Extended Lagrangian Approach for the defocusing non-linear

May 24, 2018 - Defocusing NLS equation. Extended Lagrangian approach. Dispersive Shock Waves. Results. Introduction : Euler's equation for compressible ...
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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Extended Lagrangian Approach for the defocusing non-linear Schr¨odinger Equation Firas Dhaouadi Sergey Gavrilyuk Nicolas Favrie Jean-Paul Vila Aix-Marseille Universit´ e - Universit´ e Toulouse III

24 May 2018

Firas DHAOUADI

SHARK-FV 2018, Porto

1 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Introduction : Euler’s equation for compressible fluids A Lagrangian : Z L(ρ, u) = Ωt

2

ρ2

ρ |u| − 2 2

! dΩt

A Constraint : ρt + div(ρu) = 0 =⇒ The corresponding Euler-Lagrange equation :   ρ2 (ρu)t + div ρu ⊗ u + =0 2

Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Dispersive models in mechanics 1

Surface waves with surface tension [Nikolayev, Gavrilyuk, Gouin 2006] : ! Z 2 2 2 ρ0 h |u| ρ0 gh |∇h| L(u, h, ∇h) = − −σ dΩt 2 2 2 Ωt

2

Shallow water equations described by Serre-Green-Naghdi equations [Salmon (1998)]: ! Z 2 2 2 hu gh hh˙ ˙ L(u, h, h) = − + dΩt 2 2 6 Ωt

Firas DHAOUADI

SHARK-FV 2018, Porto

3 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

The Non-Linear Schr¨odinger equation   2 2 iψt + ∆ψ − f |ψ| ψ = 0 2

;

=

~ m

A wide range of applications: Nonlinear optics, quantum fluids, surface gravity waves Advantage : the equation is integrable. [Zakharov,Manakov 1974] Construction of analytical solutions is possible. Problematic Can we solve a dispersive problem by the means of hyperbolic equations ? Firas DHAOUADI

SHARK-FV 2018, Porto

4 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Outline

1

The Defocusing NLS equation

2

Extended Lagrangian approach

3

Dispersive Shock Waves

4

Numerical results

5

Conclusions - Perspectives

Firas DHAOUADI

SHARK-FV 2018, Porto

5 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

The defocusing NLS equation 

2



In what follows we take : f |ψ| = |ψ|2 and  = 1; t 0 = : 1 iψt + ∆ψ − |ψ|2 ψ = 0 2 The Madelung transform p ψ(x, t) = ρ(x, t)e iθ(x,t) ( ρt + div(ρu) = 0

with :

t 

x0 =

x 

u = ∇θ

(ρu)t + div (ρu ⊗ u + Π) = 0  2  ρ 1 1 Π= − ∆ρ Id + ∇ρ ⊗ ∇ρ 2 4 4ρ Firas DHAOUADI

SHARK-FV 2018, Porto

6 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

A Lagrangian for DNLS equation For the previous set of equations, we can construct the Lagrangian: 2

Z L(u, ρ, ∇ρ) =

ρ Ωt

ρ2

|u| 1 |∇ρ| − − 2 2 4ρ 2

2

! dΩt

Energy conservation law: ∂E 1 + div(E u + Πu − ρ∇ρ) ˙ =0 ∂t 4 where

;

ρ˙ = ρt + u · ∇ρ

|u| 2 ρ2 1 |∇ρ| 2 E =ρ + + 2 2 4ρ 2 Firas DHAOUADI

SHARK-FV 2018, Porto

7 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Extended Lagrangian approach

The objective Obtain a new Lagrangian whose Euler-Lagrange equations : are hyperbolic approximate Schr¨odinger’s equation in a certain limit The idea Decouple ∇ρ from u and ρ, have it as an independent variable.

Firas DHAOUADI

SHARK-FV 2018, Porto

8 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Extended Lagrangian approach : Application to DNLS DNLS Lagrangian : 2

Z L(u, ρ, ∇ρ) =

ρ Ωt

ρ2

|u| 1 |∇ρ| − − 2 2 4ρ 2

2

! dΩt

’Extended’ Lagrangian approach [Favrie, Gavrilyuk, 2017] ˜ ρ, η, ∇η, η) L(u, ˙ L˜ =

Z Ωt

λ ρ 2

p = ∇η w = η˙ !  2 2 2 2 |u| ρ 1 |p| λ η βρ 2 ρ − − ρ dΩt − −1 + w 2 2 4ρ 2 2 ρ 2



η −1 ρ

2 : Penalty Firas DHAOUADI

βρ 2 η˙ : Regularizer 2 SHARK-FV 2018, Porto

9 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Extended system Euler-Lagrange equations The extended Lagrangian : L˜ =

