FRACTIONAL OPTICAL SOLITARY WAVE SOLUTIONS OF THE HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume Number /00x pp FRACTIONAL OPTICAL SOLITARY WAVE SOLUTIONS OF THE HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION Zhenya YAN Insue of Sysems Scence Key Laboraory of Mahemacs Mechanaon AMSS Chnese Academy of Scences Bejng 0090 Chna E-mal: yyan@mmrc.ss.ac.cn; yyan_mah@yahoo.com Based on a specfc fraconal ransformaon we fnd new ypes of fraconal opcal solary wave soluons of he hgher-order nonlnear Schrödnger equaon wh he group velocy dsperson selfphase modulaon hrd order dsperson self-seepenng and self-frequency shf whch descrbes he propagaon of femosecond lgh pulses n nonlnear opcal fbers. For dfferen values of a real parameer R hese new soluons can descrbe concavely W-shaped ( R > ) convexly W-shaped ( < R < 0) brgh (0 < R < ) and dark (< R ) opcal solary wave soluons. In parcular for he concavely W-shaped soluon ( R > 0 ) he larger he parameer R becomes he larger he maxmum value of he cener par of he nensy s and when R hs maxmum value approaches a consan. Whle for he convexly W-shaped soluon ( < R < 0 ) he smaller he absolue value of he parameer R becomes he larger he maxmum value of he cener par of he nensy s. Key words: hgher-order nonlnear Schrödnger equaon; fraconal ransformaon; opcal solary wave soluons; perodc wave soluons.. INTRODUCTION The nonlnear Schrödnger (NLS) equaon s used o model he propagaon of pcosecond opcal pulses and s of fundamenal mporance n opcal fber communcaons [ 6] (he NLS equaon s known as he Gross-Paevsk equaon n he sudy of Bose-Ensen condensaes [7 8]). I s well known ha he NLS equaon s a relavely smple solon equaon and s compleely negrable by usng he nverse scaerng ransform he Panlevé analyss he blnear ransformaon ec. [ ]. However a lo of expermenal and heorecal works have shown ha he NLS equaon s no longer vald for he sudy of femosecond opcal pulses and of ulrashor (few-cycle) opcal solons [ ]. In order o model he propagaon of much shorer (femosecond) opcal pulses hrd order and nonlnear dsperson erms have been added o he NLS equaon. As a resul he hgher-order nonlnear Schrödnger (HONLS) equaon was pu forward n he form [9 ] E = ( E E E) E ( E E) E( E ) () where E E( ) sands for he slowly varyng envelope of he elecrc feld he subscrps and are he emporal and spaal paral dervaves respecvely ( =... ) are he real parameers relaed o he group velocy dsperson (GVD) self-phase modulaon (SPM) hrd order dsperson (TOD) selfseepenng (SS) and self-frequency (SF) shf va smulaed Raman scaerng respecvely. Equaon () ncludes many ypes of complex nonlnear wave equaons for he dfferen parameers ( = ) n he area of nonlnear opcs: ) When =0 ( = ) Eq. () becomes he cubc NLS equaon wh GVD and SPM.e. E = ( E E E) whch adms many negrably properes [ 6].

