On Whithams modulated equations for the EulerKorteweg system S. - - PowerPoint PPT Presentation

On Whitham’s modulated equations for the Euler–Korteweg system

• S. Benzoni-Gavage
• P. Noble

L.M. Rodrigues

Institut Camille Jordan Universit´ e Claude Bernard Lyon 1

June 26, 2012

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The Euler–Korteweg system and travelling waves

Outline

1

The Euler–Korteweg system and travelling waves

2

Modulated equations and periodic waves

2 / 21

The Euler–Korteweg system and travelling waves

General model for isothermal capillary fluids

Euler–Lagrange equations for Lagrangian 1 2ρu2 − E (ρ, ∇ρ) − ρ∂tϕ − ρ u · ∇ϕ in coordinates (ρ, ϕ, p) ✇

• ∂tρ + div(ρu) = 0 ,

∂tu + (u · ∇)u + ∇(EρE ) = 0 , ρ = density, u = velocity, E = free energy density, EρE = variational derivative of E . ②

3 / 21

The Euler–Korteweg system and travelling waves

General model for isothermal capillary fluids

Euler–Lagrange equations for Lagrangian 1 2ρu2 − E (ρ, ∇ρ) − ρ∂tϕ − ρ u · ∇ϕ in coordinates (ρ, ϕ, p) ✇

• ∂tρ + div(ρu) = 0 ,

∂tu + (u · ∇)u + ∇(EρE ) = 0 , ρ = density, u = velocity, E = free energy density, EρE = variational derivative of E . Standard compressible fluids: E = E (ρ), EρE = dE

dρ .

② ‘compressible’ Euler equations.

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The Euler–Korteweg system and travelling waves

General model for isothermal capillary fluids

Euler–Lagrange equations for Lagrangian 1 2ρu2 − E (ρ, ∇ρ) − ρ∂tϕ − ρ u · ∇ϕ in coordinates (ρ, ϕ, p) ✇

• ∂tρ + div(ρu) = 0 ,

∂tu + (u · ∇)u + ∇(EρE ) = 0 , ρ = density, u = velocity, E = free energy density, EρE = variational derivative of E . Standard compressible fluids: E = E (ρ), EρE = dE

dρ .

② ‘compressible’ Euler equations. Capillary fluids: E = E (ρ, ∇ρ), EρE := ∂E ∂ρ − d

j=1 Dxj

∂E ∂ρxj

• .

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The Euler–Korteweg system and travelling waves

Special cases

Korteweg capillarity theory (after Rayleigh, van der Waals,...), see [Rowlinson & Widom’82] E (ρ, ∇ρ) = F(ρ) + 1 2 K (ρ) ∇ρ2 . ② ②

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The Euler–Korteweg system and travelling waves

Special cases

Korteweg capillarity theory (after Rayleigh, van der Waals,...), see [Rowlinson & Widom’82] E (ρ, ∇ρ) = F(ρ) + 1 2 K (ρ) ∇ρ2 . Quantum fluids (Schr¨

• dinger, Madelung, Gross–Pitaevskii)

∂tρ + div(ρu) = 0 , ∂tu + (u · ∇)u + ∇

• F ′(ρ) − ∆√ρ

2√ρ

• = 0 .

② ρ K (ρ) ≡ 1

4.

②

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The Euler–Korteweg system and travelling waves

Special cases

Korteweg capillarity theory (after Rayleigh, van der Waals,...), see [Rowlinson & Widom’82] E (ρ, ∇ρ) = F(ρ) + 1 2 K (ρ) ∇ρ2 . Quantum fluids (Schr¨

• dinger, Madelung, Gross–Pitaevskii)

∂tρ + div(ρu) = 0 , ∂tu + (u · ∇)u + ∇

• F ′(ρ) − ∆√ρ

2√ρ

• = 0 .

② ρ K (ρ) ≡ 1

4.

Vortex-filaments (Levi-Civita & Da Rios, Hasimoto), see [Arnold & Khesin’98] ∂tρ + ∂x(ρu) = 0 , ∂tu + u∂xu = ∂x ρ 4 + ∂2

x

√ρ 2√ρ

• .

② ρ K (ρ) ≡ 1

4 , F(ρ) = − 1 8ρ2 , d = 1.

