Transverse stability of periodic waves in water-wave models Mariana - - PowerPoint PPT Presentation

SLIDE 1

Transverse stability of periodic waves in water-wave models

Mariana Haragus

Institut FEMTO-ST and LMB Universit´ e Bourgogne Franche-Comt´ e, France

ICERM, April 28, 2017

SLIDE 2

Water-wave problem

gravity/gravity-capillary waves

three-dimensional inviscid fluid layer constant density gravity/gravity and surface tension irrotational flow

SLIDE 3

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

SLIDE 4

Euler equations

Laplace’s equation φ

xx + φ yy + φ zz = 0 in Dη

boundary conditions φ

y

=

• n

y = 0

η

t

= φ

y − η xφ x − η zφ z

• n

y = h + η

φ

t

= −1 2(φ2

x + φ2 y + φ2 z) − gη + σ

ρ K

• n

y = h + η

velocity potential φ; free surface

h + η

mean curvature K =

• ηx

√

1+η2

x+η2 z

• x

+

• ηz

√

1+η2

x+η2 z

• z

parameters ρ,

g, σ, h

SLIDE 5

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 ρ h 2

SLIDE 6

Euler equations

φ

xx + φ yy + φ zz = 0 fo r

0 <

y < 1 + η

φ

y = 0

• n

y = 0

φ

y = η t + η x + η xφ x + η zφ z

• n

y = 1 + η

φ

t + φ x + 1

2

• φ2

x + φ2 y + φ2 z

• + αη − β K = 0
• n

y = 1 + η

SLIDE 7

Euler equations

φ

xx + φ yy + φ zz = 0 fo r

0 <

y < 1 + η

φ

y = 0

• n

y = 0

φ

y = η t + η x + η xφ x + η zφ z

• n

y = 1 + η

φ

t + φ x + 1

2

• φ2

x + φ2 y + φ2 z

• + αη − β K = 0
• n

y = 1 + η

difficulties

variable domain (free surface) nonlinear boundary conditions

very rich dynamics

symmetries, Hamiltonian structures many particular solutions

SLIDE 8

Focus on . . .

traveling periodic 2D waves transverse stability/instability analytical results long-wave models

SLIDE 9

Two-dimensional periodic waves

exist in different parameter regimes

β

1 3

1 α

SLIDE 10

Two-dimensional periodic waves

transverse (in)stability

β

1 3

1 α

SLIDE 11

Large surface tension

transverse linear instability

longitudinal co-periodic perturbations transverse periodic perturbations

Euler equations

[H., 2015]

SLIDE 12

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

SLIDE 13

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 + L U ,

L =

F ′( U∗)

possesses a solution of the form

U( x, z, t) = eλ t Vλ( x, z)

with λ ∈ C,

Reλ > 0, Vλ bounded function.

SLIDE 14

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);

SLIDE 15

Main result

Theorem

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

U z = DU t + L U

possesses a solution of the form

U(·, z, t) = eλ t Vλ(·, z)

with

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

2

U∗ is transversely linearly unstable.

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

SLIDE 16

Euler equations

Hamiltonian formulation of the 3D problem:

U z = DU t + F( U)

boundary conditions

φ

y = b( U) t + g( U)

• n

y = 0, 1

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

SLIDE 17

Periodic waves

β > 1 3, α = 1 + ǫ , ǫ small

model: Kadomtsev-Petviashvili-I equation instability ∂x∂tu =∂x∂x(∂2

xu + u + 1

2u2)−∂2

yu

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

SLIDE 18

Periodic waves

β > 1 3, α = 1 + ǫ , ǫ small

model: Kadomtsev-Petviashvili-I equation instability ∂x∂tu =∂x∂x(∂2

xu + 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) = ε

p a(ε1/2 x, ε),

ϕǫ, a(x) = ε1/2

q a(ε1/2 x, ε) p a(ξ, 0) = ∂ξ q a(ξ, 0), p a(ξ, 0) satisfies the Korteweg de

Vries equation

[Kirchg¨ assner, 1989]

SLIDE 19

Linearized system

linearized system (rescaled)