2 9 Z. Yan ) When =0 ( = ) Eq. () becomes he dervave NLS equaon ncludng GVD and SS.e. E = E E( E ) whch descrbes he propagaon of crcularly polared nonlnear van Alfvén waves n plasmas []; s negrable va he nverse scaerng ransform and adms mul-solon soluons []. ) When =0 ( = ) Eq. () becomes he exended dervave NLS equaon ncludng GVD SPM and SS.e. E = ( E E E) ( E E) [6 7] whose N-solon soluons have been obaned see Ref. [6]. v) When = 0 Eq. () becomes he Hroa-ype HONLS equaon wh GVD SPM TOD SS and SF.e. E = ( E E E) E E E whose N-solon soluons were gven n Ref. [8]. v) When = 0 Eq. () becomes he generaled NLS equaon wh GVD SPM TOD and SS.e. E = ( E E E) E ( E E) ; solary wave soluons and N-solon soluons of hs equaon were also deduced [9 0]. v) When = = 0 Eq. () reduces o he complex modfed KdV equaon wh TOD SS and SF.e. E = E ( E E) E( E ) [9 ]. For Eq. () some ypes of exac soluons have been also nvesgaed usng dfferen mehods [9 0 6]. Here we wll sudy new ypes of opcal solary wave soluons of () whch mgh adequaely descrbe some specfc physcal phenomena for he chosen values of he parameers. The srucure of he paper s as follows. In Secon we consder wo dsnc fraconal ransformaons nvolvng (a) hyperbolc funcons and (b) Jacob ellpc funcons n order o oban new opcal solary wave soluons of Eq. (). Fnally Secon concludes he paper. We consder he ansae. FRACTIONAL TRANSFORMATION AND EXACT SOLUTIONS E() = A ()exp[()] ϕ ϕ () = k Ω () where A( ) s a complex funcon and k Ω are boh real consans. Then he subsuon of hs expresson no Eq. () yelds he complex nonlnear wave equaon: A + b A b A A b A A+ b A A + b A A + b A= 0 () * 6 * A s he complex conjugae of A and where b = Ω+ Ω b = Ω b = Ω b = b = b6 = k Ω Ω. L e al. [0] used he followng polynomal ransformaon of hyperbolc funcons A ( ) =β+λ anh[ η( χ )] +ρ sech[ η( χ )] () o sudy soluons of Eq. () such ha hey gave some exac soluons of Eq. () for ceran values of a se of parameers defnng he solary wave soluons... Solary waves due o he use of hyperbolc funcons In wha follows we would lke o fnd some new ypes of opcal solary wave soluons of Eq. () va he fraconal ransformaon [7]:

3 Fraconal opcal solary wave soluons of he hgher-order nonlnear Schrödnger equaon 9 β+λanh[ η( χ )] + ρ sech[ η( χ)] A () = + R sech[ η( χ)] () where β λ ρ η χ R are real-valued consans. Thus he correspondng amplude s wren as / {( ρ λ )sech [ η( χ )] + βρ sech[ η( χ )] +λ +β } A ( ) =. + R sech[ η( χ)] In parcular f R = 0 hen he fraconal ransformaon () reduces o he known form (). Here we do no consder hs case. We wll consder he case R 0 whch leads o new ypes of opcal solary wave soluons of HONLS equaon () va he ansae (). Wh he ad of symbolc compuaon we subsue Eq. () no Eq. () and separae he real and magnary pars and furher balance he coeffcens of hese lnearly ndependen erms j snh [ η( χ)]cosh [ η( χ )] ( = 0; j = 0...) o yeld a se of algebrac equaons wh respec o unknowns. Here we om he dervaon of he se of hese raher cumbersome algebrac equaons. In he followng when R 0 we presen new opcal solary wave soluons of Eq. () usng he fraconal ransformaon (): Case. =. We oban he followng opcal solary wave soluon of HONLS equaon () (6) β β( R / R) sech[ η( χ)] E( ) = exp[( k Ω)] + R sech[ η( χ)] and s nensy s defned by { β