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The Euler–Korteweg system and travelling waves

Two formulations of 1D model

Eulerian coordinates

• ∂tρ + ∂x(ρu) = 0 ,

∂tu + u∂xu + ∂x(EρE ) = 0 , ρ = density, u = velocity, E = E (ρ, ρx) energy density, EρE := ∂E ∂ρ − Dx ∂E ∂ρx

• ⑨❡

⑨❡ ⑨❡ ⑨❡ ⑨❡ ⑨❡

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The Euler–Korteweg system and travelling waves

Two formulations of 1D model

Eulerian coordinates

• ∂tρ + ∂x(ρu) = 0 ,

∂tu + u∂xu + ∂x(EρE ) = 0 , ρ = density, u = velocity, E = E (ρ, ρx) energy density, EρE := ∂E ∂ρ − Dx ∂E ∂ρx

• Mass Lagrangian coordinates
• dtv = ∂yu ,

dtu = ∂y(Ev⑨❡) , v = specific volume, u = velocity, ⑨❡ = ⑨❡(v, vy) specific energy, Ev⑨❡ := ∂⑨❡ ∂v − Dy ∂⑨❡ ∂vy

• 5 / 21

The Euler–Korteweg system and travelling waves

Two formulations of 1D model

Eulerian coordinates

• ∂tρ + ∂x(ρu) = 0 ,

(∂t + u∂x)u + ∂x(EρE ) = 0 , ρ = density, u = velocity, E = E (ρ, ρx) energy density, EρE := ∂E ∂ρ − Dx ∂E ∂ρx

• Mass Lagrangian coordinates
• dtv = ∂yu ,

dtu = ∂y(Ev⑨❡) , v = specific volume, u = velocity, ⑨❡ = ⑨❡(v, vy) specific energy, Ev⑨❡ := ∂⑨❡ ∂v − Dy ∂⑨❡ ∂vy

• dy = ρ dx − ρu dt ⇔ dx = v dy + u dt

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The Euler–Korteweg system and travelling waves

Two formulations of 1D model

Eulerian coordinates

• ∂tρ + ∂x(ρu) = 0 ,

∂tu + u∂xu + ∂x(EρE ) = 0 , ρ = density, u = velocity, E = E (ρ, ρx) energy density, EρE := ∂E ∂ρ − Dx ∂E ∂ρx

• Mass Lagrangian coordinates
• dtv = ∂yu ,

dtu = ∂y(Ev⑨❡) , v = specific volume, u = velocity, ⑨❡ = ⑨❡(v, vy) specific energy, Ev⑨❡ := ∂⑨❡ ∂v − Dy ∂⑨❡ ∂vy

• dy = ρ dx − ρu dt ⇔ dx = v dy + u dt

∂x(EρE ) = − ∂y(Ev⑨❡)

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The Euler–Korteweg system and travelling waves

More special cases

Korteweg / Cahn–Hilliard energy again E (ρ, ρx) = F(ρ) + 1 2K (ρ) ρ2

x

⇔ ⑨❡(v, vy) = f (v) + 1 2 κ(v) v2

y ,

with F(ρ) = ρf (v) , κ(v) := ρ5 K (ρ) . Water waves [Boussinesq’72], [Bona & Sachs’88]: ∂tv = ∂yu , ∂tu − gH ∂y(v +

3 2H v2) = 1 3 gH3 ∂3 yv ,

② κ = − 1

3 gH3 , f (v) = 1 2 gH v2 (1 + v H ).

H y g v 6 / 21

The Euler–Korteweg system and travelling waves

More special cases

Korteweg / Cahn–Hilliard energy again E (ρ, ρx) = F(ρ) + 1 2K (ρ) ρ2

x

⇔ ⑨❡(v, vy) = f (v) + 1 2 κ(v) v2

y ,

with F(ρ) = ρf (v) , κ(v) := ρ5 K (ρ) . Water waves [Boussinesq’72], [Bona & Sachs’88]: ∂tv = ∂yu , ∂tu − gH ∂y(v +

3 2H v2) = 1 3 gH3 ∂3 yv ,

② κ = − 1

3 gH3 , f (v) = 1 2 gH v2 (1 + v H ).

