A Generalized Sub-Equation Expansion Method and Some Analytical Solutions to the Inhomogeneous Higher-Order Nonlinear Schrödinger Equation

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1 A Generalied Sub-Euaion Expansion Mehod and Some Analyical Soluions o he Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion Biao Li ac Yong Chen ab andyu-qili ac a Nonlinear Science Cener Ningbo Universiy Ningbo China b Insiue of Theoreical Compuing Eas China Normal Universiy Shanghai China c Key Laboraory of Mahemaics Mechaniaion Chinese Academy of Sciences Beijing China Reprin reuess o Dr. B. L.; biaolee@yahoo.com.cn Z. Naurforsch. a (); received May On he basis of symbolic compuaion a generalied sub-euaion expansion mehod is presened for consrucing some exac analyical soluions of nonlinear parial differenial euaions. To illusrae he validiy of he mehod we invesigae he exac analyical soluions of he inhomogeneous high-order nonlinear Schrödinger euaion (IHNLSE) including no only he group velociy dispersion self-phase-modulaion bu also various high-order effecs such as he hird-order dispersion self-seepening and self-freuency shif. As a resul a broad class of exac analyical soluions of he IHNLSE are obained. From our resuls many previous soluions of some nonlinear Schrödinger-ype euaions can be recovered by means of suiable selecions of he arbirary funcions and arbirary consans. Wih he aid of compuer simulaion he abundan srucure of brigh and dark soliary wave soluions combined-ype soliary wave soluions dispersion-managed soliary wave soluions Jacobi ellipic funcion soluions and Weiersrass ellipic funcion soluions are shown by some figures. Key words: Inhomogeneous High-Order NLS Euaion; Ellipic Funcion Soluions; Soliary Wave Soluions. PACS numbers:..yv..tg..dp. Inroducion The search for a new mahemaical algorihm o discover exac soluions of nonlinear parial differenial euaions (NLPDEs) is an imporan and essenial ask in nonlinear science. Many mehods have been developed by mahemaicians and physiciss o find special soluions of NLPDEs such as he inverse scaering ransformaion [] and Darboux ransformaion [] he formal variable separaion approach [] he Jacobi ellipic funcion mehod [] he anh mehod [ ] various exended anh mehods [9 ] and he generalied projecive Riccai euaions mehod [ 7]. Recenly Fan [] developed a new algebraic mehod which furher exceeds he applicabiliy of he anh mehod in obaining a series of exac soluions of nonlinear euaions. More recenly Yan [9] Chen e al. [] and Yomba [] have furher developed his mehod and obained some new and more general soluions for some NLPDEs. In his paper firsly on he basis of he work done in [ ] we will propose a generalied sub-euaion mehod by a more general ansa so ha i can be used o obain more ypes and general formal soluions which include a series of nonravelling wave and coefficien funcion soluions such as brigh-like soliary wave soluions dark-like soliary wave soluions W-shaped soliary wave soluions combined brigh and dark soliary wave soluions single and combined nondegenerae Jacobi ellipic funcion soluions and Weiersrass ellipic doubly periodic soluions for some NLPDEs. Secondly o illusrae he validiy of he proposed mehod we will invesigae some exac analyical soluions of he following inhomogeneous higherorder nonlinear Schrödinger euaion (IHNLSE) [] which governs he envelope wave euaion for femosecond opical pulse propagaion in inhomogeneous fibers: = i(α () + α () )+α () + α ()( ) + α ()( ) + Γ () () 9 7 / / 7 $. c Verlag der Zeischrif für Naurforschung Tübingen hp://naurforsch.com

