2,2 + ıkH

ǫ1 )dζ −

∫

1

0Gζǫ

2Gǫ2,1dζ

+ǫ3 ıµG |ζ=1(h

ǫ2 + ıkβhǫ1 )

1 + ǫ + βq2+

∫

1

0ǫq

2Gp

ǫ2 dζ + ǫp

ǫ2 −

ǫ2µ2G |ζ=1pǫ2 |ζ=1

1 + ǫ + βq2,



Transverse dynamics


. . . . . . . . .

ıµhǫ2 = ıµω

†+ ıµF

[

−1

ǫ2

∫

1

0

{

ǫΦ⋆x Φx −

Φ⋆y Φy

(1 + ǫη⋆)2+

ǫΦ⋆2y η

(1 + ǫη⋆)3−

ǫ3y2η⋆2x Φ⋆

y Φy

(1 + ǫη⋆)2−

ǫ3y2η⋆x Φ⋆2

y ηx

(1 + ǫη⋆)2+

ǫ4y

(1

+µ2

ǫF

[

∫ 1

0

{

yΦ⋆y Φx + yΦ

⋆x Φy −

2ǫy2η⋆x ΦyΦ

⋆y

1 + ǫη⋆−

ǫy2Φ⋆2y ηx

1 + ǫη⋆+

ǫ2y2η⋆x Φ⋆2

y η

(1 + ǫη⋆)2

}

dy

]

−βµ2

ǫF

[

(1 +

F−1

[

ǫıµhǫ2

1 + ǫ + βq2

]

= −F−1[

1

1 + ǫ + βq2F [(Φ

⋆1xΦ1x )x ]

]

+ F−1

[

µ2

1 + ǫ + βq2F

[∫ 1

0yΦ

⋆x Φ2y dy

]

]

+

{

F−1

[

−1

1 + ǫ + βq2F

[

∫ 1

0

(

Φ⋆2xΦ1x + Φ

⋆x Φ2x −

Φ⋆y Φy

ǫ(1 + ǫη⋆)2+

Φ⋆2y η

(1 + ǫη⋆)3

−ǫ2y2η⋆2

x Φ⋆y Φy

(1 + ǫη⋆)2−

ǫ2y2η⋆x Φ

⋆2y ηx

(1 + ǫη⋆)2+

ǫ3y2η⋆x Φ⋆2

y η

(1 + ǫη⋆)3

)

dy

]

−ıµ

1 + ǫ + βq2F

[

∫

1

0

(

yΦ⋆y Φx −

2ǫy2η⋆x Φ

⋆y Φy

1 + ǫη⋆−

ǫy2Φ⋆2y ηx

1 + ǫη⋆+

ǫ2y2η⋆x Φ

⋆2y η

(1 + ǫη⋆)2

)

dy

]

+βıµ

1 + ǫ + βq2F

[

ηx

(1 + ǫ3η⋆2x )3/2

− ηx

]]}

x

+ F−1

[

ǫıµω†

1 + ǫ + βq2

]

= −F−1[

1

1 + ǫ + βq2F [(Φ

⋆1xΦ1x )x ]

]

+ F−1

[

µ2

1 + ǫ + βq2F

[∫

1

0yΦ

⋆x Φ2y dy

]

]

+ (L(ǫΦ1x ,Φ2x ,Φ2y , ǫ2η, ǫ

4ηx ))x + ǫ

−1/2(L(ǫΦx , ǫ

2Φ2y , ǫ

4η, ǫ

3ηx ))x + ǫ

1/2L(ω

†),



Transverse dynamics


. . . . . . . . .

F−1

[

µ2

1 + ǫ + βq2F

[∫ 1

0yΦ

⋆x Φ2y dy

]

]

= F−1

[

µ2

1 + ǫ + βq2F

[

Φ⋆1xΦ2|y=1 −

∫ 1

0Φ⋆1xΦ2dy +

∫ 1

0yΦ

⋆2xΦ2y

=

[

F−1

[

µ1/2

1 + ǫ + βq2µ1/2

F [Φ⋆1xΦ2|y=1 ] −

1

1 + ǫ + βq2

∫ 1

0(Φ

⋆1xΦ2)xdy +

µ

1 + ǫ + βq2

∫ 1

0yΦ

⋆2xΦ2y dy

]]

x

= ǫ−1/4

(L(Φ2))x + (L(Φ2,Φ2x , ǫ1/2

Φ2y ))x ,

F−1

[

ǫıµhǫ2

1 + ǫ + βq2

]

= −F−1[

1

1 + ǫ + βq2F [(Φ

⋆1xΦ1x )x ]

]

+ ǫ−1/4

(L(Φ2))x

+ ǫ−1/2

(L(ǫΦx , ǫ2Φ2y , ǫ

4η, ǫ

3ηx )x + (L(ǫΦ1x ,Φ2,Φ2x ,Φ2y , ǫ

2η, ǫ

4ηx )x + H.

