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
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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
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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
15 / 30
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
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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
24 / 30
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
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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