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

SLIDE 1

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

Geometry, Geometry, Integrability Integrability and and Quatization Quatization − − June 8 June 8-

• 13, 2007

13, 2007

SLIDE 2

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

SLIDE 3

Symmetry Group of Δ

{ }

, Ω ∈ ⊂ Ω ∈ =

r a r

R a T G

System of PDE

( ) ( )

( )

l k u u u x F

n k

, , 2 , 1 , ,..., , ,

1

K = =

( )

Δ Creating Defining System

( ) ( )

( ) [ ]

z F pr

n

= X

n

for

Solving Defining System

( ) ( )

u x u x

i i

, , ,

α α

η η ξ ξ = =

( )

F n

z Δ ∈

MATHEMATICA

( ) ( )

u f da d x f f da df

a a

= = = =

= =

, , , , ϕ ϕ η ϕ ϕ ξ

Solving the Lie Equation

( ) ( )

∑ ∑

= =

∂ ∂ + ∂ ∂ =

q p i i i

u u x x u x X

1 1

, ,

α α α ν ν ν

η ξ

Basic Infinitesimal Generators

SLIDE 4

Each solution of after transformation of the group remains a solution of .

G Δ Δ

Lie Group of Symmetry Transformations

( ) ( )

( )

l k u u u x F

n k

, , 2 , 1 , ,..., , ,

1

K = =

{ }

, δ δ ∈ ⊂ ∈ = R a T G

a

( )

Δ u x

( )

x f u =

u′ x′

( )

x f u ′ ′ = ′

a

T

If is a solution of then is also a solution of .

f T f

a ⋅

= ′ Δ f Δ

SLIDE 5

The system of PDE and the Prolonged Space

( ) ( )

( )

l k u u u x F

n k

, , 2 , 1 , ,..., , ,

1

K = =

( )

p p

R x x x ∈ = ,...,

1

( )

q q

R u u u ∈ = ,...,

1

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ = = = ∂ ∂ ∂ ≡ = s k p j q x x u u u

k j j j j s

s s

,..., 1 ; ,..., 1 ; ,..., 1 ,...,

1 1,...,

α

α α

( )

Δ

( ) ( ) ( ) ( )

( )

n n n

Z u u u u x z ∈ = ,..., , , ,

2 1 q p

R R Z z × = ∈

is the prolongation of the space Z

( )

n

Z

th

n

( ) ( ) ( )

( )

{ }

( )

n n n n F

Z z F Z z ⊂ = ∈ = Δ The system is considered as a sub-manifold in the prolonged space .

F

Δ Δ

( )

n

Z

SLIDE 6

prolongation of the Infinitesimal Generator

th

n

X

( )

∑∑ ∑ ∑∑

= = = = =

∂ ∂ + + ∂ ∂ + =

p j q j j j j p j i p i q i n

n n n

u u X X

1 1 1 1 1

1 1 1

pr

α α α α α α

ς ς

K

K

K L

( ) ( )

∑

=

− =

p s s i s i i

D u D

1

ξ η ς

α α α

( ) ( )

s j p s s j j j j j j j

k k k k k

D u D ξ ς ς

α α α

∑

=

− −

− =

1

1 1 1 1 1

K K K

∑ ∑ ∑ ∑∑ ∑

= = = = =

− − − ∂

∂ + + ∂ ∂ + ∂ ∂ + ∂ ∂ =

p j p j q j j i j j p j q j ji q i i i

n n n

u u u u u u x D

1 1 1 1 1 1

1 1 1 1 1 α α α α α α α α α K K

K L

( ) ( )

α α α

η ξ u u x x u x X

q i p i i

∂ ∂ + ∂ ∂ =

∑ ∑

= =

, ,

1 1

SLIDE 7

The Infinitesimal Criterion and the Defining System

( )

∑∑ ∑ ∑∑

= = = = =

∂ ∂ + + ∂ ∂ + =

p j q j j j j p j i p i q i n

n n n

u u X X

1 1 1 1 1

1 1 1

pr

α α α α α α

ς ς

K

K

K L

( ) ( )

α α α

η ξ u u x x u x X

q i p i i

∂ ∂ + ∂ ∂ =

∑ ∑

= =

, ,

1 1

is a Lie group of symmetry transformations of the system

• f PDE with the infinitesimal

generator .