U z = Dε U t + DFε( u a) U

boundary conditions

φ

y =Dbε( u a) U t + Dgε( u a) U

• n

y = 0, 1

SLIDE 20

Linearized system

linearized system (rescaled)

U z = Dε U t + DFε( u a) U

boundary conditions

φ

y =Dbε( u a) U t + Dgε( u a) U

• n

y = 0, 1

linear operator Lε :=

DFε( u a)

boundary conditions

φ

y = Dgε( u a) U

• n

y = 0, 1

space of symmetric functions (

x → − x)

Xs = H1

e (0, 2π) × L2 e(0, 2π) × H1

• ((0, 2π) × (0, 1)) × L2
• ((0, 2π) × (0, 1))

SLIDE 21

Linear operator Lε

Lε = L0

ε + L1 ε

L0

ε

      η ω φ ξ       =        ω β −εk2

a βηxx + (1 + ǫ)η − kaφx |y=1

ξ −εk2

a φxx − φyy

       , L1

ε

      η ω φ ξ       =       g1 g2 G1 G2       g1 = (1 + εk2

a η2 ax )1/2

β

• ω +

1 1 + εηa 1 yφay ξ dy

• −

ω β g2 = 1

• εk2

a φax φx −

φay φy (1 + εηa)2 + εφ2

ay η

(1 + εηa)3 − ε3k2

a y2η2 ax φay φy

(1 + εηa)2 − ε3k2

a y2ηax φ2 ay ηx

(1 + εηa)2 + ε3k2

a y2η2 ax φ2 ay η

(1 + εηa)3 +

• εk2

a yφay φx + εk2 a yφax φy −

2ε2k2

a y2ηax φay φy

1 + εηa − ε2k2

a y2φ2 ay ηx

1 + εηa + ε3k2

a y2ηax φ2 ay η

(1 + εηa)2

• x
• dy

+ εk2

a βηxx − εk2 a β

• ηx

(1 + ε3k2

a η2 ax )3/2

• x

G1 = − εηaξ 1 + εηa + (1 + ε3k2

a η2 ax )1/2

β(1 + εηa)

• ω +

1 1 + εηa 1 yφay ξ dy

• yφay

G2 =

• εηaφ

(1 + εηa) + εφaη (1 + ηa)2

• yy

− ε2k2

a [ηaφx + φax η − yφay ηx − yηax φy ]x

+ ε2k2

a

• yηax φx + yφax ηx +

ε2y2η2

ax φay η

(1 + εηa)2 − εy2η2

ax φy

1 + εηa − 2εy2ηax φay ηx 1 + εηa

• y

SLIDE 22

Check hypotheses . . .

Main difficulty: spectrum of Lε . . .

• perator with compact resolvent

− → pure point spectrum spectral analysis |λ| ≥ λ∗ |λ| ≤ λ∗

|λ| ≤ εℓ∗

SLIDE 23

Key step

Reduction to a scalar operator Bε,ℓ in L2

• (0, 2π)

scaling λ = εℓ,

ω = ε˜ ω, ξ = ε˜ ξ

decomposition φ( x,

y) = φ1( x) + φ2( x, y)

• λ = εℓ eigenvalue iff Bε,ℓφ1 = 0

Bε,ℓφ1 =

• β − 1

3

• k4

aφ1xxxx − k2 aφ1xx + ℓ2(1 + ǫ)φ1 − 3 k2 a( P aφ1x) x + . . .

SLIDE 24

. . . . . . . . .

ω = β (1 + ǫ3η⋆2

x )1/2 (η† + ikη) −

1 1 + ǫη⋆ 1 yΦ⋆

y ξdy,

ξ = (1 + ǫη⋆)(Φ† + ikΦ) − ǫyΦ⋆

y (η† + ikη)

(1 + ǫ) ǫ2 η − 1 ǫ2 Φx |y=1 − 1 ǫ βηxx − ikβ(hǫ

1 + ikη) = hǫ 2

− 1 ǫ Φxx − 1 ǫ2 Φyy − ik(Hǫ

1 + ikΦ) = Hǫ 2 ,

hǫ

2 = ω† − gǫ 2 ,

Hǫ

2 = ξ† − Gǫ 2

hǫ

1 =

ω β − ikη = − 1 β(1 + ǫη⋆) 1 yΦ⋆

y [−ǫyΦ⋆ y (ikη + η†) + (1 + ǫη⋆)(ikΦ + Φ†)]dy

+

• 1

(1 + ǫ3η⋆2

x )1/2 − 1

• ikη +

η† (1 + ǫ3η⋆2

x )1/2 ,

Hǫ

1 = ξ − ikΦ

= (1 + ǫη⋆)Φ† + ikǫη⋆Φ − ǫyΦ⋆

y (η† + ikη).

