transverse stability of periodic waves in water-wave models...transverse stability of periodic waves...

Transverse stability of periodic

waves in water-wave models

Mariana Haragus

Institut FEMTO-ST and LMB

Universite Bourgogne Franche-Comte, France

ICERM, April 28, 2017

Water-wave problem

gravity/gravity-capillary waves

� three-dimensional inviscid fluid layer

� constant density

� gravity/gravity and surface tension

� irrotational flow

Water-wave problem

x

y

z

y = 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

Euler equations

Laplace’s equation

φxx

+ φyy

+ φzz

= 0 in Dη

boundary conditions

φy

= 0 on y = 0

ηt

= φy

− ηx

φx

− ηz

φz

on y = h + η

φt

= −1

2(φ2

x

+ φ2y

+ φ2z

) − gη +σ

ρK on y = h + η

� velocity potential φ; free surface h + η

� mean curvature K =

[ηx√

1+η2x+η2

z

]

x

+

[ηz√

1+η2x+η2

z

]

z� parameters ρ, g , σ, h

Euler equations

moving coordinate system, speed −

dimensionless variables

� characteristic length h

� characteristic velocity

parameters

� inverse square of the Froude number α =gh

2

� Weber number β =σ

ρh 2

Euler equations

φxx

+ φyy

+ φzz

= 0 for 0 < y < 1 + η

φy

= 0 on y = 0

φy

= ηt

+ ηx

+ ηx

φx

+ ηz

φz

on y = 1 + η

φt

+ φx

+ 12

(φ2x

+ φ2y

+ φ2z

)+ αη − βK = 0 on y = 1 + η

Euler equations

φxx

+ φyy

+ φzz

= 0 for 0 < y < 1 + η

φy

= 0 on y = 0

φy

= ηt

+ ηx

+ ηx

φx

+ ηz

φz

on y = 1 + η

φt

+ φx

+ 12

(φ2x

+ φ2y

+ φ2z

)+ αη − βK = 0 on y = 1 + η

difficulties

� variable domain (free surface)

� nonlinear boundary conditions

very rich dynamics

� symmetries, Hamiltonian structures

� many particular solutions

Focus on . . .

traveling periodic 2D waves

transverse stability/instability

analytical results

long-wave models

Two-dimensional periodic waves

exist in different parameter regimes

β13

1

α

Two-dimensional periodic waves

transverse (in)stability

β13

1

α

Large surface tension

transverse linear instability

� longitudinal co-periodic perturbations

� transverse periodic perturbations

Euler equations

[H., 2015]

Transverse instability problem

Transverse spatial dynamics

U

z

= DU

t

+ F (U)

�U(x , z, t), D linear operator, F nonlinear map

� a periodic wave U∗(x) is an equilibrium

z

x

Transverse linear instability

Transverse spatial dynamics

U

z

= DU

t

+ F (U)

U∗(x) is transversely linearly unstable if the linearized system

U

z

= DU

t

+ LU , L = F

′(U∗)

possesses a solution of the form U(x , z, t) = eλtVλ(x , z)

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

Hypotheses

1 the system U

z

= DU

t

+ F (U) is reversible/Hamiltonian;

2 the linear operator L = F

′(U∗) possesses a pair of

simple purely imaginary eigenvalues ±iκ∗;

3 the operators D and L are closed in X with D(L) ⊂ D(D);

Main result

Theorem

1 For any λ ∈ R sufficiently small, the linearized system

U

z

= DU

t

+ LU

possesses a solution of the form U(·, z, t) = eλtVλ(·, z)

with Vλ(·, z) ∈ D(L) a periodic function in z .

