Finding Lie Symmetries of PDEs with MATHEMATICA : Applications to - PowerPoint PPT Presentation

Quatization − − June 8 Geometry, Integrability Integrability and and Quatization June 8- -13, 2007 13, 2007 Geometry, Finding Lie Symmetries of PDEs with MATHEMATICA : Applications to Nonlinear Fiber Optics Vladimir Pulov Department of Physics, Technical University-Varna, Bulgaria Ivan Uzunov Department of Applied Physics, Technical University-Sofia, Bulgaria Eddy Chacarov Department of Informatics and Mathematics, Varna Free University, Bulgaria

Plan of Presentation 1. MATHEMATICA package for finding Lie symmetries of PDE 1.1. Block-scheme and algorithm 1.2. Input and output 1.3. Tracing the evaluation 1.4. Trial run 2. Applications to nonlinear fiber optics 2.1. Physical model 2.2. Results obtained 3. Conclusion

Symmetry Group of Δ System of PDE { } ( ) ( ) ( ) ( ) = = Δ = ∈ Ω ⊂ ∈ Ω 1 n K r r , , ,..., 0 , 1 , 2 , , F x u u u k l , 0 G T a R k a Solving the Lie Equation Creating Defining System [ ] ( ) ( ) df MATHEMATICA ( ) ( ) ( ) = ξ ϕ = = ∈ Δ n , , n for n f f x pr F z 0 X z = a 0 F da ϕ ( ) d = η ϕ ϕ = , , f u = a 0 Solving Defining System da ( ) ( ) ξ = ξ η α = η α i i , , , x u x u Basic Infinitesimal Generators ∂ ∂ p q ( ) ( ) ∑ ∑ α = ξ + η i , , X x u x u ν ν ν ∂ ∂ α i x u = α = 1 1 i

Lie Group of Symmetry Transformations ( ) ( ) ( ) ( ) = = Δ 1 n K , , ,..., 0 , 1 , 2 , , F x u u u k l k { } = ∈ δ ⊂ ∈ δ , 0 G T a R a Δ Δ Each solution of after transformation of the group remains a solution of . G T u ′ a u ( ) ′ ′ ′ = ( ) u f x u = f x x ′ x ′ = a ⋅ Δ Δ f T f If is a solution of then is also a solution of . f

The system of PDE and the Prolonged Space ( ) ( ) ( ) ( ) Δ = = 1 n K , , ,..., 0 , 1 , 2 , , F x u u u k l k ( ) ( ) = ∈ = ∈ 1 1 p p q q ,..., ,..., x x x R u u u R ⎧ ⎫ α ∂ ⎪ ⎪ u ( ) α = ≡ α = = = s ⎨ ⎬ 1 ,..., ; 1 ,..., ; 1 ,..., u u q j p k s ∂ ∂ j 1 ,..., j k ⎪ ⎪ ,..., s x x ⎩ ⎭ j j 1 s ( ) ( ) ( ) ( ) ( ) = ∈ 1 2 n n n ∈ = × , , , ,..., z x u u u u Z p q z Z R R ( ) th n is the prolongation of the space Z Z n { } ( ) ( ) ( ) ( ) ( ) Δ = ∈ = ⊂ n n n n 0 z Z F z Z F ( ) Δ Δ n The system is considered as a sub-manifold in the prolonged space . Z F

th X prolongation of the Infinitesimal Generator n ∂ ∂ p q ( ) ( ) ∑ ∑ = ξ + η α i , , X x u x u ∂ ∂ α i x u = α = 1 1 i ∂ ∂ p q p p q ∑∑ ∑ ∑∑ ( ) α α = + ς + + ς n L K pr X X ∂ α ∂ α i j j K u 1 n u = α = = = α = 1 1 1 1 1 i j j K i j j 1 n 1 n ( ) ( ) p ∑ α α α ς = η − ξ s D u D i i s i = s 1 ( ) ( ) p ∑ ς α = ς α − α ξ s D u D K K K j j j j j j j s j − − 1 k k 1 k 1 1 k 1 k = 1 s ∂ ∂ ∂ ∂ q p q p p q ∑ ∑∑ ∑ ∑ ∑ α α α = + + + + L K D u u u − ∂ ∂ ∂ α ∂ α α K i i ji j j i i 1 1 x u u n u α = = α = = α = 1 j 1 1 j j 1 1 K j j j − − 1 n 1 1 1 n

The Infinitesimal Criterion and the Defining System ∂ ∂ p q ( ) ( ) ∑ ∑ = ξ + η α i , , X x u x u ∂ ∂ α i x u = α = 1 1 i ∂ ∂ p q p p q ∑∑ ∑ ∑∑ ( ) α α = + ς + + ς n L K pr X X α α ∂ ∂ i j j K u 1 n u = α = = = α = 1 1 1 1 1 i j j K i j j 1 n 1 n 1 G is a Lie group of symmetry The infinitesimal criterion holds. transformations of the system [ ] ( ) ( ) ( ) ( ) Δ ∈ Δ = n n n for pr F z 0 of PDE with the infinitesimal X z F generator . X Defining System