Bǫ(η, Φ) = −ǫηx + Bǫ

0 + Bǫ 1 ,

Bǫ = ǫη⋆Φy 1 + ǫη⋆ + ǫΦ⋆

y η

(1 + ǫη⋆)2

• y=1

, Bǫ

1

= ǫ2η⋆

x Φx + ǫ2Φ⋆ x ηx +

ǫ4η⋆2

x Φ⋆ y η

(1 + ǫη⋆)2 − ǫ3η⋆2

x Φy

1 + ǫη⋆ − −ˆ Φyy + q2 ˆ Φ = ǫ2( ˆ Hǫ

2 + ik ˆ

Hǫ

1 ),

0 < y < 1 ˆ Φy = 0, y = 0 ˆ Φy − ǫµ2 ˆ Φ 1 + ǫ + βq2 = − ǫ3iµ(ˆ hǫ

2 + ikβˆ

hǫ

1)

1 + ǫ + βq2 + ˆ Bǫ

0 + ˆ

Bǫ

1 , 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 −

SLIDE 25

. . . . . . . . .

ˆ Φ1 = 1 + ǫ ǫ2(k2(1 + ǫ) + µ2 + (β − 1/3)µ4) × 1 ǫ2( ˆ ξ† − iµ ˆ Gǫ

2,2 + ik ˆ

Hǫ

1 )dζ − ǫq2

1 ˆ pǫ

2 dζ

− ǫ3iµ(ˆ hǫ

2 + ikβˆ

hǫ

1 )

1 + ǫ + βq2 + ǫ2µ2 ˆ pǫ

2 |ζ=1

1 + ǫ + βq2

• ,

ˆ Φ2 = − 1 G1( ˆ ξ† − iµ ˆ Gǫ

2,2 + ik ˆ

Hǫ

1 )dζ −

1 G1ζ ˆ Gǫ

2,1dζ +

1 (ǫk2 + µ2)G1 ˆ pǫ

2 dζ + ǫˆ

pǫ

2

− G1|ζ=1

• −

ǫiµ(ˆ hǫ

2 + ikβˆ

hǫ

1)

1 + ǫ + βq2 + µ2 ˆ pǫ

2 |ζ=1

1 + ǫ + βq2

• ,

ˆ Φ = − 1 Gǫ2( ˆ ξ† − ıµ ˆ Gǫ

2,2 + ık ˆ

Hǫ

1 )dζ −

1 Gζǫ2 ˆ Gǫ

2,1dζ +

ǫ3ıµG|ζ=1(ˆ hǫ

2 + ıkβˆ

hǫ

1 )

1 + ǫ + βq2 + 1 ǫq2G ˆ pǫ

2 dζ + ǫˆ

pǫ

2 −

ǫ2µ2 1 1 ǫq2G ˆ pǫ

2 dζ + ǫˆ

pǫ

2 −

ǫ2µ2G|ζ=1 ˆ pǫ

2 |ζ=1

1 + ǫ + βq2 = 1 Gǫˆ pǫ

2ζζdζ − ǫG|ζ=1 ˆ

pǫ

2ζ|ζ=1 =

1 Gǫ2( ˆ Gǫ

2,0)ζζdζ − G|ζ=1 ˆ

Bǫ

0 ,

ˆ Φ1 + ˆ Φ2 = − 1 Gǫ2( ˆ ξ† − ıµ ˆ Gǫ

2,2 + ık ˆ

Hǫ

1 )dζ −

1 Gζǫ2 ˆ Gǫ

2,1dζ

+ ǫ3ıµG|ζ=1(ˆ hǫ

2 + ıkβˆ

hǫ

1)

1 + ǫ + βq2 + 1 ǫq2G ˆ pǫ

2 dζ + ǫˆ

pǫ

2 −

ǫ2µ2G|ζ=1 ˆ pǫ

2 |ζ=1

1 + ǫ + βq2 ,

SLIDE 26

. . . . . . . . .