1

G Δ X

The infinitesimal criterion holds. for

( )

F n

z Δ ∈

( ) ( )

( ) [ ]

z F pr

n

= X

n

Defining System

SLIDE 8

Symmetry Group of Δ

{ }

, Ω ∈ ⊂ Ω ∈ =

r a r

R a T G

System of PDE

( ) ( )

( )

l k u u u x F

n k

, , 2 , 1 , ,..., , ,

1

K = =

( )

Δ Creating Defining System

( ) ( )

( ) [ ]

z F pr

n

= X

n

for

Solving Defining System

( ) ( )

u x u x

i i

, , ,

α α

η η ξ ξ = =

( )

F n

z Δ ∈

MATHEMATICA

( ) ( )

u f da d x f f da df

a a

= = = =

= =

, , , , ϕ ϕ η ϕ ϕ ξ

Solving the Lie Equation

( ) ( )

∑ ∑

= =

∂ ∂ + ∂ ∂ =

q p i i i

u u x x u x X

1 1

, ,

α α α ν ν ν

η ξ

Basic Infinitesimal Generators

SLIDE 9

Data Input Basic Setup

Equivalent Transformations Block Solvers Block

Data Output

At least one equation has been solved.

Creating Defining System Solving Procedure

False True

M A T H E M A T I C A

SLIDE 10

Data Input

PDE indvar depvar deriv _________________________________________

• Data Input is data about the considered PDE.

{ }

α

s

j j

u

, ,

1

K

≡

{ }

, ,

1

= = ≡

l

F F K

{ }

p

x x , ,

1 K

≡

{ }

q

u u , ,

1 K

≡

SLIDE 11

Basic Set-Up

LHS Man InfGen ProlGen

(InfGen)

_________________________________________________________

• are unknown

functions that are to be determined and given at the package

• utput as solutions of the defining system.

{ }

l

F F , ,

1 K

≡

F

Δ ≡

( ) ( ) ( ) ( )

{ }

u x u x u x u x

p

, , , , , , , , , ,

q 1 1

η η ξ ξ K K K ≡

( ) ( ) ( ) ( )

{ }

u x u x u x u x

p

, , , , , , , , , ,

q 1 1

η η ξ ξ K K K

n

pr ≡

SLIDE 12

Creating 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.

Defining System Infinitesimal Criterion

SLIDE 13

At least one equation has been solved.

Data Output

False True

Solving Procedure

Transforming Defining System Solving Defining System

Equivalent Transformations Block Solvers Block Hints

SLIDE 14

Equivalent Transformations Block

_______________________________________________________

• The block is open for adding new modules of equivalent

transformations.

for breaking the equations into parts

Module-4

for differentiating of the equations

Module-3

for adding and subtracting of two equations

Module-1

SLIDE 15

Solvers Block

_______________________________________________________

• The block is open for adding new modules for solving

equations.

solver of

Module-4

solver of

Module-3

solver of

Module-2

solver of

Module-1

2 1

= + C x C

Module-5

solver of

2 1

= + y C x C

2 1

= + ′ C y C

2 1

= + ′ ′ C y C

2 1

= + ′ ′ ′ C y C

SLIDE 16

Interactive Mode

user level commands

SLIDE 17

SLIDE 18

Heat Equation

= −

xx t

u u

LieInfGenerator {u[t]}, {u[x, x]}, {x, t}, {u}, { infgenx , infgent }, { infgenu } ] Input Output {infgenx c[1] t + c[4] x t + c[5] x + c[2], infgent c[4] t + c[5] t - c[6] }, {infgenu

• c[4] xu - c[4] t u - c[4] x u - c[3] u + [x, t] }

{ [x, t] - [x, t] == 0}

→ → →

( )

1 , 1

f

( )

, 2 1

f

1

f

SLIDE 19

Heat equation

= −

xx t

u u

Tracing the Evaluation

2 1

= + C x C

2 1

= + ′ C y C

2 1

= + ′ ′ C y C

2 1

= + ′ ′ ′ C y C

2 1

= + y C x C

SLIDE 20

Tracing the Evaluation

Coupled Nonlinear Schrödinger Equations

( )

