Finding Lie Symmetries of PDEs with MATHEMATICA: Applications to Nonlinear Fiber Optics

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Κλωθώ Μοσχοβάκης
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1 Geomery Inegrably and Qazaon Jne Fndng Le Symmeres of PDEs wh MTHEMTIC: lcaons o Nonlnear Fber Ocs Vladmr Plov Dearmen of Physcs Techncal Unversy-Varna lgara Ivan Uznov Dearmen of led Physcs Techncal Unversy-Sofa lgara Eddy Chacarov Dearmen of Informacs and Mahemacs Varna Free Unversy lgara

2 Plan of Presenaon. MTHEMTIC ackage for fndng Le symmeres of PDE.. lock-scheme and algorhm.. In and o.. Tracng he evalaon.4. Tral rn. lcaons o nonlnear fber ocs.. Physcal model.. Resls obaned. Conclson

3 F Sysem of PDE ( ) ( n) (... ) k K l ( Δ) k G r Symmery Gro of Δ r { T a Ω R Ω } a MTHEMTIC r Creang Defnng Sysem [ F( z )] ( n) ( n ) for Solvng Defnng Sysem ( ) η η ( ) ξ ξ z ( n) Δ F Solvng he Le Eaon df ξ da dϕ η da ( f ϕ ) f a ( f ϕ ) ϕ a asc Infnesmal Generaors ν ξν ( ) η ( ) ν

4 Le Gro of Symmery Transformaons F G ( ) ( n) (... ) k K l k { T a δ δ } R a ( Δ) Each solon of Δ afer ransformaon of he gro G remans a solon of Δ. f ( ) T a ( ) f If f s a solon of Δ hen f T f s also a solon of Δ. a

5 The sysem of PDE and he Prolonged Sace F () ( n) (... ) k K l k ( Δ) ( ) R (... ) R... ( s )... ; k... ; k... s... s... s z Z R R z n ( ) Z ( n) ( ) ( ) ( n)... Z ( n) h s he n rolongaon of he sace Z Δ F ( n) ( n) ( n) ( n) { z Z F( z ) } Z n The sysem Δ s consdered as a sb-manfold n he rolonged sace Z. Δ F ( )

6 rolongaon of he Infnesmal Generaor h n ( ) n n n n r ς ς K K K L ( ) ( ) s s s D D ξ η ς ( ) ( ) s s s k k k k k D D ξ ς ς K K K n n n D K K K L ( ) ( ) η ξ

7 The Infnesmal Creron and he Defnng Sysem ξ ( ) η ( ) r ( n) ς L K ς K n n K n G s a Le gro of symmery ransformaons of he sysem of PDE Δ wh he nfnesmal generaor. The nfnesmal creron holds. r [ F( z )] ( n) ( n ) for z ( n) Δ F Defnng Sysem

8 F Sysem of PDE ( ) ( n) (... ) k K l ( Δ) k G r Symmery Gro of Δ r { T a Ω R Ω } a MTHEMTIC r Creang Defnng Sysem [ F( z )] ( n) ( n ) for Solvng Defnng Sysem ( ) η η ( ) ξ ξ z ( n) Δ F Solvng he Le Eaon df ξ da dϕ η da ( f ϕ ) f a ( f ϕ ) ϕ a asc Infnesmal Generaors ν ξν ( ) η ( ) ν

9 Daa In M T H E M T I C asc Se Solvers lock Evalen Transformaons lock Creang Defnng Sysem Solvng Procedre leas one eaon has been solved. False Tre Daa O

10 PDE ndvar devar derv { F K F } l { K } { K } { } K s Daa In Daa In s daa abo he consdered PDE.

11 asc Se-U LHS { } F K F l Man InfGen Δ F { ξ ( ) K ξ ( ) K η ( ) K η ( ) } ProlGen n r (InfGen) { ξ ( ) K ξ ( ) K η ( ) K η ( ) } are nknown fncons ha are o be deermned and gven a he ackage o as solons of he defnng sysem.

12 Creang Defnng Sysem Infnesmal Creron Defnng Sysem Defnng Sysem s he maor obec n he rogram. Defnng Sysem s creaed by alyng he nfnesmal creron InfGen (LHS) Man. Defnng Sysem consss of lnear aral dfferenal eaons.