2 7 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion () represens he complex envelope of he elecrical field is he normalied propagaion disance is he normalied rearded ime and α () α () α () α () andα () are he disribued parameers which are funcions of he propagaion disance relaed o he group velociy dispersion (GVD) self-phase-modulaion (SPM) hird-order dispersion (TOD) self-seepening and he delayed nonlinear response effec respecively. Γ () denoes he amplificaion or absorpion coefficien. The sudy of he IHNLSE () is of grea ineres due o is wide range of applicaions. Is use is no only resriced o opical pulse propagaion in inhomogeneous fiber media bu also o he core of dispersion-managed solions and combined managed solions. In [] exac mulisolion soluions of () are presened by employing he Darboux ransformaion based on he Lax pair. An exac dark solion soluion of he IHNLSE () has been obained in []. When Γ () = in he IHNLSE () wo exac analyical soluions ha describe he modulaion insabiliy and he solion propagaion on a coninuous wave background are obained by using he Darboux ransformaion []. Using a direc assumpion mehod Yang e al. [] presened hree exac combined soliary wave soluions for he IHNLSE () analyed he feaures of he soluions and numerically discussed he sabiliies of hese soluions under sligh violaions of he parameer condiions and finie iniial perurbaions for some exac analyical soluions. Besides he resuls menioned above in recen years many auhors have analyed he nonlinear Schrödinger-ype euaion from differen poins of view and have obained some ineresing and significan resuls such as in [7 ]. The presen paper is organied as follows: In Secion he generalied sub-euaion expansion mehod is presened. In Secion he proposed mehod is applied o invesigae he IHNLSE () and a series of exac analyical soluions of he IHNLSE () are obained. Furhermore some figures are given wih compuer simulaions o show he properies of he kink profile soliary wave soluions brigh-like and darklike soliary wave soluions Jacobi ellipic funcion soluions and Weiersrass ellipic funcion soluions. Finally some conclusions are given briefly.. The Generalied Sub-Euaion Expansion Mehod Now we esablish he generalied sub-euaion expansion mehod as follows: Given a NLPDE wih say wo variables {}: E(uu u u u u )=. () Sep. We assume ha he soluions of () are as follows: u()= A + M i= φ i (ξ )[A i φ(ξ )+B i φ (ξ )] a + N j= φ j (ξ )[a j φ(ξ )+b j φ (ξ )] () A = A () A i = A i () B i = B i () (i = M) a = a () a j = a j () b j = b j ()(j = N) ξ = ξ () are all differeniable funcions of {} and he new variable φ = φ(ξ ) saisfies ( ) dφ(ξ ) φ (ξ )= = h + h φ(ξ )+h φ (ξ ) dξ () + h φ (ξ )+h φ (ξ ) h i (i = ) are consans. The parameers M and N can be deermined by balancing he highes-order derivaive erm and he nonlinear erms in (). M N are usually posiive inegers; if no some proper ransformaion u() u m () may be used in order o saisfy his reuiremen. Sep. Subsiuing () along wih () ino () exracing he numeraor of he obained sysem separaing he sysem ino a real and imaginary par hen seing he coefficiens of φ i (ξ )(φ (ξ )) j (i = ; j = ) o ero we obain a se of over-deermined PDEs [or ordinary differenial euaions (ODEs)] wih regard o he differenial funcions A A i B i (i = M) a a j b j ( j = N) ξ and he varying differenial funcion coefficiens included in (). Sep. Solving he overdeermined PDEs (or ODEs) by a symbolic compuaion sysem (Maple) we would end up wih he explici expressions for A A i B i (i = M) a a j b j ( j = N) ξ and he funcions coefficiens or he consrains among hem. Sep. By using he resuls obained in he above seps and he various soluions of () a rich se of soluions for () can be expeced. By considering he differen values of h h h h and h we can derive ha () admis a series of fundamenal soluions. For simpliciy we only lis some hyperbolic funcion soluions Jacobi ellipic funcion soluions and Weiersrass ellipic doubly periodic soluions as follows.