β( R / R) sech[ η( χ)]} { + R sech[ η( χ)]} E ( ) = (7) (8) where he parameers are gven by R Ω= = η = β ( )( + ) k R β ( R + )( + ) χ= Ω+ Ω. R From Eq. (9) wh η 0 we know ha R ±. When R < he soluon (7) has a sngular surface wh η( χ ) = arcsech( R ). Therefore f R > and R hen soluon (7) s a new regular opcal solary wave soluon of HONLS equaon (). For dfferen values of he parameer R Eq. (7) adms abundan srucures whch nclude he W-shaped opcal solary wave brgh opcal solary wave and dark opcal solary wave: ) We ake parameers as β= ρ= χ=. When R > we have he W-shaped opcal solary wave soluon of () and he maxmum value of he cener par of he nensy E ( ) s ypcally less han he maxmum values of he wo addonal nensy shoulders (Fgs. a b). Ths knd of W-shaped solary wave s called a concavely W-shaped solary wave. Moreove Fg. b shows ha he maxmum value of he cener par of he nensy approaches he consan value β = when R. ) We ake parameers as β= 0. ρ= χ=. If < R < 0 hen we also ge a W-shaped opcal solary wave whch dffers from he prevous W-shaped solary wave because n hs case he maxmum value of he cener par of he nensy E ( ) s larger han he maxmum values of he wo addonal shoulders (Fgs. a b). We call hs knd of solary wave as a convexly W-shaped solary wave. Moreover Fg. b shows ha he larger R becomes he larger he maxmum value of he solon s cener par s. (9)

4 96 Z. Yan ) We ake parameers as β= ρ= χ=. When 0 < R < we have from Eq. (8) he brgh solary wave soluon whch s shown n Fg. a. When 0 < R we have he dark solary wave soluon whch s shown n Fg. b. a b Fg. The nensy of W-shaped opcal solary wave soluon (7) wh R > : a) he hree-dmensonal nensy plo for R = ; b) he nensy plo a = 0 for R = 0 0 and 00. a Fg. The nensy of W-shaped opcal solary wave soluon (7) for < R < 0: a) The hree-dmensonal nensy plo for R = 0.9; b) he nensy plo a = 0 for R = and 0.9. Case. = = 0. We oban he dark opcal solary wave soluon of Eq. () b λanh[ η( χ)] E( ) = β+ exp[( k Ω)] + sech[ η( χ)] and s nensy s defned by λ anh [ η( χ)] E ( ) =β + { + sech[ η( χ)]} (0) () where he parameers are deermned by

5 Fraconal opcal solary wave soluons of he hgher-order nonlnear Schrödnger equaon 97 λ Ω= = η = χ= (/ λ +β ) Ω+ Ω k () a Fg. a) The nensy of brgh opcal solary wave soluon (7) for R = 0.8; b) he nensy of dark opcal solary wave soluon (7) for R =.. I follows from Eq. () ha we need he condon < 0. From he expresson of nensy () we see ha he soluon (0) s a dark opcal solary wave soluon of Eq. () whch s dfferen from he usual dark solary wave soluons. Case. = 0 = 0. We oban he opcal solary wave soluon of Eq. () b λanh[ η( χ)] λ sech[ η( χ)] E( ) = exp[( k Ω)] + sech[ η( χ)] () and s nensy s defned by λ { sech [ η( χ)] E ( ) = { + sech[ η( χ)]} () where he parameers are deermned by ( Ω) ( Ω ) k = η= Ω χ= Ω λ = () Case. = = + = 0. We oban he opcal solary wave soluon of Eq. () β+λanh[ η( χ )] + ρ sech[ η( χ)] E() = exp[( k Ω)] + R sech[ η( χ)] (6) and s nensy s defned by ( ρ λ ) sech [ η( χ )] + βρ sech [ η( χ )] +λ +β { + R sech[ η( χ)]} E ( ) = (7) where he parameers are deermned by