H y g v

van der Waals fluids

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The Euler–Korteweg system and travelling waves

Travelling wave profiles

Eulerian coordinates (ρ, u) = (R, U)(x − σt) solution of

• ∂tρ + ∂x(ρu) = 0 ,

∂tu + u∂xu + ∂x(EρE ) = 0 , iff

• ∂ξ(R(U − σ)) = 0 ,

(U − σ)∂ξU + ∂ξ(EρE ) = 0 . Lagrangian coordinates (v, u) = (V , W )(y +jt) solution of

• dtv = ∂yu ,

dtu = ∂y(Ev⑨❡) , iff

• ∂ζ(W − j V ) = 0 ,

∂ζ(Ev⑨❡ − j W ) = 0 .

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The Euler–Korteweg system and travelling waves

Travelling wave profiles

Eulerian coordinates (ρ, u) = (R, U)(x − σt) solution of

• ∂tρ + ∂x(ρu) = 0 ,

∂tu + u∂xu + ∂x(EρE ) = 0 , iff

• R(U − σ) ≡ j ,

(U − σ)∂ξU + ∂ξ(EρE ) = 0 . Lagrangian coordinates (v, u) = (V , W )(y +jt) solution of

• dtv = ∂yu ,

dtu = ∂y(Ev⑨❡) , iff

• W − j V ≡ σ ,

∂ζ(Ev⑨❡ − j W ) = 0 .

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The Euler–Korteweg system and travelling waves

Travelling wave profiles

Eulerian coordinates (ρ, u) = (R, U)(x − σt) solution of

• ∂tρ + ∂x(ρu) = 0 ,

∂tu + u∂xu + ∂x(EρE ) = 0 , iff

• R(U − σ) ≡ j ,

(U − σ)∂ξU + ∂ξ(EρE ) = 0 . Lagrangian coordinates (v, u) = (V , W )(y +jt) solution of

• dtv = ∂yu ,

dtu = ∂y(Ev⑨❡) , iff

• W − j V ≡ σ ,

∂ζ(Ev⑨❡ − j W ) = 0 . R(ξ)V (Z(ξ)) = 1 , U(ξ) = W (Z(ξ)) , dZ dξ = R = 1 V (Z) .

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The Euler–Korteweg system and travelling waves

Travelling wave profiles

Eulerian coordinates (ρ, u) = (R, U)(x − σt) solution of

• ∂tρ + ∂x(ρu) = 0 ,

∂tu + u∂xu + ∂x(EρE ) = 0 , iff

• R(U − σ) ≡ j ,

(U − σ)∂ξU + ∂ξ(EρE ) = 0 . Lagrangian coordinates (v, u) = (V , W )(y +jt) solution of

• dtv = ∂yu ,

dtu = ∂y(Ev⑨❡) , iff

• W − j V ≡ σ ,

∂ζ(Ev⑨❡ − j W ) = 0 . R(ξ)V (Z(ξ)) = 1 , U(ξ) = W (Z(ξ)) , dZ dξ = R = 1 V (Z) . Proposition Relationships above yield a one-to-one correspondence between Eulerian travelling waves s.t. R(R) is compact in R+∗, and Lagrangian travelling waves s.t. V (R) is compact in R+∗.

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The Euler–Korteweg system and travelling waves

Key to one-to-one correspondence

• PDEs solutions, Eulerian vs mass Lagrangian coordinates: x ↔ ξ ↔ y.

ρ(χ(ξ, t), t) v(y(χ(ξ, t), t), t) = 1 , u(χ(ξ, t), t) = w(y(χ(ξ, t), t), t) , ∂tχ = u(χ, t) , χ(ξ, 0) = ξ , y(χ(ξ, t), t) = y0(ξ) , y0′(ξ) = ρ(ξ, 0) .

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The Euler–Korteweg system and travelling waves

Key to one-to-one correspondence

• PDEs solutions, Eulerian vs mass Lagrangian coordinates: x ↔ ξ ↔ y.

ρ(χ(ξ, t), t) v(y(χ(ξ, t), t), t) = 1 , u(χ(ξ, t), t) = w(y(χ(ξ, t), t), t) , ∂tχ = u(χ, t) , χ(ξ, 0) = ξ , y(χ(ξ, t), t) = y0(ξ) , y0′(ξ) = ρ(ξ, 0) .

• A travelling wave solution in Eulerian coordinates is such that

u(χ, t) = U(χ − σt) ⇒ ∂t(χ − σt) = U(χ − σt) − σ = j/R(χ − σt)

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The Euler–Korteweg system and travelling waves

Key to one-to-one correspondence

• PDEs solutions, Eulerian vs mass Lagrangian coordinates: x ↔ ξ ↔ y.