2 1

2 2 2 2

= + + ∂ ∂ + ∂ ∂ A B h A t A x A i

( )

2 1

2 2 2 2

= + + ∂ ∂ + ∂ ∂ B A h B t B x B i

Length of Solved System = 131

2 1

= + ′ C y C

2 1

= + C x C

2 1

= + y C x C

SLIDE 21

Trial Run

Heat equation

= −

xx t

u u

is an arbitrary solution of the Heat Equation

( )

t x, α

x

X ∂ =

1 t

X ∂ =

2 u

u X ∂ =

3 t x

t x X ∂ + ∂ = 2

4 u x

xu t X ∂ − ∂ = 2

5

( )

u t x

u t x t tx X ∂ + − ∂ + ∂ = 2 4 4

2 2 6

( )

u

t x X ∂ = , α

α

SLIDE 22

Trial Run

KdV equation

= + +

x xxx t

uu u u

time translation

x

X ∂ =

1 t

X ∂ =

2 u x

t X ∂ + ∂ =

3 u t x

u t x X ∂ − ∂ + ∂ = 2 3

4

space translation

( )

( )

ε − = t x f u ,

2

( )

( )

ε ε + − = t t x f u ,

3

( )

( )

t e x e f e u

ε ε ε 3 2 4

,

− − −

=

dilation Galilean boost

is an arbitrary solution of the KdV Equation is the group parameter

( )

t x f u , = R ∈ ε

( )

( )

t x f u ,

1

ε − =

SLIDE 23

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.

SLIDE 24

Application to Fiber Optics

(physical model)

Coupled Nonlinear Schrödinger Equations (CNSEs)

2 1

2 2 2 2 2 2

= + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ − + + ∂ ∂ + ∂ ∂ B A t B t A B A t A x A i σ θ θ γ 2

2 2 2 2 2 2

= + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ − + + ∂ ∂ + ∂ ∂ A B t B t A A B t B x B i σ θ θ γ ν

weak birefringent fibers two-mode fibers strong birefringent fibers Raman gain coefficient

≠ σ 2 , = = γ σ 3 2 , = = γ σ θ

SLIDE 25

Lie Group Analysis Lie Group Analysis

t X ∂ ∂ =

1

x X ∂ ∂ =

2

α ∂ ∂ =

3

X β ν α ∂ ∂ + ∂ ∂ + ∂ ∂ = t t t x X 5

ς ς ∂ ∂ + ∂ ∂ + ∂ ∂ − ∂ ∂ − = z z x x t t X 2

6

β ∂ ∂ =

4

X

1

a t t + = ′

2

a x x + = ′

3

a + = ′ α α

4

a + = ′ β β x a t t

5

+ = ′ x a t a 2

2 5 5 +

+ = ′ α α x a t a 2

2 5 5

ν ν β β + + = ′

( )

6

exp a t t − = ′

( )

6

2 exp a x x − = ′

( )

6

exp a z z = ′

( )

6

exp a ξ ξ = ′

groups groups algebras algebras

2

T

3

T

4

T

5

T

6

T

1

T

( ) ( )

β ς α i B i z A exp exp = =

Admitted Lie point symmetries Coupled nonlinear Schrödinger equations

( ) ( )

B B B 2 B 2 1

2 2 2 2 2 2 2 2

= + + ∂ ∂ + ∂ ∂ = + + ∂ ∂ + ∂ ∂ A t x i A B A t A x A i γ ν γ

group velocity dispersion

positive 1 − = ν negative 1 + = ν

• ptical fiber

two mode 2 = γ strong birefringent 3 2 = γ

SLIDE 26

t X ∂ ∂ =

1

x X ∂ ∂ =

2

α ∂ ∂ =

3

X ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ = β α t t x X 5

β ∂ ∂ =

4

X

1

a t t + = ′

2

a x x + = ′

3

a + = ′ α α

4

a + = ′ β β x a t t

5

+ = ′ x a t a 2

2 5 5 +

+ = ′ α α x a t a 2

2 5 5 +

+ = ′ β β

groups groups algebras algebras

2

T

3

T

4

T

5

T

1

T

( ) ( )

