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

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

Water-wave problem gravity/gravity-capillary waves � three-dimensional inviscid fluid layer � constant density � gravity/gravity and surface tension � irrotational flow

Water-wave problem 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

Euler equations Laplace’s equation φ xx + φ yy + φ zz = 0 D η in boundary conditions φ = 0 y = 0 on y η = φ y − η x φ x − η z φ y = h + η on t z − 1 g η + σ 2( φ 2 x + φ 2 y + φ 2 φ = z ) − ρ K y = h + η on t � velocity potential φ ; free surface h + η � � � � � mean curvature K = √ √ η x + η z 1+ η 2 x + η 2 1+ η 2 x + η 2 z z x 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 σ � Weber number β = ρ h 2

Euler equations φ xx + φ yy + φ zz = 0 0 < y < 1 + η fo r φ y = 0 y = 0 on φ y = η t + η x + η x φ x + η z φ y = 1 + η � � on z x + 1 φ 2 x + φ 2 y + φ 2 t + φ + αη − β K = 0 y = 1 + η φ 2 on z

Euler equations φ xx + φ yy + φ zz = 0 0 < y < 1 + η fo r φ y = 0 y = 0 on φ y = η t + η x + η x φ x + η z φ y = 1 + η � � on z x + 1 φ 2 x + φ 2 y + φ 2 t + φ + αη − β K = 0 y = 1 + η φ 2 on z 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 α 1 1 β 3

Two-dimensional periodic waves � transverse (in)stability α 1 1 β 3

Large surface tension transverse linear instability � longitudinal co-periodic perturbations � transverse periodic perturbations Euler equations [H., 2015]

Transverse instability problem Transverse spatial dynamics z = t + F ( U ) U DU 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 z = t + F ( U ) U DU U ∗ ( x ) is transversely linearly unstable if the linearized system F ′ ( U ∗ ) z = t + L U , L = U DU t ) = e λ possesses a solution of the form U ( x , z , V λ ( x , z ) t with λ ∈ C , Re λ > 0 , V λ bounded function.

Hypotheses 1 the system z = t + F ( U ) is reversible/Hamiltonian ; U DU F ′ ( U ∗ ) possesses a pair of 2 the linear operator L = simple purely imaginary eigenvalues ± i κ ∗ ; 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 z = t + L U U DU t ) = e λ possesses a solution of the form U ( · , V λ ( · , z ) z , t with V λ ( · , z ) ∈ D ( L ) a periodic function in z . U ∗ is transversely linearly unstable . 2 [Godey, 2016; see also Rousset & Tzvetkov, 2010]

Euler equations Hamiltonian formulation of the 3D problem: z = t + F ( U ) U DU � boundary conditions y = b ( U ) t + g ( U ) y = 0 , 1 φ on ( e.g. [Groves, H., Sun, 2002])

Periodic waves β > 1 3 , α = 1 + ǫ , ǫ small model: Kadomtsev-Petviashvili-I equation � instability x u + u + 1 ∂ x ∂ t u = ∂ x ∂ x ( ∂ 2 2 u 2 ) − ∂ 2 y u [H.; Johnson & Zumbrun; Hakkaev, Stanislavova & Stefanov, . . . ]

Periodic waves β > 1 3 , α = 1 + ǫ , ǫ small model: Kadomtsev-Petviashvili-I equation � instability x u + u + 1 ∂ x ∂ t u = ∂ x ∂ x ( ∂ 2 2 u 2 ) − ∂ 2 y u [H.; Johnson & Zumbrun; Hakkaev, Stanislavova & Stefanov, . . . ] The Euler equations possess a one-parameter family of symmetric periodic waves a ( ε 1 / 2 ϕ ǫ, a ( x ) = ε 1 / 2 a ( ε 1 / 2 η ǫ, a ( x ) = ε x , ε ) , x , ε ) p q a ( ξ, 0) = ∂ ξ a ( ξ, 0) , a ( ξ, 0) satisfies the Korteweg de p q p Vries equation [Kirchg¨ assner, 1989]

Linearized system linearized system (rescaled) z = t + DF ε ( u a ) U D ε U U � boundary conditions φ y = Db ε ( u a ) U t + Dg ε ( u a ) U y = 0 , 1 on

Linearized system linearized system (rescaled) z = t + DF ε ( u a ) U D ε U U � boundary conditions φ y = Db ε ( u a ) U t + Dg ε ( u a ) U y = 0 , 1 on linear operator L ε := DF ε ( u a ) � boundary conditions φ y = Dg ε ( u a ) U y = 0 , 1 on � space of symmetric functions ( x → − x ) X s = H 1 e (0 , 2 π ) × L 2 e (0 , 2 π ) × H 1 o ((0 , 2 π ) × (0 , 1)) × L 2 o ((0 , 2 π ) × (0 , 1))