ρ(χ(ξ, t), t) v(y(χ(ξ, t), t), t) = 1 , u(χ(ξ, t), t) = w(y(χ(ξ, t), t), t) , ∂tχ = u(χ, t) , χ(ξ, 0) = ξ , y(χ(ξ, t), t) = y0(ξ) , y0′(ξ) = ρ(ξ, 0) .

• A travelling wave solution in Eulerian coordinates is such that

u(χ, t) = U(χ − σt) ⇒ ∂t(χ − σt) = U(χ − σt) − σ = j/R(χ − σt) ⇒ ∂t(Z(χ − σt)) = j with Z ′ = R. Z(χ(ξ, t) − σt) = Z(ξ) + j t

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The Euler–Korteweg system and travelling waves

One-to-one correspondence in practice

Profile equations are Euler–Lagrange equations [Benjamin’72].

• R(U − σ) ≡ j ,

(U − σ)∂ξU + ∂ξ(EρE ) = 0 .

• W − j V ≡ σ ,

∂ζ(Ev⑨❡ − j W ) = 0 . ⑨❡

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The Euler–Korteweg system and travelling waves

One-to-one correspondence in practice

Profile equations are Euler–Lagrange equations [Benjamin’72].

• R(U − σ) ≡ j ,

EρL = 0 , L : = E − j2 2ρ − µρ .

• W − j V ≡ σ ,

Evℓ = 0 , ℓ := ⑨❡ − j2v2 2 − λv .

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The Euler–Korteweg system and travelling waves

One-to-one correspondence in practice

Profile equations are Euler–Lagrange equations [Benjamin’72].

• R(U − σ) ≡ j ,

EρL = 0 , L : = E − j2 2ρ − µρ .

• W − j V ≡ σ ,

Evℓ = 0 , ℓ := ⑨❡ − j2v2 2 − λv . First integrals LρL := ρx ∂L

∂ρx − L ,

Lvℓ := vy

∂ℓ ∂vy − ℓ .

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The Euler–Korteweg system and travelling waves

One-to-one correspondence in practice

Profile equations are Euler–Lagrange equations [Benjamin’72].

• R(U − σ) ≡ j ,

EρL = 0 , L : = E − j2 2ρ − µρ .

• W − j V ≡ σ ,

Evℓ = 0 , ℓ := ⑨❡ − j2v2 2 − λv . First integrals LρL := ρx ∂L

∂ρx − L ,

Lvℓ := vy

∂ℓ ∂vy − ℓ .

EρL = − v Evℓ − Lvℓ − µ , Evℓ = − ρ EρL − LρL − λ EρL = 0 LρL = − λ

• ⇔

Evℓ = 0 Lvℓ = − µ

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The Euler–Korteweg system and travelling waves

Phase portraits for van der Waals pressure law

Convex pressure

2

slope j v v v’ p=f’(v)

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The Euler–Korteweg system and travelling waves

Phase portraits for van der Waals pressure law

Convex pressure

2

slope j v v v’ p=f’(v)

Nonconvex pressure

v

v0 v3 v2 v

v0 v1 v2 v3

p

v’

v1

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The Euler–Korteweg system and travelling waves

Phase portraits for van der Waals pressure law

Convex pressure

2

slope j v v v’ p=f’(v)

Nonconvex pressure

1

v0 v3 v2

v0 v1 v3 v2

v p v v’ v

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The Euler–Korteweg system and travelling waves

Phase portraits for van der Waals pressure law

Convex pressure

2

slope j v v v’ p=f’(v)

Nonconvex pressure

v

v0 v3

v0 v1 v3 v2

p v

v2 v1

v’

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Modulated equations and periodic waves

Outline

1

The Euler–Korteweg system and travelling waves

2

Modulated equations and periodic waves

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Modulated equations and periodic waves

General Hamiltonian framework

(H) ∂tU = J (EH [U]) , J = ∂xJ , H = H (U, Ux) .

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Modulated equations and periodic waves

General Hamiltonian framework

(H) ∂tU = J (EH [U]) , J = ∂xJ , H = H (U, Ux) . Benjamin’s impulse Q(U) := 1

2 U · J−1U is such that J EQ[U] = ∂xU ,

and solutions of (H) satisfy local conservation law (C) ∂tQ(U) = ∂x(S [U]) , S [U] := U · EH [U] + LH [U] .