13 Solvng Procedre Transformng Defnng Sysem Solvng Defnng Sysem Solvers lock Evalen Transformaons lock Hns leas one eaon has been solved. False Tre Daa O

14 Evalen Transformaons lock Modle- for addng and sbracng of wo eaons Modle- for dfferenang of he eaons Modle-4 for breakng he eaons no ars The block s oen for addng new modles of evalen ransformaons.

15 Solvers lock Modle- C C solver of Modle- solver of C C y Modle- solver of C y C Modle-4 solver of C y C Modle-5 solver of C y C The block s oen for addng new modles for solvng eaons.

16 ser level commands Ineracve Mode

18 Hea Eaon In LeInfGeneraor {[]} {[ ]} { } {} { nfgen nfgen } { nfgen } ] O {nfgen c[] c[4] c[5] c[] nfgen c[4] c[5] - c[6] } {nfgen - c[4] - c[4] - c[4] - c[] [ ] } f { ( ) [ ] - ( ) [ ] } f f

19 Tracng he Evalaon Hea eaon C C y C C C y C C C y C y C

20 Tracng he Evalaon C C C C y C y C Coled Nonlnear Schrödnger Eaons ( h ) ( h ) Lengh of Solved Sysem

21 Tral Rn Hea eaon ( ) ( ) ( ) s an arbrary solon of he Hea Eaon

22 Tral Rn KdV eaon sace ranslaon ( ) f ( ) me ranslaon ( ) f ( ) Gallean boos ( ) f ( ) 4 dlaon ( 4) ( e f e e ) ( ) f s an arbrary solon of he KdV Eaon R s he gro arameer

23 References [] Schwarz F. Comng 4 (985) 9. [] amann G. Mah. Com. Smlaon 48 (998) 5. [] amann G. Le Symmeres of Dfferenal eaons: a MTHEMTIC Program o Deermne Le Symmeres a

24 lcaon o Fber Ocs (hyscal model) Coled Nonlnear Schrödnger Eaons (CNSEs) σ θ θ γ σ θ θ γ ν weak brefrngen fbers wo-mode fbers srong brefrngen fbers Raman gan coeffcen σ γ σ γ σ θ

25 Le Gro nalyss Coled nonlnear Schrödnger eaons ν ( γ ) ( γ ) wo mode γ ocal fber srong brefrngen γ gro velocy dserson dmed Le on symmeres ( ) ς e( β ) z e negave ν osve ν gros algebras T a 4 T T T4 β a a β β a4 5 T 5 ν β a5 a5 a5 νa β β νa5 5 6 T 6 z z ( ) ( ) e a 6 e a z z e( a 6 ) ξ ξ e( a 6 ) ς ς 6

26 β 5 β 4 a a a 4 a β β a 5 a a 5 5 a a 5 5 β β gros gros algebras algebras T T 4 T T 5 T ( ) ( ) β ς z e e ( ) ( ) ( ) ( ) θ θ γ θ θ γ srong brefrngen fber γ srong brefrngen fber wh arallel Raman scaerng θ Le Gro nalyss Le Gro nalyss Coled nonlnear Schrödnger eaons dmed Le on symmeres

27 Le Gro nalyss Coled nonlnear Schrödnger eaons ( γ ) dmed Le on symmeres ( γ ) k k fber weak brefrngen γ k nonlnear dreconal coler γ k ( ) ς e( β ) z e gros algebras T a T T a a β β a T 4 4 β a4 a4 β β a4 a4 a4

28 SYMMETRY GROUP REDUCTION symmery gro adon reresenaons classfcaon omal se of sbalgebras omal se of redced ODEs omal se of gro nvaran solons

29 INTERIOR UTOMORPHISMS wo mode fbers srong brefrngen fbers ν ( γ ) ( γ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ν ) 4 e ( ν ) 4 e ( ν ) e ( ) [ ] [ [ ] L

30 OPTIML SET OF SULGERS OPTIML SET OF SULGERS Case Case C Case ( ) β ν 5 4 β δ δ 4 Case D ( ) β ν δ δ 5 4 Case E β δ ς ς δ z z 6 4 Case F β δ δ 4 ± ± R ± ± δ or R ± δ R δ or δ δ R wo mode fbers wo mode fbers srong brefrngen fbers srong brefrngen fbers ( ) ( ) γ ν γ