2

Z ρ Ωt

2

ρ2

βρ 2 1 |p| λ |u| + w − − − ρ 2 2 2 4ρ 2 2



2 ! η −1 dΩt ρ

The constraint : ρt + div(ρu) = 0 =⇒ We apply Hamilton’s principle : Z

t1

a=

L˜ dt

=⇒

δa = 0

t0

Firas DHAOUADI

SHARK-FV 2018, Porto

10 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Types of variations Two types of variations will be considered : I

z }| { ˜ L(u, ρ, η, ˙ η, ∇η ) | {z }

η˙ = ηt + u · ∇η

II

Type I : Virtual displacement of the continuum: ˆ = −div(ρδx) δρ

ˆ = δx ˙ − ∇u · δx δu

ˆ · ∇η δ η˙ = δu

Type II : Variations with respect to η δ∇η = ∇δη Firas DHAOUADI

δ η˙ = (δη)t + u · ∇δη SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Extended system Euler-Lagrange Equations Type I : Virtual displacement of the continuum: (ρu)t + div (ρu ⊗ u + P) = 0  with :

P=

 ρ2 1 η 1 − |p|2 + ηλ(1 − ) Id + p ⊗ p 2 4ρ ρ 4ρ

Type II : Variations with respect to η:     λ 1 η (ρw )t + div ρw u − p = 1− 4ρβ β ρ

Firas DHAOUADI

SHARK-FV 2018, Porto

12 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Closure of the system

w = η˙ = ηt + u · ∇η

=⇒

(ρw )t + div (ρηu) = 0

∇w = ∇(ηt + u · ∇η) = (∇η)t + ∇(u · ∇η) =⇒

(∇η)t + ∇(u · ∇η − w ) = 0

=⇒

pt + div((p · u − w )Id) = 0

Firas DHAOUADI

SHARK-FV 2018, Porto

13 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

The full extended system   ρt + div(ρu) = 0        (ρu)t + div (ρu ⊗ u + P) = 0 (ρη)t + div(ρηu) = ρw       η 1 λ  (ρw )t + div ρw u − 4ρβ p = β 1 − ρ      p + div ((p · u − w ) Id) = 0; curl(p) = 0 t

 P=

 ρ2 1 η 1 − |p|2 + ηλ(1 − ) Id + p ⊗ p 2 4ρ ρ 4ρ

Closed system. What about hyperbolicity ? Values of λ and β ? Firas DHAOUADI

SHARK-FV 2018, Porto

14 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

One dimensional case: In 1-d, the system reduces to : ρt + (ρu)x = 0   2 ρ η (ρu)t + ρu 2 + + ηλ(1 − ) = 0 2 ρ x (ρη)t + (ρηu)x = ρw     1 λ η (ρw )t + ρwu − p = 1− 4ρβ x β ρ pt + (pu − w )x = 0 Remainder : The original DNLS equations :   ρt + (ρu)  x = 0 2 ρ 1 1 − ρxx + ρx ρx =0  (ρu)t + ρu 2 + 2 4 4ρ x Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

One dimensional case: Relaxation

  2 ρ η (ρu)t + ρu 2 + + ηλ(1 − ) = 0 2 ρ x     1 λ η (ρw )t + ρwu − p = 1− 4ρβ x β ρ

(1) (2)

Injecting (2) in (1) yields:   2 ρ ρ 1 xx = (ρu)t + ρu 2 + − + ρx ρx 2 4 4ρ x 1 β β −β(ρ2 ρ¨)x + ρxxxxx + O(β 2 ) + O( 2 ) + O( ) 16λ λ λ Firas DHAOUADI

SHARK-FV 2018, Porto

16 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

One-Dimensional case : variables : ρ, u, η, p = ηx , w = η˙ 1-D system : ∂ρ ∂ρ ∂u +u +ρ =0 ∂t ∂x ∂x     ∂u ∂ρ λ η 2 ∂ρ 2η ∂η ∂u +u + + + 1− =0 ∂t ∂x ∂x ρ ρ2 ∂x ρ ∂x     ∂w ∂w 1 1 ∂p p ∂ρ λ η +u − − 2 = 1− ∂t ∂x 4βρ ρ ∂x ρ ∂x βρ ρ ∂p ∂p ∂u ∂w +u +p − =0 ∂t ∂x ∂x ∂x ∂η ∂η +u =w ∂t ∂x Firas DHAOUADI

SHARK-FV 2018, Porto

17 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

One-Dimensional case : Hyperbolicity In order to study the hyperbolicity of this system, we write it in quasi-linear form : ∂U ∂U + A(U) =q ∂t ∂x where:    T T η 1λ q = 0, 0, βρ 1 − ρ , 0, w U = ρ, u, w , p, η 

u

 1 + λη2  ρ3  p A(U) =   4βρ3  0 0

ρ u

0 0

0 0

 0 λ ρ 1−

1 0 u − 4βρ 2 p −1 u 0 0 0

Firas DHAOUADI

SHARK-FV 2018, Porto

0 0 u

 2η ρ

      

18 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

One-Dimensional case : Hyperbolicity The eigenvalues c of the matrix A are :  c = u, (c −

u)2±

=

1 4βρ2

+ρ+

λη 2 ρ2



±

r

1 − 4βρ 2

2

+ρ+

λη 2 ρ2

2 .