β ς α i B i z A exp exp = =

( ) ( ) ( ) ( )

B B B 2 1 B 2 1

2 2 2 2 2 2 2 2 2 2 2 2

= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ − + + ∂ ∂ + ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ − + + ∂ ∂ + ∂ ∂ t B t A A t x i A t B t A B A t A x A i θ θ γ θ θ γ

strong birefringent fiber 3 2 = γ strong birefringent fiber with parallel Raman scattering ≠ θ

Lie Group Analysis Lie Group Analysis

Coupled nonlinear Schrödinger equations Admitted Lie point symmetries

SLIDE 27

t X ∂ ∂ =

1

α α ∂ ∂ + ∂ ∂ =

3

X ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ = β α t t x X 4

1

a t t + = ′

2

a x x + = ′

3

a + = ′ α α

3

a + = ′ β β x a t t

4

+ = ′ x a t a 2

2 4 4 +

+ = ′ α α x a t a 2

2 4 4 +

+ = ′ β β

groups groups algebras algebras

2

T

3

T

4

T

1

T

( ) ( )

β ς α i B i z A exp exp = =

( ) ( )

B B B 2 1 B 2 1

2 2 2 2 2 2 2 2

= + + + ∂ ∂ + ∂ ∂ = + + + ∂ ∂ + ∂ ∂ kA A t x i kB A B A t A x A i γ γ

fiber

weak birefringent ≠ ≠ k γ nonlinear directional coupler ≠ = k γ

x X ∂ ∂ =

2

Admitted Lie point symmetries Coupled nonlinear Schrödinger equations

Lie Group Analysis Lie Group Analysis

SLIDE 28

SYMMETRY SYMMETRY GROUP GROUP REDUCTION REDUCTION

• ptimal set
• f subalgebras
• ptimal set
• f reduced ODEs
• ptimal set
• f group invariant solutions

classification

adjoint representations symmetry group

SLIDE 29

INTERIOR AUTOMORPHISMS INTERIOR AUTOMORPHISMS

( )

ε

1

A

( )

ε

2

A

( )

ε

4

A

( )

ε

5

A

( )

ε

6

A

( )

ε

3

A

( )

ε

i

A

1

X

1

X

1

X

1

X

1

X ( )

4 3 1

X X X ν ε + +

1

e X

ε − 2

X

2

X

2

X

2

X

2

X

( )

4 3 1 2

X X X X ν ε ε + + +

2 2

e X

ε −

3

X

3

X

3

X

3

X

3

X

3

X

3

X

4

X

4

X

4

X

4

X

4

X

4

X

4

X

5

X

5

X

5

X

1 5

X X ε −

5

X

( )

4 3 5

X X X ν ε + +

5

e X

ε 6

X

6

X

5 6

X X ε −

6

X

1 6

X X ε +

6

X

2 6

2 X X ε + ( )

[ ] [ ] [ ] L

− + − =

j i i j i j j i

X X X X X X X A , , 2 ,

2

ε ε ε

• two mode fibers

two mode fibers

• strong birefringent fibers

strong birefringent fibers

( ) ( )

B B B 2 B 2 1

2 2 2 2 2 2 2 2

= + + ∂ ∂ + ∂ ∂ = + + ∂ ∂ + ∂ ∂ A t x i A B A t A x A i γ ν γ

SLIDE 30

OPTIMAL SET OF SUBALGEBRAS OPTIMAL SET OF SUBALGEBRAS

α ε ε ∂ ∂ + ∂ ∂ = + t X X

3 1

Case A Case C Case B

( ) β

ν ε α ε ∂ ∂ + + ∂ ∂ + ∂ ∂ = + t t t x X X

5 4

β ε α δ ε δ ∂ ∂ + ∂ ∂ + ∂ ∂ = + + x X X X

4 3 2

Case D

( ) β

ν δ α ε δ ε ∂ ∂ + + ∂ ∂ + ∂ ∂ + ∂ ∂ = + + t t x t x X X X

5 4 2

Case E

β δ α ε ς ς δ ε ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ − ∂ ∂ − = + + z z x x t t X X X 2