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Modulated equations and periodic waves

General Hamiltonian framework

(H) ∂tU = J (EH [U]) , J = ∂xJ , H = H (U, Ux) . Benjamin’s impulse Q(U) := 1

2 U · J−1U is such that J EQ[U] = ∂xU ,

and solutions of (H) satisfy local conservation law (C) ∂tQ(U) = ∂x(S [U]) , S [U] := U · EH [U] + LH [U] . Examples (gKdV) ∂tv + ∂xp(v) = −∂3

xv,

U = v, Q = 1

2 v2, H = f (v) + 1 2v2 x ,

f ′ = −p . (EK1) U = (ρ, u)T , Q = ρu , H = 1

2ρu2 + F(ρ) + 1 2K (ρ)ρ2 x .

(EKℓ) U = (v, u)T , Q = vu , H = 1

2u2 + f (v) + 1 2κ(v)u2 x .

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Modulated equations and periodic waves

Wave trains

The profile U of a periodic travelling wave solution to (H) of speed c is a stationary solution in co-moving frame to (Hc) ∂tU = J (E(H + cQ)[U]) . It is characterized by E(H + cQ)[U] = λ , ✍ as well as the integrated equation S [U] + cQ(U) = µ . ✍

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Modulated equations and periodic waves

Wave trains

The profile U of a periodic travelling wave solution to (H) of speed c is a stationary solution in co-moving frame to (Hc) ∂tU = J (E(H + cQ)[U]) . It is characterized by E(H + cQ)[U] = λ ,✍ N parameters if U(t, x) ∈ RN as well as the integrated equation S [U] + cQ(U) = µ .✍ one additional parameter. Slowly varying wave trains generically evolve in a (N + 2)-dimensional manifold parametrized by (c, λ, µ).

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Modulated equations and periodic waves

Derivation of modulated equations

‘Two-timing’ method. Look for solutions admitting an asymptotic expansion of the form U(t, x) = U0( εt

• T

, εx

• X

, φ(εt, εx)/ε

• θ

) + ε U1(εt, εx, φ(εt, εx)/ε, ε) + o(ε) , with U0 and U1 one-periodic in θ.

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Modulated equations and periodic waves

Derivation of modulated equations

‘Two-timing’ method. Look for solutions admitting an asymptotic expansion of the form U(t, x) = U0( εt

• T

, εx

• X

, φ(εt, εx)/ε

• θ

) + ε U1(εt, εx, φ(εt, εx)/ε, ε) + o(ε) , with U0 and U1 one-periodic in θ. k := φX , ω := φT , c := −ω/k ⇒ ∂Tk + ∂X(ck) = 0.

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Modulated equations and periodic waves

Derivation of modulated equations

‘Two-timing’ method. Look for solutions admitting an asymptotic expansion of the form U(t, x) = U0( εt

• T

, εx

• X

, φ(εt, εx)/ε

• θ

) + ε U1(εt, εx, φ(εt, εx)/ε, ε) + o(ε) , with U0 and U1 one-periodic in θ. k := φX , ω := φT , c := −ω/k ⇒ ∂Tk + ∂X(ck) = 0.

• Averaging. U(x) := U0(T, X, kx) is a 1/k-periodic travelling wave of

speed c profile such that ∂TU0 = J ∂XG0 , G0 := EHk[U0] , ∂TQ0 = ∂XS0 , S0 := U0 · G0 + LHk[U0] .

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Modulated equations and periodic waves

Derivation of modulated equations

‘Two-timing’ method. Look for solutions admitting an asymptotic expansion of the form U(t, x) = U0( εt

• T

, εx

• X

, φ(εt, εx)/ε

• θ

) + ε U1(εt, εx, φ(εt, εx)/ε, ε) + o(ε) , with U0 and U1 one-periodic in θ. k := φX , ω := φT , c := −ω/k ⇒ ∂Tk + ∂X(ck) = 0.

• Averaging. U(x) := U0(T, X, kx) is a 1/k-periodic travelling wave of

speed c profile such that ∂TU0 = J ∂XG0 , G0 := EHk[U0] , ∂TQ0 = ∂XS0 , S0 := U0 · G0 + LHk[U0] . [Whitham’70]: ‘the relation of the stability of the periodic wave with the type of the [modulated equations is] given in the previous papers’ ...

