transverse dynamics of gravity-capillary periodic water waves...transverse dynamics water-wave...
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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
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Water waves
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Water waves
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Water waves
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Water-wave problem
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Water-wave problem
gravity-capillary water waves
three-dimensional inviscid fluid layer
constant density ρ
gravity and surface tension
irrotational flow
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Euler equations
moving coordinate system, speed −c
dimensionless variables
characteristic length h
characteristic velocity c
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional 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
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Euler equations
very rich dynamics
difficulties
variable domain (free surface)
nonlinear boundary conditions
symmetries, Hamiltonian structure
many particular solutions
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Three-dimensional traveling waves
[Groves, Mielke, Craig, Nicholls, H., Kirchgassner, Deng, Sun, Sandstede,
Iooss, Plotnikov, Wahlen, . . . ]
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Solitary wave
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Periodic waves
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Questions
Existence two- and three-dimensional waves
Dynamics
2D stability
3D stability
new solutions (bifurcations)
(Numerical results ; Model equations ; Cauchy problem ; . . .)
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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]
capillary-gravity waves 0 < β < 13
2D stability
3D instability (linear)
bifurcations : dimension-breaking
[Buffoni ; Groves & Wahlen ; Groves, Wahlen & Sun]
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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]
capillary-gravity waves 0 < β < 13
2D stability
3D instability (linear)
bifurcations : dimension-breaking
[Buffoni ; Groves & Wahlen ; Groves, Wahlen & Sun]
gravity waves β = 02D stability [Pego & Sun]
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Dynamics of periodic waves
gravity waves β = 0
Benjamin-Feir instability [Bridges & Mielke]
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
Dynamics of periodic waves
gravity waves β = 0
Benjamin-Feir instability [Bridges & Mielke]
gravity-capillary waves β > 13
3D instability (linear)
bifurcations : dimension-breaking
(see [Groves, H. & Sun, 2001, 2002])
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Water-wave problemTwo-dimensional waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
Questions
Transverse dynamics
3D instability
bifurcations : new solutions
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
Spatial dynamics : Hamiltonian formulation
time-like variable z [Kirchgassner, 1982]
fixed domain R × (0, 1) : variable y′ = y/(1 + η)
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
Hamiltonian system
Hamilton’s equations u = (η, ω, φ, ξ)
uz = Dut + F(u)
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
Hamiltonian system
Hamilton’s equations u = (η, ω, φ, ξ)
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
Hamiltonian system
Hamilton’s equations u = (η, ω, φ, ξ)
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
2D periodic waves
scaling
x = kax, η = εη, φ = ε1/2φ, ω = εω, ξ = ε1/2ξ
Hamilton’s equations u = (η, ω, φ, ξ)
uz = Dεut + Fε(u)
boundary conditions
φy = bε(u)t + gε(u) on y = 0, 1
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spatial dynamics2D periodic waves
2D periodic waves
scaling
x = kax, η = εη, φ = ε1/2φ, ω = εω, ξ = ε1/2ξ
Hamilton’s equations u = (η, ω, φ, ξ)
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Questions
Transverse dynamics
3D instability
bifurcations : new solutions
– analysis of the linearized problem –
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Linear operator
linearized system
uz = Dεut + DFε(ua)u
boundary conditions
φy =Dbε(ua)ut + Dgε(ua)u on y = 0, 1
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Linear operator
linearized system
uz = Dεut + DFε(ua)u
boundary conditions
φy =Dbε(ua)ut + Dgε(ua)u on y = 0, 1
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Proof
operator with compact resolvent
−→ pure point spectrum
spectral analysis
|λ| ≥ λ∗
|λ| ≤ λ∗
|λ| ≤ εℓ∗
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Proof
STEP I : |λ| ≥ λ∗
no eigenvalues
Lε small relatively bounded perturbation of L0ε
L0ε operator with constant coefficients
a priori estimates
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
. . . . . . . . .
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
. . . . . . . . .
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
. . . . . . . . .
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
. . . . . . . . .
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Transverse linear instability
linearized system
uz = Dεut + DFε(ua)u
boundary conditions
φy =Dbε(ua)ut + Dgε(ua)u on y = 0, 1
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Transverse linear instability
linearized system
uz = Dεut + DFε(ua)u
boundary conditions
φy =Dbε(ua)ut + Dgε(ua)u on y = 0, 1
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Transverse linear instability
bounded solutions of
vz = λDεv + DFε(ua)v
boundary conditions
φy =λDbε(ua)v + Dgε(ua)v on y = 0, 1
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Transverse linear instability
bounded solutions of
vz = λDεv + DFε(ua)v
boundary conditions
φy =λDbε(ua)v + Dgε(ua)v on y = 0, 1
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Proof
bounded solutions ofvz = λDεv + DFε(ua)v
boundary conditions
φy =λDbε(ua)v + Dgε(ua)v on y = 0, 1
—————————————————————————–
the linear operator Lε,λ := λDε + DFε(ua) possesses
two purely imaginary eigenvalues
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Proof
bounded solutions ofvz = λDεv + DFε(ua)v
boundary conditions
φy =λDbε(ua)v + Dgε(ua)v on y = 0, 1
—————————————————————————–
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Transverse linear instability
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
3D solutions
Hamiltonian system
uz = Fε(u)
boundary conditions
φy = gε(u) on y = 0, 1
Transverse dynamics of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
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 of periodic water waves
Water wavesThe hydrodynamic problem
Transverse dynamics
Spectral analysisTransverse linear instabilityBifurcations : dimension-breaking
Proof
Lyapunov center theorem
Hamiltonian formulation
spectrum of Lε : spec(Lε) ∩ iR = {−iεκε, iεκε}
boundary conditions . . . . . .
Transverse dynamics of periodic water waves
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)
. . . . . .
Transverse dynamics of periodic water waves
Transverse dynamics of periodic water waves
Q.E.D.
Transverse dynamics of periodic water waves