6 4 3

Case F

β δ α ε δ ε ∂ ∂ + ∂ ∂ = +

4 3

X X 1 , 0 ± = ε 1 , 0 ± = ε R ∈ ± = ± = δ ε ε , 1

• r

1 , R ∈ ± = δ ε , 1 R ∈ δ ε, 1 ,

• r

, 1 = ∈ = = δ ε δ ε R

• two mode fibers

two mode fibers

• strong birefringent fibers

strong birefringent fibers

( ) ( )

B B B 2 B 2 1

2 2 2 2 2 2 2 2

= + + ∂ ∂ + ∂ ∂ = + + ∂ ∂ + ∂ ∂ A t x i A B A t A x A i γ ν γ

SLIDE 31

B 2 1 B 2 1

2 2 2 2 2 2

= + + ∂ ∂ + ∂ ∂ = + + ∂ ∂ + ∂ ∂ A B B t x i B A A t A x A i σ σ

Nonlinear directional coupler Nonlinear directional coupler Reduced system Reduced system

( ) ( ) ( ) ( )

g f q p q gf f g p q p f g f p q f g q p − + = − + − = ′ = − + ′ = − + ′ cos cos 2 sin sin

2 2 2

σ σ δ σ σ

( ) ( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − + = 2 | 2 dn arcsin 4 3 exp 2 | 2 cn

2 2

h x Ex i h x E E A σ σ

( ) ( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − = 2 | 2 dn arcsin 4 3 exp 2 | 2 cn

2 2

h x Ex i h x E E B σ σ σ 4 const,

2 2

E h E B A = = = +

Exact solution Exact solution

Exact solution for Case A

SLIDE 32

REDUCTION PROCES REDUCTION PROCES

(Case C) (Case C)

Generator

β ε α δ ε δ ∂ ∂ + ∂ ∂ + ∂ ∂ = + + x X X X

4 3 2

R ∈ ± = ± = δ ε ε , 1

• r

1 ,

• two mode fibers

two mode fibers

• strong birefringent fibers

strong birefringent fibers

( ) ( )

B B B 2 B 2 1

2 2 2 2 2 2 2 2

= + + ∂ ∂ + ∂ ∂ = + + ∂ ∂ + ∂ ∂ A t x i A B A t A x A i γ ν γ

Invariants

t J =

1

z J =

2

ς =

3

J x J δ α − =

3

x J ε β − =

4

New variables

( )

x p z = ( ) ( )

β ς α i B i z A exp exp = =

( )

x q = ς

( )

x t f δ α + =

( )

x t g ε β + = Reduced system

( ) ( )

2 2 2 2 2 2 2 2

2 3 2 2 3 2

= − + + ′ − ′ ′ = − + + ′ − ′ ′ = ′ ′ + ′ ′ = ′ ′ + ′ ′ q qp q g q q p pq p f p p g q g q f p f p ε ν γ ν ν δ γ

SLIDE 33

Exact solution for Case C

(two-mode fibers and strongly birefringent fibers)

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + Π + = x m j n b h C i U A ε λ | ; 1 2 exp

1 1

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + Π + ± = x m j n b h C i U B ε λ | ; 1 2 exp

1 1

( ) ( )

2 1 2 2 2 1

2 1 , 1 2 , | cn U b b t h j b m j b b − = + = + − = λ λ

( ) ( )

[ ] dw

m w n m j n

j

∫

−

− = Π

1 2

| sn 1 | ; 1 0, , ,

1 2 1 3 1 2 1

± = − = − − = ε b b b n b b b b m

are the roots of the polynomial

3 2 1

b b b > >

( ) ( )

1 1 4 1 2

2 1 2 2 3

+ + + − + − = h C h C h Q θ θ ε θ θ

and are the Jacobean sine and cosine elliptic functions

( )

m j | sn

( )

m j | cn

SLIDE 34

Approximate vector solitary waves Approximate vector solitary waves

• Strong birefringent fibers with Raman scattering

Strong birefringent fibers with Raman scattering

• A generalized version of previously obtained scalar solitary-wave solution
• A generalized version of previously obtained scalar solitary-wave solution