3 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion 7 Case. derived: If h = h = wo soluions of () are φ I (ξ )= h C sech( h ξ ) { C h sech( h ξ ) ( + )anh( h ξ )+( ) } () φi (ξ )= h C sech( h ξ ) { C h sech( h ξ ) () +( +C )anh( h ξ )+( C )} h > C = exp( h C ) is an arbirary consan and = C (h h h ) =(h h h ). If furher choosing he parameers h h h and C in () and () we can derive he sech-ype soluions φi (ξ )= h ( ) sech h h ξ (7) h = h = h = = (or = C ) φi (ξ )= h ( ) sech h ξ h = h = h = h h < = (or = C ). () Case. If h = h = he Jacobi ellipic funcion soluions for () can be derived: ( ) φii (ξ )= h m h (m ) cn h m ξ m (9) h < h > h = h m ( m ) h (m ) ( ) φ h II(ξ )= h ( m ) dn h m ξ m () h < h > h = h ( m ) h ( m ) ( ) φ h m II(ξ )= h (m + ) sn h m + ξ m () h h > h < h = m h (m + ) m is a modulus. If m andm he Jacobi funcions degenerae o he hyperbolic funcions and he rigonomeric funcions respecively i. e. m snξ anhξ cnξ sechξ dnξ sechξ ; () m snξ sinξ cnξ cosξ dnξ. () Therefore when m he soluions (9) and () degenerae o he soluion () and he soluion () degeneraes o he following soliary wave soluion: ( ) φ II(ξ )= h anh h h ξ h = h () h = h = h < h >. h Case. If h = h = () has he following Weiersrass ellipic doubly periodic ype soluion: ( ) h φ III (ξ )= ξ g g h > g = h g = h (). h h Remark. ) The ansa () presened in his paper is more general han he ansa by some auhors [ ]. (i) When a = a i = b i = (i = N) A i B i (i = M) are all consans and ξ = ξ () is only a linear funcion of {} our ansa in he paper is changed ino he ansa in [ ]. By his ansa we only obain he ravelling wave soluions of given PDEs. (ii) When B i = (i = M) a = and a j = b j = ( j = N) A i (i = M)and ξ are undeermined funcions of {} he ansa is changed ino he ansa in []. ) Theoreically speaking one could inroduce erms up o h o h ec. order or even consider erms of infinie order which migh be summed up o give rise o a funcionally compleely new ansa. Due o complexiy of he compuaion he PDEs (or ODEs) sysems derived may no be solved in mos cases. ) In order ha he PDEs (or ODEs) derived in sep can be solved by symbolic compuaion we usually choose some special forms of A i B i a j b j and ξ.. Exac Analyical Soluions of he Inhomogeneous Nonlinear Schrödinger Euaion We now invesigae he IHNLSE () wih he generalied sub-euaion expansion mehod of Secion. According o he mehod we assume ha he soluions of he IHNLSE () are of he special forms ()= A ()+A ()φ(ξ )+B ()φ (ξ ) + a ()φ(ξ )+b ()φ (ξ ) exp { i[ λ ()+λ ()+λ ()] } ()

4 7 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion c) d) F F ω Fig.. Evoluion plos of he soluion () wih he parameers: α () =α () =sin() h = h = c = c = c =. c = µ =.. ( Γ () =.; ( Γ () =.sin()/(.sin ); (c d) spaial Fourier specra of he soluion () corresponding o Fig. a and Fig. b respecively. ω ξ = ξ ()=p() + η() (7) and A () A () B () a () b () λ () λ () λ () p() andη() are real funcions of o be deermined and φ(ξ ) saisfies (). Subsiuing () and (7) along wih () ino he IHNLSE () removing he exponenial erm collecing he coefficiens of he polynomial of φ(ξ ) φ (ξ ) and of he resuling sysem s numeraor hen separaing each coefficien ino he real par and imaginary par and seing each par o ero we obain an ODE sysem wih respec o he differeniable funcions α () α () α () α () α () A () A () B () a () b () λ () λ () λ () p()andη(). For simpliciy we omi he ODE sysem in his paper. Solving he ODE sysem wih he symbolic compuaion sysem Maple we obain a rich se of soluions of he sysem. For simpliciy we omi he oo complex and he rivial soluions. Then ogeher wih he soluions of he ODE sysem and () (7) he following