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Modulated equations and periodic waves

Stability of periodic waves vs type of modulated equations

[Serre’05] [Bronski–Johnson–Zumbrun’09-11] [Pogan–Scheel–Zumbrun’12]

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Modulated equations and periodic waves

Stability of periodic waves vs type of modulated equations

[Serre’05] [Bronski–Johnson–Zumbrun’09-11] [Pogan–Scheel–Zumbrun’12] Theorem (SBG–Noble–Rodrigues) If the set of periodic travelling wave profiles is a (N + 2)-dimensional manifold parametrized by (k, M := U, P := Q) near a reference profile U (of period 1/k and speed c), and the linearized operator A about U has an (N + 2)-dimensional generalized kernel in space of 1/k-periodic functions, then a necessary condition for U to be stable is that the modulated system be ‘weakly hyperbolic’ at (k, M, P).

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Modulated equations and periodic waves

Sketch of proof

By Floquet–Bloch theory, the spectrum of A := J Hess(H + cQ)[U] in Lp is made of the eigenvalues of A ν := A (k(∂θ + iν)) in L∞

per, ν ∈ R/2πZ.

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Modulated equations and periodic waves

Sketch of proof

By Floquet–Bloch theory, the spectrum of A := J Hess(H + cQ)[U] in Lp is made of the eigenvalues of A ν := A (k(∂θ + iν)) in L∞

per, ν ∈ R/2πZ.

Expanding A ν in powers of ν, we see the leading operator A 0 has a generalized kernel spanned by ∂θU0, ∂MαU0, ∂PU0.

16 / 21

Modulated equations and periodic waves

Sketch of proof

By Floquet–Bloch theory, the spectrum of A := J Hess(H + cQ)[U] in Lp is made of the eigenvalues of A ν := A (k(∂θ + iν)) in L∞

per, ν ∈ R/2πZ.

Expanding A ν in powers of ν, we see the leading operator A 0 has a generalized kernel spanned by ∂θU0, ∂MαU0, ∂PU0. Spectral projector Π0 onto span(∂θU0, ∂MαU0, ∂PU0) extends to spectral projector Πν for A ν. Kato’s perturbation argument then yields dual bases (Φν

β) and (Ψν β) of g-ker A ν and g-ker(A ν)∗

depending analytically on ν.

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Modulated equations and periodic waves

Sketch of proof

By Floquet–Bloch theory, the spectrum of A := J Hess(H + cQ)[U] in Lp is made of the eigenvalues of A ν := A (k(∂θ + iν)) in L∞

per, ν ∈ R/2πZ.

Expanding A ν in powers of ν, we see the leading operator A 0 has a generalized kernel spanned by ∂θU0, ∂MαU0, ∂PU0. Spectral projector Π0 onto span(∂θU0, ∂MαU0, ∂PU0) extends to spectral projector Πν for A ν. Kato’s perturbation argument then yields dual bases (Φν

β) and (Ψν β) of g-ker A ν and g-ker(A ν)∗

depending analytically on ν. Matrix Dν := ikν

• Ψν

α · A νΦν β

• is similar to a matrix whose leading
• rder part turns out to coincide with A + cIN+2, where A is the

matrix of linearized modulated equations about (k, M, P).

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Modulated equations and periodic waves

Modulated equations for Euler–Korteweg systems

Theorem (SBG–Noble–Rodrigues) The following diagram is commutative mass Lagrangian change of coordinates (EK1) − → (EKℓ) Whitham’s averaging ↓ ↓ EK1 − → EKℓ

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Modulated equations and periodic waves

Proof

Recall that (R, U) is a 1/K-periodic solution of (EK1) if and only if (V , W ) is a 1/k-periodic solution of (EKℓ), with R(ξ)V (Z(ξ)) = 1, U(ξ) = W (Z(ξ)), dZ dξ = R = 1 V (Z), 1 k = Z 1 K

• .

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Modulated equations and periodic waves

Proof

Recall that (R, U) is a 1/K-periodic solution of (EK1) if and only if (V , W ) is a 1/k-periodic solution of (EKℓ), with R(ξ)V (Z(ξ)) = 1, U(ξ) = W (Z(ξ)), dZ dξ = R = 1 V (Z), 1 k = Z 1 K

• .