F−1

[

ǫıµ.ıkhǫ1

1 + ǫ + βq2

]

= (L(Φ2, ǫ2η))x + ǫ

2k2(L(Φ1))x + H, F

−1

[

µ2 pǫ2 |ζ=1

1 + ǫ + βq2

]

= ǫ−1/4

(L(Φ2, ǫη))x

F−1[

(ǫk2+ µ

2)

∫

1

0pǫ2 dζ

]

= k2L(ǫΦ2, ǫ

2η) + (L(Φ2,Φ2x , ǫη, ǫηx ))x

∫

1

0(ξ

†− (G

ǫ2,2)x + ıkH

ǫ1 )dζ = (η

⋆Φ1x )x + (Φ

⋆1xη)x + (L(Φ2x ,Φ2y , ǫη, ǫηx ))x +

(β − 1/3)Φ1xxxx − Φ1xx + k2(1 + ǫ)Φ1 = (η

⋆Φ1x )x + (Φ

⋆1xη)x + F

−1[

1

1 + ǫ + βq2F [(Φ

⋆1xΦ1x )x ]

]

+ (L(ǫ1/2

Φ1x , ǫ−1/4

Φ2,Φ2x ,Φ2y , ǫ3/4

η, ǫηx ))x + k2[L(ǫΦ1, ǫΦ2, ǫ

2η) + ǫ

2L(Φ1)x ] + H,

η = F−1

[

ıµΦ1

1 + ǫ + βq2

]

+ L(ǫΦ1x , ǫ3/4

Φ2,Φ2x ,Φ2y , ǫ3η, ǫ

7/2ηx ) + k

2ǫ3L(Φ1) + H.



Transverse dynamics


Proof

STEP II a : εℓ∗ ≤ |λ| ≤ λ∗

no eigenvalues

Bε,ℓ small relatively bounded perturbation of

Cε,ℓ =

(

β −1

3

)

k4aφ1xxxx − k2aφ1xx + ℓ2(1 + ǫ)φ1

Bε,ℓ selfadjoint operator with constant coefficients

a priori estimates



Transverse dynamics


Proof

STEP II b : |λ| ≤ εℓ∗

two simple eigenvalues ±iεκε

Bε,ℓ small relatively bounded perturbation of B0,ℓ

B0,ℓ = k2a∂x A∂x + ℓ2 A =

(

β −1

3

)

k2a∂xx − 1 − 3Pa



Transverse dynamics


Proof

STEP II b : |λ| ≤ εℓ∗

two simple eigenvalues ±iεκε

Bε,ℓ small relatively bounded perturbation of B0,ℓ

B0,ℓ = k2a∂x A∂x + ℓ2 A =

(

β −1

3

)

k2a∂xx − 1 − 3Pa

spectrum of A is known (KdV !)

∂xA∂x : one simple negative eigenvalue −ω2a

perturbation arguments . . . . . .



Transverse dynamics


Spectrum of Lε

Theorem

The linear operator Lε has the following properties (ε small) :

pure point spectrum spec(Lε) ;

spec(Lε) ∩ iR = {−iεkε, iεkε} ;

±iεkε are simple eigenvalues ;

resolvent estimate

‖(Lε − iλI)−1‖ ≤c

|λ|, ∀ |λ| ≥ λ⋆



Transverse dynamics


Transverse linear instability

linearized system


boundary conditions




Transverse dynamics



linearized system


boundary conditions


Definition

The periodic wave ua is linearly unstable if the linearized system

possesses a solution

u(t, x, y, z) = eλtvλ(x, y, z)

with λ ∈ C, Reλ > 0, vλ bounded function.



Transverse dynamics



bounded solutions of

vz = λDεv + DFε(ua)v

boundary conditions

φy =λDbε(ua)v + Dgε(ua)v on y = 0, 1



Transverse dynamics



bounded solutions of

vz = λDεv + DFε(ua)v

boundary conditions


Theorem

For any λ ∈ R sufficiently small, there exists a solution vλ

which is 2π– periodic in x and periodic in z.

The periodic wave ua is linearly unstable with respect to 3D

periodic perturbations.



Transverse dynamics


Proof

bounded solutions ofvz = λDεv + DFε(ua)v

boundary conditions


—————————————————————————–

the linear operator Lε,λ := λDε + DFε(ua) possesses

two purely imaginary eigenvalues



Transverse dynamics


Proof

bounded solutions ofvz = λDεv + DFε(ua)v

boundary conditions


—————————————————————————–

the linear operator Lε,λ := λDε + DFε(ua) possesses

two purely imaginary eigenvalues

for small and real λ, Lε,λ is a small relatively bounded

perturbation of Lε ;

Lε possesses two simple eigenvalues ±iεκε ;

reversibility z → −z ;

boundary conditions . . . . . .



Transverse dynamics



Theorem

For λ ∈ R sufficiently small, there exists a solution vλ, 2π–

periodic in x and periodic in z.

The periodic wave ua is linearly unstable with respect to 3D

periodic perturbations.



Transverse dynamics


3D solutions

Hamiltonian system

uz = Fε(u)

boundary conditions

φy = gε(u) on y = 0, 1



Transverse dynamics


3D solutions

Hamiltonian system

uz = Fε(u)

boundary conditions

φy = gε(u) on y = 0, 1

family of equilibria (Fε(ua) = 0) Qa =

∫

x

0Pa(ζ)dζ

ua = (ηa, 0, φa, 0) = (Pa, 0,Qa, 0) + O(ε)

3D solutions : u = ua + v



Transverse dynamics


Dimension-breaking

Theorem

A family of 3D doubly periodic waves ua,b(x, y, z), b small,

emerges from the 2D periodic wave ua(x, y) in a

“dimension-breaking” bifurcation :

ua,b(x, y, z) = ua(x, y) + O(|b|) ;

ua,b and ua have the same period in x ;

ua,b is periodic in z with period 2π/κ, κ = εκε + O(|b|2).

−→



Transverse dynamics


Proof

Lyapunov center theorem

Hamiltonian formulation

spectrum of Lε : spec(Lε) ∩ iR = {−iεκε, iεκε}

boundary conditions . . . . . .


Transverse dynamics

2D periodic water waves β > 13

transverse linear instability

dimension-breaking −→

Questions

transverse nonlinear instability

other periods in the direction of propagation

parameter β < 13

2D stability (spectral, linear, nonlinear)

. . . . . .


Q.E.D.


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