2 U∗ is transversely linearly unstable.

[Godey, 2016; see also Rousset & Tzvetkov, 2010]

Euler equations

Hamiltonian formulation of the 3D problem:

U

z

= DU

t

+ F (U)

� boundary conditions

φy

= b(U)t

+ g(U) on y = 0, 1

(e.g. [Groves, H., Sun, 2002])

Periodic waves

β >1

3, α = 1 + ǫ , ǫ small

model: Kadomtsev-Petviashvili-I equation instability

∂x∂tu =∂x∂x(∂2xu + u +

1

2u2)−∂2

yu

[H.; Johnson & Zumbrun; Hakkaev, Stanislavova & Stefanov, . . . ]

Periodic waves

β >1

3, α = 1 + ǫ , ǫ small

model: Kadomtsev-Petviashvili-I equation instability

∂x∂tu =∂x∂x(∂2xu + u +

1

2u2)−∂2

yu

[H.; Johnson & Zumbrun; Hakkaev, Stanislavova & Stefanov, . . . ]

The Euler equations possess a one-parameter family of

symmetric periodic waves

ηǫ,a(x) = εpa

(ε1/2x, ε), ϕǫ,a(x) = ε1/2

q

a

(ε1/2x, ε)

p

a

(ξ, 0) = ∂ξqa(ξ, 0), pa(ξ, 0) satisfies the Korteweg de

Vries equation[Kirchgassner, 1989]

Linearized system

linearized system (rescaled)

U

z

= DεUt

+ DFε(ua)U


φy

=Dbε(ua)Ut

+ Dgε(ua)U on y = 0, 1

Linearized system

linearized system (rescaled)

U

z

= DεUt

+ DFε(ua)U


φy

=Dbε(ua)Ut

+ Dgε(ua)U on y = 0, 1

linear operator Lε := DFε(ua)


φy

= Dgε(ua)U on y = 0, 1

� space of symmetric functions (x → −x)

Xs = H1e (0, 2π)× L

2e(0, 2π)× H

1o ((0, 2π)× (0, 1))× L

2o((0, 2π)× (0, 1))

Linear operator Lε

Lε = L0ε + L1

ε L0ε

η

ω

φ

ξ

=

ω

β−εk

2aβηxx + (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

{εk

2aφaxφx −

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

+

[εk

2a 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 +

ε2y2η2axφayη

(1 + εηa)2−

εy2η2axφy

1 + εηa−

2εy2ηaxφayηx

1 + εηa

]

y

Check hypotheses . . .

Main difficulty: spectrum of Lε . . .

operator with compact resolvent −→ pure point spectrum

spectral analysis

|λ| ≥ λ∗

|λ| ≤ λ∗

|λ| ≤ εℓ∗

Key step

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

)k

4a

φ1xxxx − k

2a

φ1xx + ℓ2(1 + ǫ)φ1 − 3k2a

(Pa

φ1x)x + . . .

. . . . . . . . .

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 −

. . . . . . . . .

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,

. . . . . . . . .

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

†),

. . . . . . . . .

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, ǫ

2η))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[(ǫk

2+ µ

2)

∫ 1

0pǫ2 dζ

]= k

2L(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 , ǫ

3/4Φ2,Φ2x ,Φ2y , ǫ

3η, ǫ

7/2ηx ) + k

2ǫ3L(Φ1) + H.

Locate eigenvalues

|λ| ≤ εℓ∗

two simple eigenvalues ±iεκε

� Bε,ℓ small relatively bounded perturbation of B0,ℓ

B0,ℓ = k

2a

∂x

A ∂x

+ ℓ2 A =

(β −

1

3

)k

2a

∂xx

− 1 − 3Pa

Locate eigenvalues

|λ| ≤ εℓ∗

two simple eigenvalues ±iεκε

� Bε,ℓ small relatively bounded perturbation of B0,ℓ

B0,ℓ = k

2a

∂x

A ∂x

+ ℓ2 A =

(β −

1

3

)k

2a

∂xx

− 1 − 3Pa

� spectrum of ∂x

A∂x

is known (KP-I):

one simple negative eigenvalue −ω2a

� perturbation arguments . . . . . .

