An augmented Lagrangian Approach for the defocusing non-linear Schr - PowerPoint PPT Presentation

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives An augmented 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 20 August 2019 Firas DHAOUADI CEMRACS 2019, Marseille 1 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Introduction : Euler’s equation for compressible fluids A Lagrangian : � � � ρ | u | 2 − ρ 2 L ( ρ, u ) = d Ω t 2 2 Ω t A Constraint : ρ t + div ( ρ u ) = 0 = ⇒ The corresponding Euler-Lagrange equation : � � ρ u ⊗ u + ρ 2 ( ρ u ) t + div = 0 2 Firas DHAOUADI CEMRACS 2019, Marseille 2 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Dispersive models in mechanics 1 Surface waves with surface tension [Nikolayev, Gavrilyuk, Gouin 2006] : � � � ρ 0 h | u | 2 − σ |∇ h | 2 − ρ 0 gh 2 L ( u , h , ∇ h ) = d Ω t 2 2 2 Ω t 2 Shallow water equations described by Serre-Green-Naghdi equations [Salmon (1998)]: � � � + h ˙ hu 2 − gh 2 h 2 L ( u , h , ˙ h ) = d Ω t 2 2 6 Ω t Firas DHAOUADI CEMRACS 2019, Marseille 3 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Euler-Korteweg type systems � � � ρ | u | 2 − A ( ρ ) − K ( ρ ) |∇ ρ | 2 L ( u , ρ, ∇ ρ ) = d Ω t 2 2 Ω t � ∂ t ρ + div( ρ u ) = 0 � 2 K ′ ( ρ ) |∇ ρ | 2 � K ( ρ )∆ ρ + 1 ∂ t ( ρ u ) + div( ρ u ⊗ u ) + ∇ p ( ρ ) = ρ ∇ K ( ρ ) = σ : constant capillarity ∂ t ( ρ u ) + div( ρ u ⊗ u ) + ∇ p ( ρ ) = σρ ∇ (∆ ρ ) 1 K ( ρ ) = 4 ρ : Quantum capillarity / DNLS equation � � ρ 2 ∂ t ( ρ u ) + div( ρ u ⊗ u + 1 2 − 1 4 ρ ∇ ρ ⊗ ∇ ρ ) + ∇ 4 ∆ ρ = 0 Firas DHAOUADI CEMRACS 2019, Marseille 4 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives The Non-Linear Schr¨ odinger equation i ǫψ t + ǫ 2 � | ψ | 2 � ǫ = � 2 ∆ ψ − f ψ = 0 ; 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 CEMRACS 2019, Marseille 5 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Outline 1 The Defocusing NLS equation 2 Augmented Lagrangian approach 3 Numerical results 4 Conclusions - Perspectives Firas DHAOUADI CEMRACS 2019, Marseille 6 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives The defocusing NLS equation � | ψ | 2 � = | ψ | 2 and ǫ = 1; t ′ = t ǫ x ′ = x In what follows we take : f ǫ : i ψ t + 1 2∆ ψ − | ψ | 2 ψ = 0 The Madelung transform � ρ ( x , t ) e i θ ( x , t ) ψ ( x , t ) = u = ∇ θ � ρ t + div ( ρ u ) = 0 ( ρ u ) t + div ( ρ u ⊗ u + Π) = 0 � ρ 2 � 2 − 1 Id + 1 with : Π = 4∆ ρ 4 ρ ∇ ρ ⊗ ∇ ρ Firas DHAOUADI CEMRACS 2019, Marseille 7 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives A Lagrangian for DNLS equation For the previous set of equations, we can construct the Lagrangian: � 2 � � 2 − ρ 2 ρ | u | 2 − 1 |∇ ρ | L ( u , ρ, ∇ ρ ) = d Ω t 2 4 ρ 2 Ω t Energy conservation law: ∂ E ∂ t + div ( E u + Π u − 1 4 ˙ ρ ∇ ρ ) = 0 ; ρ = ρ t + u · ∇ ρ ˙ where 2 2 + ρ 2 E = ρ | u | 2 + 1 |∇ ρ | 2 4 ρ 2 Firas DHAOUADI CEMRACS 2019, Marseille 8 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Augmented 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 CEMRACS 2019, Marseille 9 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Augmented Lagrangian approach : Application to DNLS DNLS Lagrangian : � 2 � � 2 − ρ 2 ρ | u | 2 − 1 |∇ ρ | L ( u , ρ, ∇ ρ ) = d Ω t 2 4 ρ 2 Ω t ’Augmented’ Lagrangian approach [Favrie, Gavrilyuk, 2017] ˜ L ( u , ρ, η, ∇ η, ˙ η ) p = ∇ η w = ˙ η � � � η � 2 � 2 2 − ρ 2 ρ | u | 2 − 1 | p | − λ + βρ ˜ 2 w 2 L = 2 ρ ρ − 1 d Ω t 2 4 ρ 2 Ω t � η � 2 λ βρ η 2 : Regularizer 2 ρ ρ − 1 : Penalty 2 ˙ Firas DHAOUADI CEMRACS 2019, Marseille 10 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Augmented system Euler-Lagrange equations The Augmented Lagrangian : � � 2 � � η � 2 2 2 w 2 − ρ 2 ρ | u | + βρ 2 − 1 | p | − λ ˜ L = 2 ρ ρ − 1 d Ω t 2 4 ρ 2 Ω t The constraint : ρ t + div ( ρ u ) = 0 = ⇒ We apply Hamilton’s principle : � t 1 ˜ a = L dt = ⇒ δ a = 0 t 0 Firas DHAOUADI CEMRACS 2019, Marseille 11 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Types of variations Two types of variations will be considered : I � �� � ˜ L ( u , ρ, ˙ η, η, ∇ η ) η = η t + u · ∇ η ˙ � �� � II Type I : Virtual displacement of the continuum: ˆ δ u = ˙ ˆ η = ˆ δρ = − div ( ρδ x ) δ x − ∇ u · δ x δ ˙ δ u · ∇ η Type II : Variations with respect to η δ ∇ η = ∇ δη δ ˙ η = ( δη ) t + u · ∇ δη Firas DHAOUADI CEMRACS 2019, Marseille 12 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Augmented system Euler-Lagrange Equations Type I : Virtual displacement of the continuum: ( ρ u ) t + div ( ρ u ⊗ u + P ) = 0 � ρ 2 � 2 − 1 4 ρ | p | 2 + ηλ (1 − η Id + 1 with : P = ρ ) 4 ρ p ⊗ p Type II : Variations with respect to η : � � � � 1 = λ 1 − η ( ρ w ) t + div ρ w u − 4 ρβ p β ρ Firas DHAOUADI CEMRACS 2019, Marseille 13 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives Closure of the system 1. Definition of w = ˙ η w = ˙ η = η t + u · ∇ η = ⇒ ( ρη ) t + div ( ρη u ) = ρ w 2. Evolution of p = ∇ η ∇ w = ∇ ( η t + u · ∇ η ) = ( ∇ η ) t + ∇ ( u · ∇ η ) = ⇒ ( ∇ η ) t + ∇ ( u · ∇ η − w ) = 0 = ⇒ p t + div (( p · u − w ) Id ) = 0 2’. Initial condition for p : p t =0 = ( ∇ η ) t =0 Firas DHAOUADI CEMRACS 2019, Marseille 14 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives The full Augmented system  ρ t + div ( ρ u ) = 0      ( ρ u ) t + div ( ρ u ⊗ u + P ) = 0    ( ρη ) t + div ( ρη u ) = ρ w � � � �  1 − η 1 = λ  ( ρ w ) t + div ρ w u − 4 ρβ p   β ρ     p t + div (( p · u − w ) Id ) = 0; curl ( p ) = 0 � ρ 2 � 2 − 1 4 ρ | p | 2 + ηλ (1 − η Id + 1 P = ρ ) 4 ρ p ⊗ p Closed system. What about hyperbolicity ? Values of λ and β ? Firas DHAOUADI CEMRACS 2019, Marseille 15 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives One-Dimensional case : Hyperbolicity In order to study the hyperbolicity of this system, we write it in quasi-linear form : ∂ U ∂ t + A ( U ) ∂ U ∂ x = q where: � � � � T � � T 1 − η 0 , 0 , 1 λ U = ρ, u , w , p , η q = , 0 , w βρ ρ   ρ 0 0 0 u � � 1 + λη 2 1 − 2 η λ  0 0  u ρ 3  ρ ρ    A ( U ) = p 1 0 u − 0   4 βρ 3 4 βρ 2   0 − 1 0 p u   0 0 0 0 u Firas DHAOUADI CEMRACS 2019, Marseille 16 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives One-Dimensional case : Hyperbolicity The eigenvalues c of the matrix A are : �� � � � 2 4 βρ 2 + ρ + λη 2 4 βρ 2 + ρ + λη 2 1 1 ± − ρ 2 ρ 2 c = u , ( c − u ) 2 ± = . 2 The right-hand side is always positive. However, the roots can be multiple if 4 βρ 2 = ρ + λη 2 1 ρ 2 . In the case of multiple roots : We still get five linear independent eigenvectors. = ⇒ the system is always hyperbolic Firas DHAOUADI CEMRACS 2019, Marseille 17 / 27

Introduction Defocusing NLS equation Augmented Lagrangian approach Conclusions and Perspectives 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 p = ρ 0 + k 2 c 2 4 Augmented DNLS dispersion relation �� � 2 � � 1 λ 1 λ βρ 0 k 2 + ρ 0 + λ λ + ρ 0 + λ + + ρ 0 + λ + − 4 0 k 2 − 4 βρ 2 βρ 2 4 βρ 2 βρ 2 0 k 2 4 βρ 2 ( c p ) 2 = 0 0 0 2 Firas DHAOUADI CEMRACS 2019, Marseille 18 / 27