5 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion 77 families of soluions o he IHNLSE () can be derived. Family. ()= a ()φ I (ξ ) + µφ I (ξ ) exp(i(c + λ ())) () [ a ()=c exp ] Γ ()d h = µh h = h = ξ = c ( + (c h c )α () c α ()d)+c α ()= ( µ h + h )c α () (a ()) α () α ()= [ ( µ (a ()) h + h )c α () +( c µ h + c h )c α () ] α ()c λ ()= (c h c )α () +(c c h c )α ()d + c. (9) Here µ h h c c c c c c are arbirary consans and α () α () α () Γ () are arbirary funcions of. If furhermore h = from φi and () he following soluion of he IHNLSE () can be derived: a ()h ()= ) h h cosh( ξ + µh exp(i(c + λ ())). () In his siuaion he soluion () describes a brigh-like soliary wave wih he ampliude and he widh deermined by a () =c exp[ Γ ()d] and h c / respecively. We noe ha he ime shif [χ()= (c h c )α () c α ()d] and he group velociy [V() =dχ/d = c α () (c h c )α ()] of he soliary wave are dependen on which leads o a change of he cener posiion of he soliary wave along he propagaion direcion of he fiber and means ha we may design a fiber sysem o conrol he ime shif and he velociy of he soliary wave. In order o undersand he evoluion of hese soluions expressed by () le us consider a solion managemen sysem he sysem parameers are rigonomeric and hyperbolic funcions. For simpliciy we only consider some examples for each soluion obained in Family under some special parameers. Figure a presens he evoluion plos of he soluion () wih Γ () =. > which corresponds o a dispersion increasing fiber. From i we can see ha he inensiy of he soliary wave increases when Γ () > and he ime shif and he group velociy of he soliary wave are changing while he soliary wave keeps is shape in propagaing along he fiber. Figure b presens he evoluion plos of he soluion () wih Γ () =.sin()/(.sin ) which corresponds o a fiber whose dispersion is periodically changing. From i we can see ha he inensiy of he soliary wave changes periodically and he ime shif and he group velociy of he soliary wave are also changing while he soliary wave keeps is shape in propagaing along he fiber. Especially when Γ () we can derive ha he ampliude of () keeps invariable unlike in Fig. he ampliude of () eiher increases exponenially or changes periodically. Figures c and d are he spaial Fourier specra of he soluion () corresponding o Figs. a and b respecively. From he Fourier specra of he soluion () we can derive ha he soluion () is dominan for low freuencies. Family. ] ()=c exp[ Γ ()d (c + c φ I + c 7 φ I(ξ )) () exp(i(c + c )) ξ = c + c α ()= c α () α ()=α ()= α ()= α () () and h = h = c 7 = (orc 7 = ) and h h h c c c c c c c are arbirary consans and α () Γ () are arbirary funcions of. From φi and () one obains a soluion of he IHNLSE () as follows: ] ()=c exp[ Γ ()d [ c h ( c ( )) ( )] h h anh h ξ sech h ξ exp(i(c + c )) () he corresponding parameers are deermined by () and c 7 =.

6 7 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion.... c) d) Fig.. Evoluion plos of he soluion () wih he parameers: Γ ()=. h = c = c = c =. ( h =. c = c = ; ( h = c = c = ; (c) h =. c = c = ; (d) h =. c = c =.. Because he soluion () includes boh funcion anh(ξ ) and funcion sech(ξ ) he soluion may presen differen shapes. If anh(ξ ) [sech(ξ )] is dominan he soluion describes he dark soliary wave (he brigh soliary wave). If boh effecs are nearly eual he soluion presens a combined brigh and dark soliary wave. As shown in Fig. under differen parameers {h h c c } he shapes of he inensiy () ake a brigh-like soliary wave dark-like soliary wave W-shaped soliary wave and a combined brigh and dark soliary wave respecively. I is worh noing ha he soliary velociy does no change and here is only a shape-presening ime shif in he propagaion due o he consan c. In paricular as shown in Fig. d a brigh soliary wave and a dark soliary wave are found o combine under some condiions and propagae simulaneously in an inhomogeneous fiber. Family. ( ) h ()=a () φ I h exp(i(c + λ ())) ] a ()=c exp[ Γ ()d h = h = h = h h h h ξ = c c () α ()c +(c h +c )α ()d+c 7 [ α ()= α (a ()) ()(a ()) c + α ()c h +α ()c c h (a ()) ]