Critical surface tension

transverse linear instability

� longitudinal co-periodic perturbations

� transverse periodic perturbations

5th order KP model

[H. & Wahlen, 2017]

A 5th order KP model

∂t

∂x

u = ∂2x

(∂4x

u + ∂2x

u +1

2u

2

)+ ∂2

y

u

traveling generalized solitary waves

� solutions of the Kawahara equation

∂t

u = ∂x

(∂4x

u + ∂2x

u − u +1

2u

2

)

Periodic traveling waves

small periodic traveling waves: a two-parameter family

ϕa, (x) = p

a, (ka, x)

� depend analytically upon (a, c) ∈ (−a0, a0)× (−c0, c0)

� ka,c = k0(c) + ck(a, c),

k0(c) =(

1+√

1+4c2

)1/2, k(a, c) =

∑n≥1 k2n(c)a

2n

� pa,c(z) = ac cos(z) + c∑

m,n pn,m(c)ei(n−m)zan+m,

(n,m ≥ 0, n + m ≥ 2, n − m 6= ±1)

� explicit Taylor expansions for k(a, c), pn,m(c)

[Lombardi, 2000]


one-dimensional periodic wave u∗

u∗ is transversely linearly unstable if the linearized equation

∂t

∂x

u = ∂2x

(∂4x

u + ∂2x

u − u + u∗u

)+ ∂2

y

u

possesses a solution of the form u(t, x , y) = eλtv(x , y) ,

for some Reλ > 0

(v belongs to the set of the allowed perturbations)


linearized equation

∂t

∂x

u = A∗u + ∂2y

u , A∗ = ∂2x

(∂4x

+ ∂2x

− + u∗

)

� Fourier transform in y

∂t

∂x

u = A∗u − ω2u


linearized equation

∂t

∂x

u = A∗u + ∂2y

u , A∗ = ∂2x

(∂4x

+ ∂2x

− + u∗

)

� Fourier transform in y

∂t

∂x

u = A∗u − ω2u

u∗ is transversely unstable if there exists a solution of the

form u(t, x) = eλtv(x) , for some Reλ > 0, and ω ∈ R

∗

�v ∈ H , a space of functions depending upon the longitudinal

spatial variable x , e.g., H = L

2(R) or H = L

2(0, L), and

λ∂x

v = A∗v − ω2v


for some Reλ > 0 and ω ∈ R∗ , there exists a solution

λ∂x

v = A∗v − ω2v , v ∈ H

�u∗ is transversely spectrally unstable if the linear operator

λ∂x

− A∗ + ω2 is not invertible in H

u∗ is transversely spectrally unstable if the spectrum of

the linear operator λ∂x

− A∗ contains a negative value

−ω2 < 0 for some Reλ > 0.


u∗ is transversely spectrally unstable if the spectrum

of the linear operator λ∂x

− A∗ contains a negative value

−ω2 < 0 for some Reλ > 0.

� if −ω2 is an isolated eigenvalue then u∗ is transversely

linearly unstable

� if −ω2 belongs to the essential spectrum

σess(λ∂x−A∗) = {ν ∈ C ; λ∂x−A∗−ν is not Fredholm with index 0}

then u∗ is transversely essentially unstable

Periodic waves

small periodic waves: ϕa, (x) = p

a, (ka, x)

� scaling: z = k

a, x , λ = k

a, Λ

rescaled operator

Λ∂z

− Ba, , B

a, = ∂2z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, )

with 2π-periodic coefficients

Co-periodic perturbations

Λ∂z

− Ba, , B

a, = ∂2z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, )

closed operator in H = L

2(0, 2π)

Theorem

1 the linear operator Λ∂z

− Ba, acting in L2(0, 2π) has a

simple negative eigenvalue.

2 the periodic wave ϕa, is transversely linearly unstable

with respect to co-periodic longitudinal perturbations.

Proof

Λ∂z

− Ba, , B

a, = ∂2z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, )

show that Ba, has a simple positive eigenvalue

� the operator Λ∂z

− Ba, is real

� perturbation argument: the negative eigenvalue of −Ba,

persists for small real Λ

(point) spectrum of Ba, ?