ıµˆ hǫ

2

= ıµˆ ω† + ıµF

• −

1 ǫ2 1

• ǫΦ⋆

x Φx −

Φ⋆

y Φy

(1 + ǫη⋆)2 + ǫΦ⋆2

y η

(1 + ǫη⋆)3 − ǫ3y2η⋆2

x Φ⋆ y Φy

(1 + ǫη⋆)2 − ǫ3y2η⋆

x Φ⋆2 y ηx

(1 + ǫη⋆)2 + ǫ4y (1 + µ2 ǫ F 1

• yΦ⋆

y Φx + yΦ⋆ x Φy −

2ǫy2η⋆

x Φy Φ⋆ y

1 + ǫη⋆ − ǫy2Φ⋆2

y ηx

1 + ǫη⋆ + ǫ2y2η⋆

x Φ⋆2 y η

(1 + ǫη⋆)2

• dy
• −

βµ2 ǫ F

• (1 +

F−1

• ǫıµˆ

hǫ

2

1 + ǫ + βq2

• = −F−1
• 1

1 + ǫ + βq2 F[(Φ⋆

1x Φ1x )x ]

• + F−1
• µ2

1 + ǫ + βq2 F 1 yΦ⋆

x Φ2y dy

• +
• F−1
• −

1 1 + ǫ + βq2 F 1

• Φ⋆

2x Φ1x + Φ⋆ x Φ2x −

Φ⋆

y Φy

ǫ(1 + ǫη⋆)2 + Φ⋆2

y η

(1 + ǫη⋆)3 − ǫ2y2η⋆2

x Φ⋆ y Φy

(1 + ǫη⋆)2 − ǫ2y2η⋆

x Φ⋆2 y ηx

(1 + ǫη⋆)2 + ǫ3y2η⋆

x Φ⋆2 y η

(1 + ǫη⋆)3

• dy
• −

ıµ 1 + ǫ + βq2 F 1

• yΦ⋆

y Φx −

2ǫy2η⋆

x Φ⋆ y Φy

1 + ǫη⋆ − ǫy2Φ⋆2

y ηx

1 + ǫη⋆ + ǫ2y2η⋆

x Φ⋆2 y η

(1 + ǫη⋆)2

• dy
• +

βıµ 1 + ǫ + βq2 F

• ηx

(1 + ǫ3η⋆2

x )3/2 − ηx

• x

+ F−1

• ǫıµˆ

ω† 1 + ǫ + βq2

• =

−F−1

• 1

1 + ǫ + βq2 F[(Φ⋆

1x Φ1x )x ]

• + F−1
• µ2

1 + ǫ + βq2 F 1 yΦ⋆

x Φ2y dy

• + (L(ǫΦ1x, Φ2x , Φ2y , ǫ2η, ǫ4ηx ))x + ǫ−1/2(L(ǫΦx, ǫ2Φ2y , ǫ4η, ǫ3ηx ))x + ǫ1/2L(ω†),

SLIDE 27

. . . . . . . . .

F−1

• µ2

1 + ǫ + βq2 F 1 yΦ⋆

x Φ2y dy

• = F−1
• µ2

1 + ǫ + βq2 F

• Φ⋆

1x Φ2|y=1 −

1 Φ⋆

1x Φ2dy +

1 yΦ⋆

2x Φ2y

=

• F−1
• µ1/2

1 + ǫ + βq2 µ1/2F[Φ⋆

1x Φ2|y=1] −

1 1 + ǫ + βq2 1 (Φ⋆

1x Φ2)xdy

+ µ 1 + ǫ + βq2 1 yΦ⋆

2x Φ2y dy

• x

= ǫ−1/4(L(Φ2))x + (L(Φ2, Φ2x , ǫ1/2Φ2y ))x , F−1

• ǫıµˆ

hǫ

2

1 + ǫ + βq2

• = −F−1
• 1

1 + ǫ + βq2 F[(Φ⋆

1x Φ1x )x ]