Symmetry Group of Δ System of PDE { } ( ) ( ) ( ) ( ) = = Δ = ∈ Ω ⊂ ∈ Ω 1 n K , , ,..., 0 , 1 , 2 , , r r F x u u u k l , 0 G T a R k a Solving the Lie Equation Creating Defining System [ ] ( ) ( ) df MATHEMATICA ( ) ( ) ( ) = ξ ϕ = = ∈ Δ n , , n for n f f x pr F z 0 X z = 0 a F da ϕ ( ) d = η ϕ ϕ = , , f u = 0 Solving Defining System a da ( ) ( ) ξ = ξ η α = η α i i , , , x u x u Basic Infinitesimal Generators ∂ ∂ p q ( ) ( ) ∑ ∑ α = ξ + η i , , X x u x u ν ν ν ∂ ∂ α i x u = α = 1 1 i

Data Input M A T H E M A T I C A Basic Setup Creating Defining System Solving Procedure True Solvers Block At least one equation has Equivalent been solved. Transformations Block False Data Output

Data Input { } ≡ = = K 0 , , 0 F F PDE 1 l { } 1 K ≡ p , , x x indvar { } 1 K ≡ q , , u u depvar { } α ≡ u deriv K , , j j 1 s _________________________________________ • Data Input is data about the considered PDE.

Basic Set-Up { } ≡ 1 K , , LHS F F l ≡ Δ Man F { } ( ) ( ) ( ) ( ) ≡ ξ ξ η η 1 p 1 q K K K , , , , , , , , , , InfGen x u x u x u x u ≡ n ( InfGen) pr ProlGen _________________________________________________________ { } ( ) ( ) ( ) ( ) ξ ξ η η 1 p 1 q K K K , , , , , , , , , , • x u x u x u x u are unknown functions that are to be determined and given at the package output as solutions of the defining system.

Creating Defining System Infinitesimal Criterion Defining System ________________________________________________________________ • Defining System is the major object in the program. • Defining System is created by applying the infinitesimal criterion InfGen (LHS) | Man =0. • Defining System consists of linear partial differential equations.

Solving Procedure Solving Transforming Defining System Defining System Solvers Block True At least one equation has Equivalent been solved. Transformations Block Hints False Data Output

Equivalent Transformations Block Module-1 for adding and subtracting of two equations for differentiating of the equations Module-3 Module-4 for breaking the equations into parts _______________________________________________________ • The block is open for adding new modules of equivalent transformations.

Solvers Block + C = 0 solver of C x Module-1 1 2 + = 0 solver of C x C y Module-2 1 2 ′ + = 0 solver of C y C Module-3 1 2 ′ ′ + = 0 Module-4 solver of C y C 1 2 ′ ′ ′ + = Module-5 solver of 0 C y C 1 2 _______________________________________________________ • The block is open for adding new modules for solving equations.

Interactive Mode user level commands

− = Heat Equation 0 u u t xx Input LieInfGenerator {u[t]}, {u[x, x]}, {x, t}, {u}, { infgenx , infgent }, { infgenu } ] Output → {infgenx c[1] t + c[4] x t + c[5] x + c[2], → infgent c[4] t + c[5] t - c[6] }, → f {infgenu - c[4] xu - c[4] t u - c[4] x u - c[3] u + [x, t] } 1 ( ) ( ) { [x, t] - 0 , 1 2 , 0 [x, t] == 0} f f 1 1

Tracing the Evaluation Heat equation − = 0 u u t xx + = 0 C x C y 1 2 + C = 0 C x 1 2 ′ + = 0 C y C 1 2 ′ ′ + = 0 C y C 1 2 ′ ′ ′ + = 0 C y C 1 2

Tracing the Evaluation + = 0 C x C y 1 2 Coupled Nonlinear + C = 0 C x Schrödinger Equations 1 2 ( ) ′ + = ∂ ∂ 2 1 0 A A C y C 2 2 + + + = 1 2 0 i A h B A ∂ ∂ 2 2 x t ( ) ∂ ∂ 2 1 B B + + 2 + 2 = 0 i B h A B ∂ ∂ 2 2 x t Length of Solved System = 131

Trial Run = ∂ Heat equation X 1 x − = 0 u u = ∂ X t xx 2 t = ∂ X u 3 u = ∂ + ∂ 2 X x t 4 x t = 2 ∂ − ∂ X t xu 5 x u ( ) = ∂ + ∂ − + ∂ 2 2 4 4 2 X tx t x t u 6 x t u ( ) = α ∂ , X x t α u ( ) α x , t is an arbitrary solution of the Heat Equation

KdV equation Trial Run + + = 0 u u uu t xxx x ( ) ( ) = − ε = ∂ 1 , u f x t X space translation 1 x ( ) ( ) = − ε 2 = ∂ , u f x t X time translation 2 t ( ) ( ) = − ε + ε 3 = ∂ + ∂ , u f x t t X t Galilean boost 3 x u ( ) ( ) = ∂ + ∂ − ∂ − ε − ε − ε = 4 2 3 3 2 , X x t u dilation u e f e x e t 4 x t u ( ) = , u f x t is an arbitrary solution of the KdV Equation ε ∈ R is the group parameter

References [1] Schwarz, F., Computing 34 (1985) 91. [2] Baumann, G., Math. Comp. Simulation 48 (1998) 205. [3] Baumann, G., Lie Symmetries of Differential equations: a MATHEMATICA Program to Determine Lie Symmetries, at www.library.wolfram.com/infocenter/MathSource/431.