Proof

Spectrum of Ba, = ∂2

z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, )

use perturbation arguments: small a and

Proof


z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, )


a = 0, = 0

B0,0 = ∂2z

(∂4z

+ ∂2z

), σ(B0,0) = {−n

2(n4 − n

2), n ∈ Z}

� 0 is a triple eigenvalue

� all other eigenvalues are negative

Proof


z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, )


a = 0, = 0

B0,0 = ∂2z

(∂4z

+ ∂2z

), σ(B0,0) = {−n

2(n4 − n

2), n ∈ Z}



spectral decomposition for small a and

σ(Ba, ) = σ1(Ba, ) ∪ σ2(Ba, )

� σ1(Ba, ) ⊂ V , V neighborhood of 0

� σ2(Ba, ) ⊂ {ν ∈ C ; Re ν < −m}

Proof


z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, ) :

locate the small eigenvalues

Proof


z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, ) :


a = 0

σ(B0, ) = {−n

2(k20n

4 − k

20n

2 − ), n ∈ Z}



Proof


z

(k4a, ∂

4z

+ k

2a, ∂

2z

− + p

a, ) :


a = 0

σ(B0, ) = {−n

2(k20n

4 − k

20n

2 − ), n ∈ Z}



a 6= 0

� use symmetries and show that 0 is a double eigenvalue

� third eigenvalue: compute an expansion for small a, . . .

. . . , νa, = a

2

2

(1

4X2

+ O(a2 +

2)

)> 0

Consequences

implies essential transverse instability of periodic waves

with respect to localize perturbations

implies essential transverse instability of generalized

solitary waves with respect to localize perturbations

extend to Euler equations . . . ?

Zero surface tension

transverse spectral stability

� fully localized/bounded perturbations

KP-II equation

[H., Li, & Pelinovsky, 2017]

Count unstable eigenvalues

Hamiltonian structure: linear operator of the form JL

� J skew-adjoint operator

� L self-adjoint operator

Under suitable conditions:

n

u

(JL) ≤ n

s

(L)

�n

u

(JL) = number of unstable eigenvalues of JL

�n

s

(L) = number of negative eigenvalues of L

[well-known result, extensively used in stability problems . . . ]

[does not work very well for periodic waves . . . ]

An extended eigenvalue count

Hamiltonian structure: linear operator of the form JL

� J skew-adjoint operator

� L self-adjoint operator

There exists a self-adjoint operator K such that

(JL)(JK) = (JK)(JL)

Under suitable conditions:

n

u

(JL) ≤ n

s

(K)

�n

u

(JL) = number of unstable eigenvalues of JL

�n

s

(K) = number of negative eigenvalues of K

Stability of periodic waves

classical result: allows to show (orbital) stability of periodic

waves with respect to co-periodic perturbations

particular case n

s

(K) = 0: used to show nonlinear (orbital)

stability of periodic waves with respect to subharmonic

perturbations (for the KdV and NLS equations)

[Deconinck, Kapitula, 2010; Gallay, Pelinovsky, 2015]

Stability of periodic waves

classical result: allows to show (orbital) stability of periodic

waves with respect to co-periodic perturbations

particular case n

s

(K) = 0: used to show nonlinear (orbital)

stability of periodic waves with respect to subharmonic

perturbations (for the KdV and NLS equations)

[Deconinck, Kapitula, 2010; Gallay, Pelinovsky, 2015]

key step: construction of a nonnegative operator K

� relies upon the existence of a conserved higher-order energy

functional (due to integrability)

KP-II equation

Kadomtsev-Petviashivili equation

(ut

+ 6uux

+ u

xxx

)x

+ u

yy

= 0

one-parameter family of one-dimensional periodic

traveling waves (up to symmetries)

u(x , t) = φ

(x + t)

� speed > 1

� 2π-periodic, even profile φ

satisfying the KdV equation

v

′′(x) + v(x) + 3v2(x) = 0

� known explicitly!