• + ǫ−1/4(L(Φ2))x

+ ǫ−1/2(L(ǫΦx , ǫ2Φ2y , ǫ4η, ǫ3ηx )x + (L(ǫΦ1x, Φ2, Φ2x , Φ2y , ǫ2η, ǫ4ηx )x + H. F−1

• ǫıµ.ıkˆ

hǫ

1

1 + ǫ + βq2

• =

(L(Φ2, ǫ2η))x + ǫ2k2(L(Φ1))x + H, F−1

• µ2 ˆ

pǫ

2 |ζ=1

1 + ǫ + βq2

• = ǫ−1/4(L(Φ2, ǫη))

F−1

• (ǫk2 + µ2)

1 ˆ pǫ

2 dζ

• =

k2L(ǫΦ2, ǫ2η) + (L(Φ2, Φ2x , ǫη, ǫηx ))x 1 (ξ† − (Gǫ

2,2)x + ıkHǫ 1 )dζ = (η⋆Φ1x )x + (Φ⋆ 1x η)x + (L(Φ2x, Φ2y , ǫη, ǫηx ))x +

(β − 1/3)Φ1xxxx − Φ1xx + k2(1 + ǫ)Φ1 = (η⋆Φ1x )x + (Φ⋆

1x η)x + F−1

• 1

1 + ǫ + βq2 F[(Φ⋆

1x Φ1x )x ]

• + (L(ǫ1/2Φ1x , ǫ−1/4Φ2, Φ2x , Φ2y , ǫ3/4η, ǫηx ))x + k2[L(ǫΦ1, ǫΦ2, ǫ2η) + ǫ2L(Φ1)x ] + H,

η = F−1

• ıµˆ

Φ1 1 + ǫ + βq2

• + L(ǫΦ1x , ǫ3/4Φ2, Φ2x , Φ2y , ǫ3η, ǫ7/2ηx ) + k2ǫ3L(Φ1) + H.

SLIDE 28

Locate eigenvalues

|λ| ≤ εℓ∗ two simple eigenvalues ±iεκε

Bε,ℓ small relatively bounded perturbation of B0,ℓ

B0,ℓ =

k2 a∂ x A ∂ x + ℓ2

A =

• β − 1

3

• k2

a∂ xx − 1 − 3 P a

SLIDE 29

Locate eigenvalues

|λ| ≤ εℓ∗ two simple eigenvalues ±iεκε

Bε,ℓ small relatively bounded perturbation of B0,ℓ

B0,ℓ =

k2 a∂ x A ∂ x + ℓ2

A =

• β − 1

3

• k2

a∂ xx − 1 − 3 P a

spectrum of ∂

xA∂ x is known (KP-I):

• ne simple negative eigenvalue −ω2

a

perturbation arguments . . . . . .

SLIDE 30

Critical surface tension

transverse linear instability

longitudinal co-periodic perturbations transverse periodic perturbations

5th order KP model

[H. & Wahl´ en, 2017]

SLIDE 31

A 5th order KP model

∂

t∂ x u = ∂2 x

• ∂4

x u + ∂2 x u + 1

2

u2

• + ∂2

y u

traveling generalized solitary waves

solutions of the Kawahara equation

∂

t u = ∂ x

• ∂4

x u + ∂2 x u − u + 1

2

u2

SLIDE 32

Periodic traveling waves

small periodic traveling waves: a two-parameter family ϕ

a, ( x) = p a, ( k a, x)

depend analytically upon (a, c) ∈ (−a0, a0) × (−c0, c0) ka,c = k0(c) + c

k(a, c),

k0(c) = 1+√1+4c

2

1/2 , k(a, c) =

n≥1

k2n(c)a2n 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 = ±1) explicit Taylor expansions for

k(a, c), pn,m(c)

[Lombardi, 2000]