Hence ρ0 = K k , v0 = k K , w0 = ρ0u0 ρ0 . dY = ρ0dX − ρ0u0dT S=T ⇐ ⇒ dX = v0dY + w0dS

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Modulated equations and periodic waves

Proof

Recall that (R, U) is a 1/K-periodic solution of (EK1) if and only if (V , W ) is a 1/k-periodic solution of (EKℓ), with R(ξ)V (Z(ξ)) = 1, U(ξ) = W (Z(ξ)), dZ dξ = R = 1 V (Z), 1 k = Z 1 K

• .

Hence ρ0 = K k , v0 = k K , w0 = ρ0u0 ρ0 . dY = ρ0dX − ρ0u0dT S=T ⇐ ⇒ dX = v0dY + w0dS K dX − σK dT = kdY + jkdS

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Modulated equations and periodic waves

Proof

Recall that (R, U) is a 1/K-periodic solution of (EK1) if and only if (V , W ) is a 1/k-periodic solution of (EKℓ), with R(ξ)V (Z(ξ)) = 1, U(ξ) = W (Z(ξ)), dZ dξ = R = 1 V (Z), 1 k = Z 1 K

• .

Hence ρ0 = K k , v0 = k K , w0 = ρ0u0 ρ0 . dY = ρ0dX − ρ0u0dT S=T ⇐ ⇒ dX = v0dY + w0dS K dX − σK dT = kdY + jkdS u0dX + ( 1

2σ2 − µ − σu0)dT = ( 1 2σ2 − µ + jσv0 + j2v2 0 )dS

+v0w0 dY

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Modulated equations and periodic waves

Proof

Recall that (R, U) is a 1/K-periodic solution of (EK1) if and only if (V , W ) is a 1/k-periodic solution of (EKℓ), with R(ξ)V (Z(ξ)) = 1, U(ξ) = W (Z(ξ)), dZ dξ = R = 1 V (Z), 1 k = Z 1 K

• .

Hence ρ0 = K k , v0 = k K , w0 = ρ0u0 ρ0 . dY = ρ0dX − ρ0u0dT S=T ⇐ ⇒ dX = v0dY + w0dS K dX − σK dT = kdY + jkdS u0dX + ( 1

2σ2 − µ − σu0)dT = ( 1 2σ2 − µ + jσv0 + j2v2 0 )dS

+v0w0 dY ρ0u0dX +

• −
• ρ0u2
• + λ + j2 1/ρ0
• dT = (λ + j2v02)dS

+w0 dY

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Modulated equations and periodic waves

Sufficient condition for hyperbolicity

Theorem ([Gavrilyuk–Serre’95]) The strict convexity of the averaged energy e := ⑨❡0 + 1

2 w2 0 − 1 2 w02

as a function of (v0, k, (v0w0 − v0 w0)/k) implies the hyperbolicity

• f modulated equations for (EKℓ).

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Modulated equations and periodic waves

Sufficient condition for hyperbolicity

Theorem ([Gavrilyuk–Serre’95]) The strict convexity of the averaged energy e := ⑨❡0 + 1

2 w2 0 − 1 2 w02

as a function of (v0, k, (v0w0 − v0 w0)/k) implies the hyperbolicity

• f modulated equations for (EKℓ).

Remark The convexity of e is equivalent to the convexity of ρ0 e = E0 + 1

2ρ0u2 0 − 1 2

ρ0u02 ρ0 as a function of (ρ0, K, (ρ0u0 − ρ0u0)/K).

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Conclusion

Concluding remarks

Periodic waves with j = 0 are unstable [Serre’94], but there are also stable ones, e.g. small-amplitude periodic waves for defocussing cubic (NLS) (ρK (ρ) ≡ 4, F ′(ρ) = ρ) [Gallay–H˘ ar˘ agu¸ s’07].

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Conclusion

Concluding remarks

Periodic waves with j = 0 are unstable [Serre’94], but there are also stable ones, e.g. small-amplitude periodic waves for defocussing cubic (NLS) (ρK (ρ) ≡ 4, F ′(ρ) = ρ) [Gallay–H˘ ar˘ agu¸ s’07]. For (EK) with van der Waals pressure law, numerical computation of eigenvalues display nontrivial regions of hyperbolicity for the modulated equations (away from unstable constant states)...

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Conclusion

Further reading

• S. Benzoni-Gavage. Planar travelling waves in capillary fluids.

To appear in special volume of Diff. Int. Eq.

• S. Benzoni-Gavage, P. Noble, and L.M. Rodrigues. Slow modulations of

periodic waves in Hamiltonian PDEs, with application to capillary fluids. In preparation.

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