The right-hand side is always positive. However, the roots can be multiple if 1 λη 2 =ρ+ 2 . 4βρ2 ρ In the case of multiple roots : We still get five linear independent eigenvectors. =⇒ the system is always hyperbolic

Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Values of λ and β Values have to be assigned : a criterion is needed. We can base this choice, for example, on the dispersion relation. Original DNLS dispersion relation cp2

k2 = ρ0 + 4

Extended DNLS dispersion relation

2

(cp ) =

1 λ + ρ0 + λ + − 2 4βρ0 βρ20 k 2

s

1 λ + ρ0 + λ + 2 4βρ0 βρ20 k 2

2

 −4

λ ρ0 + λ + 2 βρ0 k 4βρ20



2 Firas DHAOUADI

SHARK-FV 2018, Porto

20 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Example estimation cp 0 00 1 λ= 0 10 λ=

6 5 4 3

λ=10

2 1 0 0

2

4

6

8

10

12

k

Figure 1: The dispersion relation cp = f (k) for the original model (continuous line) and the dispersion relation for the extended model (dashed lines) for different values of λ and for β = 10−4 Firas DHAOUADI

SHARK-FV 2018, Porto

21 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Numerical scheme : Hyperbolic step 1-d system of equations to solve : ∂U ∂F + = S(U) ∂t ∂x Splitting for the source terms.  Un+1 i

Uni

∆t ∆x

F∗i+ 1 2



F∗i− 1 2



1

Godunov scheme:

2

Riemann Solver: HLL-Rusanov.  1  1 Fi+ 1 = F(Uni+1 ) − F(Uni ) − κni+ 1 Uni+1 − Uni 2 2 2 2 where κni+ 1 is obtained by using the Davis approximation :

=

+

2

κni+1/2 = max(|cj (Uni )|, |cj (Uni+1 )|), j

where cj are the eigenvalues of the extended system. Firas DHAOUADI

SHARK-FV 2018, Porto

22 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Numerical scheme : ODE step Reduced to a second order ODE with constant coefficients which can be solved exactly in our case.    dρ dρu dp dρη dρw λ η = 0; = 0; =0 = ρw = 1− dt dt dt dt dt β ρ Therefore, the exact solution is given by :    ρn+1 = ρ¯n u n+1 = u¯n p n+1 = p¯n    w ¯n n+1 n n n η = ρ¯ + (¯ η − ρ¯ ) cos(Ω∆t) + sin(Ω∆t)  Ω     w n+1 = Ω(¯ ρn − η¯n ) sin(Ω∆t) + w ¯ n cos (Ω∆t) where Ω =

λ . βρ2 Firas DHAOUADI

SHARK-FV 2018, Porto

23 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

A brief introduction to DSWs ρ ρL

ρ0 ρR

τ4

τ3

τ2

τ1

τ=x/t

Figure 2: Asymptotic profile of the solution to NLS equation (continuous line) for the Riemann problem ρL = 2, ρR = 1 , uL = uR = 0. Oscillations shown at t=70 Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Whitham’s theory of modulations

The main idea : Start from the conservation laws of the system variables and establish evolution equations for the amplitude, the wavenumber, ...

Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

DSW Numerical results : ρ ρ numerical simulation Whitham envelope

ρL

ρ0

ρR

τ4

τ3

τ2

τ1

x/t

Figure 3: Comparison of the numerical result ρ(x, t) = f (x/t) (blue line) with the asymptotic profile of the oscillations from Whitham’s theory of modulations. t=70 Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

DSW Numerical results : u u numerical simulation Whitham envelope

u0

uL

uR

τ4

τ3

τ2

τ1

x/t

Figure 4: Comparison of the numerical result u(x, t) = f (x/t) (blue line) with the asymptotic profile of the oscillations from Whitham’s theory of modulations. t=70 Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

vanishing oscillations at the left constant state

ρ

at

ρL

3/2

50 40 30 20

ρ0

t=20s t=40s t=60s

10

τ4

τ3

τx

0

25

50

75

100 125 150 175

t

Figure 5: Vanishing oscillations at the vicinity of τ = τ4 . amplitude decreases as ∝ t −1/2 .

Firas DHAOUADI

SHARK-FV 2018, Porto

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Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Conclusions - perspectives Conclusions : The defocusing nonlinear Schr¨odinger equation is solved by an extended Lagrangian method. The resulting system of equations is always hyperbolic. Tests were made for a non stationary solution (DSWs). Perspectives: Extension to the multidimensional case. Proper development of the boundary conditions. Further optimization of the numerical resolution.

Firas DHAOUADI

SHARK-FV 2018, Porto

29 / 30

Introduction Defocusing NLS equation Extended Lagrangian approach Dispersive Shock Waves Results

Current work I am working on thin films equations given by : ht + (hu)x = 0     2 h5 2 λ 1 3u κ h cosθ + = λh − + hhxxx (hu)t + hu 2 + 2F 2 K0 x εRe h F2 They can be seen as the Euler-Lagrange equation for the Lagrangian :  Z  2 2 κ hx λ2 5 u cosθ 2 L= h − A(h) − 2 dΩt A(h) = h + h 2 F 2 2F 2 4K0 Ωt

Firas DHAOUADI

SHARK-FV 2018, Porto

30 / 30