SLIDE 33

Transverse instability problem

• ne-dimensional periodic wave

u∗ u∗ is transversely linearly unstable if the linearized equation

∂

t∂ x u = ∂2 x

• ∂4

x u + ∂2 x u − u + u∗ u

• + ∂2

y u

possesses a solution of the form

u( t, x, y) = eλ t v( x, y) ,

for some Re λ > 0 (v belongs to the set of the allowed perturbations)

SLIDE 34

Transverse instability problem

linearized equation ∂

t∂ x u = A∗ u + ∂2 y u ,

A∗ = ∂2

x

• ∂4

x + ∂2 x − + u∗

• Fourier transform in

y

∂

t∂ x u = A∗ u − ω2 u

SLIDE 35

Transverse instability problem

linearized equation ∂

t∂ x u = A∗ u + ∂2 y u ,

A∗ = ∂2

x

• ∂4

x + ∂2 x − + u∗

• Fourier transform in

y

∂

t∂ x u = A∗ u − ω2 u u∗ is transversely unstable if there exists a solution of the

form

u( t, x) = eλ t v( x) , for some Re λ > 0, and ω ∈ R∗

• v ∈

H, a space of functions depending upon the longitudinal

spatial variable x, e.g.,

H = L2(R) or H = L2(0, L), and

λ∂

x v = A∗ v − ω2 v

SLIDE 36

Transverse instability problem

for some Re λ > 0 and ω ∈ R∗ , there exists a solution λ∂

x v = A∗ v − ω2 v , 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 Re λ > 0.

SLIDE 37

Transverse instability problem

u∗ is transversely spectrally unstable if the spectrum

• f the linear operator λ∂

x − A∗ contains a negative value

−ω2 < 0 for some Re λ > 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

SLIDE 38

Periodic waves

small periodic waves: ϕ

a, ( x) = p a, ( k a, x)

scaling:

z = k a, x,

λ =

k a, Λ

rescaled operator Λ∂

z − B a, ,

B

a, = ∂2 z ( k4 a, ∂4 z + k2 a, ∂2 z − + p a, )

with 2π-periodic coefficients

SLIDE 39

Co-periodic perturbations

Λ∂

z − B a, ,

B

a, = ∂2 z ( k4 a, ∂4 z + k2 a, ∂2 z − + p a, )

closed operator in

H = L2(0, 2π)

Theorem

1 the linear operator Λ∂

z − B a, acting in L2(0, 2π) has a

simple negative eigenvalue.

2 the periodic wave ϕ

a, is transversely linearly unstable

with respect to co-periodic longitudinal perturbations.

SLIDE 40

Proof

Λ∂

z − B a, ,

B

a, = ∂2 z ( k4 a, ∂4 z + k2 a, ∂2 z − + p a, )

show that B

a, has a simple positive eigenvalue

the operator Λ∂

z − B a, is real

perturbation argument: the negative eigenvalue of −B

a,

persists for small real Λ

(point) spectrum of B

a, ?

SLIDE 41

Proof

Spectrum of B

a, = ∂2 z( k4 a, ∂4 z + k2 a, ∂2 z − + p a, )

use perturbation arguments: small

a and

SLIDE 42

Proof

Spectrum of B

a, = ∂2 z( k4 a, ∂4 z + k2 a, ∂2 z − + p a, )

use perturbation arguments: small

a and a = 0, = 0

B0,0 = ∂2

z

• ∂4

z + ∂2 z

• ,

σ(B0,0) = {−

n2( n4 − n2), n ∈ Z}

0 is a triple eigenvalue all other eigenvalues are negative

SLIDE 43

Proof

Spectrum of B

a, = ∂2 z( k4 a, ∂4 z + k2 a, ∂2 z − + p a, )

use perturbation arguments: small

a and a = 0, = 0

B0,0 = ∂2

z

• ∂4

z + ∂2 z

• ,

σ(B0,0) = {−

n2( n4 − n2), n ∈ Z}

0 is a triple eigenvalue all other eigenvalues are negative

spectral decomposition for small

a and

σ(B

a, ) = σ1(B a, ) ∪ σ2(B a, )