6 98 Z. Yan 6 R R ρ( β ρ )( Ω) k Rβ ρr ρ Rβ ρr ρ ρβ β ρ + ρ β λ= = ( Rβ ρr β ρβ+ Rρ )( Ω) η= χ= 0 λ( Rβ ρr ρ) (8) From Eq. (8) we need he condon (ρβ Rβ ρ +ρr β )( Rβ ρr ρ ) > 0. In parcular when R = 0 he obaned soluon (6) reduces o he known soluon (7) n Ref. [9]. For he case R => and R 0 he soluon (6) adms abundan srucures whch conan he W-shaped opcal solary wave brgh opcal solary wave and dark opcal solary wave see he solon nensy (7). However here we do no analye hem n deal... Perodc waves due o he use of Jacob ellpc funcons In addon accordng o he dea pu forward n Refs. [8-0] we assume ha Eq. () has he Jacob ellpc funcon soluon β+λ sn[ η( χ ) m] + ρ cn[ η( χ) m] A () = + R cn[ η( χ) m] When m he soluon (9) reduces o he consdered soluon (). For he general case 0 < m < we can oban new doubly perodc opcal solary wave soluons of Eq. (). For example when = we have he doubly perodc wave soluon of Eq. (): (9) β+ ρ cn[ η( χ) m] E( ) = exp[( k Ω)] + R cn[ η( χ) m] (0) and s nensy s gven by { β +ρ cn[ η( χ) m]} { R cn[ ( ) m]} E ( ) = + η χ where he parameers of he solary wave soluon are gven by β[ R (m ) m ] Ω= k = ρ= R[ R ( m ) m + ] ρ ( R )[ R ( m ) + m ]( + ) η = [ R (m ) m ] R m m R m m m m χ= + ρ [ ( )( ) (8 8 ) + ( )]. ( R )[ R ( m ) + m ] Smlar o Case he soluon (0) conans dfferen ypes of opcal solary wave soluons for dfferen values of he parameer R and he modulus m. Moreover for oher cases we can also oban new doubly perodc wave soluons of () va he new ansa (9).. CONCLUSIONS In summary we have found some new ypes of opcal solary wave soluons of he HONLS equaon wh GVD SPM TOD SS and SF erms. The obaned soluons dffer from he known opcal solary wave soluons repored n he leraure. Moreover for dfferen values of he free parameer R he obaned soluons are he concavely W-shaped ( R > ) he convexly W-shaped ( < R < 0 ) he brgh ( 0 < R < ) and he dark ( < R ) opcal solary wave soluons. In parcular for he concavely W-

7 7 Fraconal opcal solary wave soluons of he hgher-order nonlnear Schrödnger equaon 99 shaped soluon ( R > 0 ) he larger he parameer R becomes he larger he maxmum value of he cener par of he nensy s and hs cener nensy approaches a consan value when R (see Fg. b). Whle for he convexly W-shaped soluon ( < R < 0 ) he smaller he absolue value of he parameer R becomes he larger he maxmum value of he cener par of he nensy s (Fg. b). These new solary wave soluons mgh be useful o descrbe oher nonlnear opcs phenomena. Moreover he approach developed n hs paper can also be exended o oher nonlnear wave models n order o generae new solary wave soluons descrbng ceran physcal phenomena. ACKNOWLEDGEMENTS The work was suppored by he NSFC under Gran Nos. 07 and REFERENCES. Y.S. Kvshar G. P. Agrawal Opcal Solons: from Fbers o Phoonc Crysals Academc Press New York 00.. A. Hasegawa Y. Kodama Solons n Opcal Communcaons Oxford Unversy Press Oxford 99.. G.P. Agrawal Nonlnear Fber Opcs Academc Press New York B.A. Malomed D. Mhalache F. Wse L. Torner Spaoemporal opcal solons J. Op. B: Quanum Semclasscal Op. 7 R R7 00. H. Leblond D. Mhalache Models of few opcal cycle solons beyond he slowly varyng envelope approxmaon Phys. Repors pp I.V. Melnkov D. Mhalache F. Moldoveanu N.