Linearized equation

linearized KP-II equation

(wt

+ w

xxx

+ w

x

+ 6(φ

(x)w)x

)x

+ w

yy

= 0

� 2π-periodic coefficients in x

� Ansatz

w(x , y , t) = e

λt+ipy

W (x), λ ∈ C, p ∈ R

linearized equation for W (x)

λWx

+W

xxxx

+ W

xx

+ 6(φ

(x)W )xx

− p

2W = 0

Spectral stability problem

linearized equation for W (x)

λWx

+W

xxxx

+ W

xx

+ 6(φ

(x)W )xx

− p

2W = 0

the periodic wave φ

is spectrally stable iff the linear

operator

A ,p(λ) = λ∂

x

+ ∂4x

+ ∂2x

+ 6∂2x

(φ

(x) ·) − p

2

is invertible for Reλ > 0.

� 2D bounded perturbations: space Cb

(R) and p ∈ R.

� continuous spectrum . . .

Floquet/Bloch decomposition

A ,p(λ) is invertible in C

b

(R) iff the operators

A ,p(λ, γ) = λ(∂

x

+ iγ) + (∂x

+ iγ)4 + (∂x

+ iγ)2 + 6(∂x

+ iγ)2(φ

(x) ·) − p

2

are invertible in L2per

(0, 2π), for any γ ∈ [0, 1).

� γ ∈ (0, 1) : study the spectrum of the operator

B ,p(γ) = −(∂

x

+ iγ)3 − (∂x

+ iγ) − 6(∂x

+ iγ)(φ

(x) ·) + p

2(∂x

+ iγ)−1

� γ = 0 : restrict to functions with zero mean

Counting criterion

apply the counting criterion to

B ,p(γ) = J (γ)L

,p(γ)

� skew-adjoint operator J (γ) = (∂x

+ iγ)

� self-adjoint operator

L ,p(γ) = −(∂

x

+ iγ)2 − − 6φ

(x) + p

2(∂x

+ iγ)−2

construct positive commuting operators K ,p(γ)

� find commuting operators M ,p(γ)

� show that suitable linear combination of M ,p(γ) and

L ,p(γ) is a positive operator

Commuting operators

natural candidate: use a higher-order conserved functional

� resulting operator satisfies the commutativity relation

� cannot obtain positive operators . . .

Commuting operators

natural candidate: use a higher-order conserved functional

� resulting operator satisfies the commutativity relation

� cannot obtain positive operators . . .

second option: use the operators from the KdV equation

�p = 0 corresponds to the KdV equation

� decompose:

L ,p = LKdV + p

2LKP, M ,p = MKdV + p

2MKP

� MKdV is obtained from a higher order conserved functional:

MKdV = ∂4x

+ 10∂x

φ

(x)∂x

− 10 φ

(x)−

2

� compute MKP directly from the commutativity relation:

MKP =5

3

(1 + ∂−2

x

)

Main result

Transverse spectral stability of periodic waves (with

respect to bounded perturbations):

� there exist constants b such that the operators

K ,p,b(γ) = M

,p(γ)− bL ,p(γ) are positive1

� the commutativity relation holds

� the general counting criterion implies that the spectra of

B ,p(γ) = J (γ)L

,p(γ) are purely imaginary

Main result

Transverse spectral stability of periodic waves (with

respect to bounded perturbations):

� there exist constants b such that the operators

K ,p,b(γ) = M

,p(γ)− bL ,p(γ) are positive1

� the commutativity relation holds

� the general counting criterion implies that the spectra of

B ,p(γ) = J (γ)L

,p(γ) are purely imaginary

Consequence: transverse linear stability of the periodic

waves with respect to doubly periodic perturbations

Many open problems . . .

• water waves: other parameter regimes, other types of waves

(solitary waves, three-dimensional waves) . . .

• periodic waves: nonlinear stability with respect to localized

perturbations (KdV equation?) . . .

transverse stability of periodic waves in water-wave models...transverse stability of periodic waves...

Documents