Linear operator L ε     ω     η η g 1   β         L ε = L 0 ε + L 1  ω  − ε k 2  ω   g 2    a βη xx + (1 + ǫ ) η − k a φ x | y =1 L 0   , L 1     = =   ε ε   ε       φ φ G 1        ξ  ξ − ε k 2 ξ G 2 a φ xx − φ yy � � 1 � (1 + ε k 2 a η 2 ax ) 1 / 2 1 ω = ω + y φ ay ξ dy − g 1 β 1 + εη a β 0 � � 1 εφ 2 ε 3 k 2 a y 2 η 2 ε 3 k 2 a y 2 η ax φ 2 ε 3 k 2 a y 2 η 2 ax φ 2 ay η ay η x ay η φ ay φ y ax φ ay φ y ε k 2 g 2 = a φ ax φ x − (1 + εη a ) 2 + (1 + εη a ) 3 − − + (1 + εη a ) 2 (1 + εη a ) 2 (1 + εη a ) 3 0 � � � ε 2 k 2 a y 2 φ 2 ε 3 k 2 a y 2 η ax φ 2 2 ε 2 k 2 a y 2 η ax φ ay φ y ay η x ay η ε k 2 a y φ ay φ x + ε k 2 + a y φ ax φ y − − + dy (1 + εη a ) 2 1 + εη a 1 + εη a x � � η x + ε k 2 a βη xx − ε k 2 a β (1 + ε 3 k 2 a η 2 ax ) 3 / 2 x � � 1 � (1 + ε 3 k 2 a η 2 ax ) 1 / 2 εη a ξ 1 G 1 = − + ω + y φ ay ξ dy y φ ay 1 + εη a β (1 + εη a ) 1 + εη a 0 � � εη a φ εφ a η − ε 2 k 2 G 2 = + a [ η a φ x + φ ax η − y φ ay η x − y η ax φ y ] x (1 + η a ) 2 (1 + εη a ) yy � � ε 2 y 2 η 2 ε y 2 η 2 2 ε y 2 η ax φ ay η x ax φ ay η ax φ y + ε 2 k 2 y η ax φ x + y φ ax η x + − − a (1 + εη a ) 2 1 + εη a 1 + εη a y

Check hypotheses . . . Main difficulty: spectrum of L ε . . . operator with compact resolvent − → pure point spectrum spectral analysis | λ | ≥ λ ∗ | λ | ≤ λ ∗ | λ | ≤ εℓ ∗

Key step Reduction to a scalar operator B ε,ℓ in L 2 o (0 , 2 π ) ξ = ε ˜ � scaling λ = εℓ, ω = ε ˜ ω, ξ � decomposition φ ( x , y ) = φ 1 ( x ) + φ 2 ( x , y ) λ = εℓ eigenvalue iff B ε,ℓ φ 1 = 0 � � � β − 1 k 4 k 2 a φ 1 xx + ℓ 2 (1 + ǫ ) φ 1 − 3 k 2 B ε,ℓ φ 1 = a φ 1 xxxx − a ( P a φ 1 x ) x + . . . 3

. . . . . . . . . � 1 B ǫ ( η, Φ) = − ǫη x + B ǫ 0 + B ǫ 1 , β 1 x ) 1 / 2 ( η † + i k η ) − y Φ ⋆ ω = y ξ dy , (1 + ǫ 3 η ⋆ 2 1 + ǫη ⋆ 0 ξ = (1 + ǫη ⋆ )(Φ † + i k Φ) − ǫ y Φ ⋆ y ( η † + i k η ) � ǫ Φ ⋆ ǫη ⋆ Φ y � y η � B ǫ = 1 + ǫη ⋆ + , � 0 (1 + ǫη ⋆ ) 2 � y =1 ǫ 4 η ⋆ 2 x Φ ⋆ ǫ 3 η ⋆ 2 y η x Φ y (1 + ǫ ) 1 1 B ǫ ǫ 2 η ⋆ x Φ x + ǫ 2 Φ ⋆ βη xx − i k β ( h ǫ 1 + i k η ) = h ǫ = x η x + (1 + ǫη ⋆ ) 2 − 1 + ǫη ⋆ − η − ǫ 2 Φ x | y =1 − 1 2 ǫ 2 ǫ 1 1 ǫ 2 Φ yy − i k ( H ǫ 1 + i k Φ) = H ǫ − Φ xx − 2 , ǫ Φ yy + q 2 ˆ − ˆ Φ = ǫ 2 ( ˆ H ǫ 2 + i k ˆ H ǫ 1 ) , 0 < y < 1 2 = ω † − g ǫ h ǫ 2 , ˆ Φ y = 0 , y = 0 2 = ξ † − G ǫ H ǫ 2 ǫµ 2 ˆ ǫ 3 i µ (ˆ h ǫ h ǫ 2 + i k β ˆ Φ 1 ) ω B ǫ B ǫ h ǫ ˆ + ˆ 0 + ˆ Φ y − 1 + ǫ + β q 2 = − 1 , y = 1 1 = − i k η 1 + ǫ + β q 2 β � 1 1 y Φ ⋆ y [ − ǫ y Φ ⋆ y ( i k η + η † ) + (1 + ǫη ⋆ )( i k Φ + Φ † )] dy = − β (1 + ǫη ⋆ ) 0  (1 + ǫ + β q 2 ) cosh q (1 − ζ ) + ( ǫµ 2 / q ) cosh qy � �   η †  1  q 2 − (1 + ǫ + β q 2 ) q tanh q − ǫ  cosh q + x ) 1 / 2 − 1 i k η + x ) 1 / 2 ,  (1 + ǫ 3 η ⋆ 2 (1 + ǫ 3 η ⋆ 2 G ( y , ζ ) =   (1 + ǫ + β q 2 ) cosh q (1 − y ) + ( ǫµ 2 / q )  H ǫ cosh q ζ  1 = ξ − i k Φ   q 2 − (1 + ǫ + β q 2 ) q tanh q − cosh q = (1 + ǫη ⋆ )Φ † + i k ǫη ⋆ Φ − ǫ y Φ ⋆ y ( η † + i k η ) .