( ) ( )

z z a z a z a z F tanh sech ln 5 8 15 8 15 16

2

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − =

( )

z z z G

2 1

sech sinh =

( )

z z G

2 2

sech =

( ) ( )

β α

∂ − + ∂ − + ∂ + ∂ = b ct a ct cx X

t x

( ) ( )

sech z F sech 2 z z a A θ + − =

( )

z G B θ =

Galilean-like symmetry reduced system

( ) ( ) ( ) ( )

2 2 1 ) ( 2 2 1 ) (

2 2 2 3 2 2 2 2 2 3 2 1

= + − + + − + − = + − + + − + −

y y yy y y yy

qpp q q q hp q q C q q cy b pqq p p p hq p p C p p cy a θ θ

1.

• L. Gagnon and P. A. Bélanger, Soliton self-frequency shift versus Galilean-like symmetry, Opt. Lett., Vol. 15, No. 9 (1990), pp. 466-468.

1 << θ − Raman parameter

SLIDE 35

• 3
• 2
• 1

1 2 3 z . 2 . 4 . 6 . 8 1

τ

2»A

»

2

• 3
• 2
• 1

1 2 3 z 5

×1

0-7 1

×1

0-6 1 . 5×1 0-6 2

×1

0-6

τ

2»B

»

2

• 3
• 2
• 1

1 2 3 z 2

×1

• 6

4

×1

• 6

6

×1

• 6

8

×1

• 6

τ

2»B

»

2

( ) ( )

z z a z a z a z F tanh sech ln 5 8 15 8 15 16

2

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − =

( )

z z z G

2 1

sech sinh =

( )

z z G

2 2

sech =

( ) ( )

sech z F sech 2 z z a A θ + − =

( )

z G B θ =

( )

z F

( )

z G1

( )

z G2 1.

• L. Gagnon and P. A. Bélanger, Soliton self-frequency shift versus Galilean-like symmetry, Opt. Lett., Vol. 15, No. 9 (1990), pp.

466-468. 2.

• N. Akhmediev and A. Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers, Phys. Rev. Lett., Vol.

70, No. 16 (1993), pp. 2395-2398.

SLIDE 36

LAWS LAWS OF OF CONSERVATION CONSERVATION

• Two

Two-

• mode

mode fibers and strong fibers and strong birefringent birefringent fibers fibers

( )

( ) ( ) ( )

1 2 2 5 2 4 2 3 2 2 4 4 2 2 2 * * 1

J J 2 1 2 1

• H

J J ixJ dt B A t J dt B dt A dt B A h B A B A dt B B A A

x x x x x x x x t t x x t t

+ + = = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + + = ≡ + =

∫ ∫ ∫ ∫ ∫

− − − − −

ν ν

TIME TRANSLATION

SYMMETRY LAWS OF CONSERVATION

SPACE TRANSLATION TRANSLATION OF THE PHASE α TRANSLATION OF THE PHASE β GALILEAN-LIKE SYMMETRY

SLIDE 37

References

[1] Christodoulides, D.N. and R.I. Joseph, Optics Lett., 13(1), 53-55 (1988). [2] Tratnik, M. V. and J. E. Sipe, Phys. Rev. A, 38(4), 2011-2017 (1988). [3] Christodoulides, D.N., Phys. Lett. A, 132(8, 9), 451-452 (1988). [4] Florjanczyk, M. and R. Tremblay, Phys. Lett. A, 141(1,2), 34-36 (1989). [5] Kostov, N. A. and I. M. Uzunov, Opt. Commun., 89, 389-392 (1992). [6] Florjanczyk, M. and R. Tremblay, Opt. Commun., 109, 405-409 (1994). [7] Pulov V., I. Uzunov, and E. Chacarov, Phys. Rev E, 57 (3), 3468-3477 (1998).

SLIDE 38

Conclusion

• The symbolic computational tools of MATHEMATICA have been

applied to determining the Lie symmetries of PDE.

• An algorithm for creating and solving the defining system of

the symmetry transformations has been developed and implemented in MATHEMATICA package.

• The package has been successfully applied to basic physical

equations from nonlinear fiber optics.

• Future work: The package capabilities can be extended by

adding new programming modules for transforming and solving other wider classes of differential equations.