7 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion 79 Fig.. Evoluion plos of he soluion () wih he parameers: α ()=α ()=cos() h = h = C = c = c =. c =. c 7 =. ( Γ ()=.; ( Γ ()=.sin(). α ()= c α ()h + (a ()) α () (a ()) λ ()= ( c c h )α () +( c c c h )α ()d + c () and c c c c c 7 h h are arbirary consans and α () α () α () Γ () are arbirary funcions of. From h = h h / h h we obain = = in () and (). Thus from () and () one can obain a soluion for he IHNLSE () as follows: ()=a () { [ h /h h C sech( h ξ ) ] [C h sech( h ξ ) anh( h ξ ) ] } exp(i(c + λ ())) () he corresponding parameers are deermined by (). Due o he exisence of boh funcion anh(ξ ) and funcion sech(ξ ) he soluion () may presen differen shapes. From Fig. one can see ha he inensiy of () presens mainly kink profile while he soliary wave propagaes along he fiber. Family. ()= a ()(c + φ I (ξ )) + µφ I (ξ ) (7) exp(i(λ () + c λ () + λ ())) ( ξ = c λ () + ) c + c h = h = α ()= d d λ () (λ ()) α ()= ( d d λ () ) c h (a ()) ( + µc ) h = c h c h h = + µc ( + µc ) Γ ()= λ () d d a () a () d d λ () a ()λ () λ ()= ( c + c c h ) ( + µc ) λ ()+c () and α () =α () =α () = c c c c c h µ are arbirary consans and λ () a () are arbirary funcions of. From () we can derive ha h h h =. Thus from () (7) and () a soluion of he IHNLSE () can be derived: ()= h α () α () c λ () [ h c h + µc c C (sinh( h ξ )+cosh( h ξ )) ][ µh h C (sinh( h ξ )+cosh( h ξ )) ] exp(i(λ () + c λ () + λ ())) (9)

8 77 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion Fig.. Evoluion plos of he soluion (9) wih he parameers: ( α ()=sin() α ()= sin() h = h = c = C = c =. c = c = c = µ = ; ( α ()=.sin() α = h = h = c = C = c = c = c = c = µ =. α () and α () are arbirary funcions of and he corresponding parameers are consrained by λ ()= α ()d + c Γ ()= α ()α () α ()α () α ()λ () α ()α () ( ξ = c λ () + ) c + c () h = c h c h h = + µc ( + µc ) λ ()= ( c + c c h ) ( + µc ) λ ()+c. Figure presens he evoluion plos of he soluion (9) under some special parameers. From Fig. a one can see ha if he main conrol funcions are α () =sin() α () = sin() he inensiy of he soliary wave presens he propery of he kink profile soliary wave and he ime shif is changing while he soliary wave keeps is shape in propagaing along he fiber. As shown in Fig. b when he main conrol funcions are α ()=.sin() α = he inensiy of he soluion (9) deceases periodically while he kink profile soliary wave propagaes along he fiber. Family. α ()h h m ()= c λ () α () h (m ) ( ) h cn m ξ m exp(i(λ () + c λ () + λ ())) α ()h h ()= c λ () α () h ( m ) ( ) h dn m ξ m exp(i(λ () + c λ () + λ ())) () () α ()h h m ()= c λ () α () h (m + ) ( ) sn h m + ξ m () exp(i(λ () + c λ () + λ ()))