31 σ σ Nonlnear dreconal coler Nonlnear dreconal coler Redced sysem Redced sysem ( ) ( ) ( ) ( ) g f gf f g f g f f g cos cos sn sn σ σ δ σ σ ( ) ( ) ( ) dn arcsn 4 e cn h E h E E σ σ ( ) ( ) ( ) dn arcsn 4 e cn h E h E E σ σ 4σ cons E h E Eac solon Eac solon Eac solon for Case

32 REDUCTION PROCES (Case C) wo mode fbers srong brefrngen fbers ν ( γ ) ( γ ) ( ) ς e( β ) z e Generaor δ δ β 4 ± or ± δ R Invarans J z ς J J J δ J 4 β New varables z ( ) ς ( ) f ( ) δ β g( ) Redced sysem f f g g ( f ) γ δ ( g ) ν ν γ ν

33 Eac solon for Case C (wo-mode fbers and srongly brefrngen fbers) ( ) Π m n b h C U λ ; e ( ) Π ± m n b h C U λ ; e ( ) ( ) cn U b b h b m b b λ λ ( ) ( ) [ ] dw m w n m n Π sn ; ± b b b n b b b b m are he roos of he olynomal b b b > > ( ) ( ) 4 h C h C h Q θ θ θ θ and are he Jacobean sne and cosne ellc fncons ( ) m sn ( ) m cn

34 romae vecor solary waves Srong brefrngen fbers wh Raman scaerng generalzed verson of revosly obaned scalar solary-wave solon c ( c a) ( c b) β ( a cy) ( b cy) yy yy C C ( h ) θ ( ) ( h ) θ ( ) y y y y Gallean-lke symmery redced sysem ( z) θ F( z) a sech sech z θ G( z) θ << Raman arameer a a a F anh ( z) z z ln( sech z) z ( z) snh z z G sech ( z) z G sech. L. Gagnon and P.. élanger Solon self-freency shf verss Gallean-lke symmery O. Le. Vol. 5 No. 9 (99)

35 ( z) θ F( z) a sech sech z θ G( z) a a a F anh ( z) z z ln( sech z) z ( z) snh z z G sech ( z) z G sech τ»».8.6 τ»» τ»» 8-6 F ( z) G ( z) G ( z) z z z. L. Gagnon and P.. élanger Solon self-freency shf verss Gallean-lke symmery O. Le. Vol. 5 No. 9 (99) N. khmedev and. nkewcz Novel solon saes and bfrcaon henomena n nonlnear fber colers Phys. Rev. Le. Vol. 7 No. 6 (99)

36 LWS OF CONSERVTION Two-mode fbers and srong brefrngen fbers SYMMETRY LWS OF CONSERVTION TIME TRNSLTION SPCE TRNSLTION J J * * ( ) H - d ( ) ( ) 4 4 ν h d TRNSLTION OF THE PHSE J d TRNSLTION OF THE PHSE β J 4 d GLILEN-LIKE SYMMETRY J 5 ( ) ν d J

37 References [] Chrsodoldes D.N. and R.I. Joseh Ocs Le. () 5-55 (988). [] Trank M. V. and J. E. Se Phys. Rev. 8(4) -7 (988). [] Chrsodoldes D.N. Phys. Le. (8 9) (988). [4] Floranczyk M. and R. Tremblay Phys. Le. 4() 4-6 (989). [5] Kosov N.. and I. M. Uznov O. Commn (99). [6] Floranczyk M. and R. Tremblay O. Commn (994). [7] Plov V. I. Uznov and E. Chacarov Phys. Rev E 57 () (998).

38 Conclson The symbolc comaonal ools of MTHEMTIC have been aled o deermnng he Le symmeres of PDE. n algorhm for creang and solvng he defnng sysem of he symmery ransformaons has been develoed and mlemened n MTHEMTIC ackage. The ackage has been sccessflly aled o basc hyscal eaons from nonlnear fber ocs. Fre work: The ackage caables can be eended by addng new rogrammng modles for ransformng and solvng oher wder classes of dfferenal eaons.

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