σ1(B

a, ) ⊂ V , V neighborhood of 0

σ2(B

a, ) ⊂ {ν ∈ C ; Re ν < − m}

SLIDE 44

Proof

Spectrum of B

a, = ∂2 z( k4 a, ∂4 z + k2 a, ∂2 z − + p a, ) :

locate the small eigenvalues

SLIDE 45

Proof

Spectrum of B

a, = ∂2 z( k4 a, ∂4 z + k2 a, ∂2 z − + p a, ) :

locate the small eigenvalues

a = 0

σ(B0, ) = {−

n2( k2 n4 − k2 n2 − ), n ∈ Z}

0 is a triple eigenvalue all other eigenvalues are negative

SLIDE 46

Proof

Spectrum of B

a, = ∂2 z( k4 a, ∂4 z + k2 a, ∂2 z − + p a, ) :

locate the small eigenvalues

a = 0

σ(B0, ) = {−

n2( k2 n4 − k2 n2 − ), n ∈ Z}

0 is a triple eigenvalue all other eigenvalues are negative

a = 0

use symmetries and show that 0 is a double eigenvalue third eigenvalue: compute an expansion for small

a, . . .

. . . , ν

a, = a2 2

1 4 X2 +

O( a2 + 2)

• > 0

SLIDE 47

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 . . . ?

SLIDE 48

Zero surface tension

transverse spectral stability

fully localized/bounded perturbations

KP-II equation

[H., Li, & Pelinovsky, 2017]

SLIDE 49

Count unstable eigenvalues

Hamiltonian structure: linear operator of the form J L

J skew-adjoint operator L self-adjoint operator

Under suitable conditions:

n u(J L) ≤ n s(L)

• n

u(J L) = number of unstable eigenvalues of J L

• 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 . . . ]

SLIDE 50

An extended eigenvalue count

Hamiltonian structure: linear operator of the form J L

J skew-adjoint operator L self-adjoint operator

There exists a self-adjoint operator K such that (J L)(J K) = (J K)(J L) Under suitable conditions:

n u(J L) ≤ n s(K)

• n

u(J L) = number of unstable eigenvalues of J L

• n

s(K) = number of negative eigenvalues of K

SLIDE 51

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]

SLIDE 52

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)

SLIDE 53

KP-II equation

Kadomtsev-Petviashivili equation ( u

t + 6uu x + u xxx) x + u yy = 0

• ne-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) + 3 v 2( x) = 0

known explicitly!

SLIDE 54

Linearized equation

linearized KP-II equation ( w

t + w xxx + w x + 6(φ ( x) w) x) x + w yy = 0

2π-periodic coefficients in

x

Ansatz

w( x, y, t) = eλ t+ ip y W ( x),

λ ∈ C,

p ∈ R

linearized equation for

W ( x)

λW

x + W xxxx + W xx + 6(φ ( x) W ) xx − p2 W = 0

SLIDE 55

Spectral stability problem

linearized equation for

W ( x)

λ W

x + W xxxx + W xx + 6(φ ( x) W ) xx − p2 W = 0

the periodic wave φ

is spectrally stable iff the linear

• perator

A

, p(λ) = λ∂ x + ∂4 x + ∂2 x + 6∂2 x(φ ( x) ·) − p2

is invertible for Re λ > 0.

2D bounded perturbations: space

C b(R) and p ∈ R.

continuous spectrum . . .

SLIDE 56

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) ·) − p2

are invertible in

L2 p er(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) ·) + p2(∂ x + iγ)−1

• γ = 0 : restrict to functions with zero mean

SLIDE 57

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) + p2(∂ 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

SLIDE 58

Commuting operators

natural candidate: use a higher-order conserved functional

resulting operator satisfies the commutativity relation cannot obtain positive operators . . .

SLIDE 59

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 + p2LKP, M , p = MKdV + p2MKP

MKdV is obtained from a higher order conserved functional:

MKdV = ∂4

x + 10∂ xφ ( x)∂ x − 10 φ ( x) − 2

compute MKP directly from the commutativity relation:

MKP = 5 3

• 1 +

∂−2 x

SLIDE 60

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

SLIDE 61

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

SLIDE 62

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?) . . .

SLIDE 63

50