-C. Panou Quasadabac followng of femosecond opcal pulses n a weakly exced semconducor Phys. Rev. A 6 pp M.J. Ablow P.A. Clarkson Solons Nonlnear Evoluon equaons and Inverse Scaerng Cambrdge Unversy Press Cambrdge A. Hasegawa F. Tapper Transmsson of saonary nonlnear opcal pulses n dspersve delecrc fbers. I. Anomalous dsperson Appl. Phys. Le pp L. Paevsk S. Srngar Bose-Ensen Condensaon Oxford Unversy Press Oxford C. J. Pehck H. Smh Bose-Ensen Condensaon n Dlue Gases Cambrdge Unversy Press Cambrdge M. Gedaln e al. Opcal solons n he hgher order nonlnear Schrödnger equaon Phys. Rev. Le. 78 pp D. Mhalache N. Trua L.-C. Crasovan Panlevé analyss and brgh solary waves of he hgher-order nonlnear Schrödnger equaon conanng hrd-order dsperson and self-seepenng erm Phys. Rev. E 6 pp Z. L e al. New Types of Solary Wave Soluons for he Hgher Order Nonlnear Schrödnger Equaon Phys. Rev. Le Y. Kodama Opcal Solons n a Monomode Fber J. Sa. Phys. 9 pp Y. Kodama A. Hasegawa Nonlnear pulse propagaon n a monomode delecrc gude IEEE J. Quanum Elecron. pp K. Mo e al. Modfed Nonlnear Schrödnger Equaon for Alfvén Waves Propagang along he Magnec Feld n Cold Plasmas J. Phys. Soc. Jpn. pp E. Mjolhus On he modulaonal nsably of hydromagnec waves parallel o he magnec feld J. Plasma Phys. 6 pp D.J. Kaup A.C. Newell An exac soluon for a dervave nonlnear Schrödnger equaon J. Mah. Phys. 9 pp S. Lu W. Wang Exac N-solon soluon of he modfed nonlnear Schrödnger equaon Phys. Rev. E 8 pp V.E. Vekslerchk Dark solon of he generaled nonlnear Schrödnger equaon Phys. Le. A pp R. Hroa Exac envelope-solon soluons of a nonlnear wave equaon J. Mah. Phys. pp S. Lu W. Wang Exac N-solon soluons of he exended nonlnear Schrödnger equaon Phys. Rev. E 9 pp W. Zheng An envelope solary-wave soluon for a generaled nonlnear Schrödnger equaon J. Phys. A 7 pp. L9 L N. Sasa J. Sasuma New-ype of solon soluons for a hgher-order nonlnear Schrödnger equaon J. Phys. Soc. Jpn. 60 pp B.A. Malomed N. Sasa J. Sasuma Evoluon of a damped solon n a hgher-order nonlnear Schrödnger equaon Chaos Solons & Fracals pp D. Mhalache L. Torner F. Moldoveanu N.-C. Panou N. Trua Inverse scaerng approach o femosecond solons n monomode opcal fbers Phys. Rev. E 8 pp D. Mhalache L. Torner F. Moldoveanu N.-C. Panou N. Trua Solon soluons for a perurbed nonlnear Schrödnger equaon J. Phys. A 6 pp. L77 L D. Mhalache N.-C. Panou F. Moldoveanu D.-M. Babou The Remann problem mehod for solvng a perurbed nonlnear Schrödnger equaon descrbng pulse propagaon n opc fbres J. Phys. A 7 pp D.J. Franeskaks Small-amplude solary srucures for an exended nonlnear Schrödnger equaon J. Phys. A 9 pp

8 00 Z. Yan 8. V.I. Karpman The exended hrd-order nonlnear Schrödnger equaon and Gallean ransformaon Eur. Phys. J. B 9 pp Z.Y. Yan Generaled mehod and s applcaon n he hgher-order nonlnear Schrödnger equaon n nonlnear opcal fbers Chaos Solons & Fracals 6 pp M.J. Poasek Novel Femosecond Solons n Opcal Fbers Phoonc Swchng and Compung J. Appl. Phys. 6 pp Z.Y. Yan An mproved algebra mehod and s applcaons n nonlnear wave equaons Chaos Solons & Fracals pp Z.Y. Yan Consrucve heory and applcaons n complex nonlnear waves Scence Press Bejng Z.Y. Yan A new sne-gordon equaon expanson algorhm o nvesgae some specal nonlnear dfferenal equaons Chaos Solons & Fracals pp Z.Y. Yan Envelope exac soluons for he generaled nonlnear Schrödnger equaon wh a source J. Phys. A 9 pp. L0 L Z.Y. Yan The new r-funcon mehod o mulple exac soluons of nonlnear wave equaons Phys. Scr Receved Aprl 0