9 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion 77 c) d) Fig.. Evoluion plos of he soluion () wih he parameers: α () =.sin() c = c = c = c =. (a α ()= h = h = m = ; (c d) α ()= h =. h = m =.9. ξ = c λ () ( + ) c + c h = h = λ ()= (c c h )λ ()+c λ ()= α ()d + c Γ ()= α ()α () α ()α () α ()α () α ()λ () () and h h h are consrained by he condiions in φii (ξ ) φ II (ξ ) and φ II (ξ ) respecively c c c c c are arbirary consans α ()=α ()=α ()= α () andα () are arbirary funcions of. From () we can derive ha he Jacobi ellipic funcion soluions () () and () reduce o hyperbolic funcion soluion. A he same ime when m = we can conclude ha he widh he group velociy and he heigh of he soliary wave soluion () are deermined by λ ()= α ()d+c c and

10 77 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion Fig.. Evoluion plos of he soluion () of he form φ II (ξ) wih he parameers: Γ ()=. α ()=sin() α ()=sin() h =. h = c = c = c =. c = c =. ( m = ; ( m =.9. α () α () λ () respecively. In essenially he widh and he heigh are conrolled by α () and α () he velociy is deermined by c. Similarly we can obain he conrol funcions and conrol parameers of he soliary wave soluion () and () under m =. The ime-space evoluion of he dispersion Jacobi ellipic periodic wave soluion () is shown in Figure. From Figs. a and c he inensiies of he dispersion-managed brigh soliary wave and dispersion-managed dark soliary wave are decreasing periodically while he widhs of hem are increasing in propagaing along he fiber. Figures b and d presen he periodic properies of he Jacobi ellipic funcions cn(ξ m) and sn(ξ m). Family. ()=a ()φii i exp(i(c +λ ())) i = () ] a ()=c exp[ Γ ()d ξ = c (+ ) c α ()+(c h c )α ()d +c λ ()= ( c + c h )α () +( c + c c h )α ()d + c α ()= (a ()) [ α ()(a ()) c + α ()c h + c h α ()c ] α ()= α ()c h + α ()(a ()) (a ()) h = h = () and h h h are consrained by he condiions in φii (ξ ) φ II (ξ ) and φ II (ξ ) respecively c c c c c are arbirary consans and α () α () α () Γ () are arbirary funcions of. AsshowninFig.whenakingφII (ξ ) in he soluion () he inensiy of he brigh-like soliary wave (m = ) and he Jacobi ellipic periodic wave ( < m < ) are increasing exponenially due o Γ ()=. >. If α () and α () ake rigonomeric periodic funcions he ime shif and he group velociy of he soliary wave and Jacobi ellipic wave are changing while he waves keeps heir shape in propagaing along he fiber. Family 7. ] 7 ()=c exp[ Γ ()d (c + c φ i II(ξ )+c (φ i II(ξ )) ) exp(i(c + c )) i = (7)

11 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion c) d) 7 7 Fig. 7. Evoluion plos of he soluion (7) wih he parameers: Γ ()=. h = h = c = c = c = c = m = c =. ( c =. c = ; ( c =. c =.; (c) c = c =.; (d) c = c =.. ξ = c + c α ()= c α () α ()=α ()= α ()= α () () and h = h = h h h are consrained by he condiions in φii (ξ ) φ II (ξ ) and φ II (ξ ) respecively c = (orc = ) c c c c c c are arbirary consans and α () Γ () are arbirary funcions of. As shown in Fig. 7 when aking φii (ξ ) in () he shapes of he inensiy () are deermined by c and c under h = h = c = m =. A brigh-like soliary wave and dark-like soliary wave are shown in Figs. 7a and 7b. In paricular as shown in Figs. 7c and 7d a brigh soliary wave a dark soliary wave and wo brigh soliary waves are found o be combined under some condiions; hey propagae simulaneously in an inhomogeneous fiber. I is worh noing ha he soliary velociy does no change and here is only a consan ime shif in propagaion due o consan c.a he same ime i is necessary o poin ou ha we only consider some special cases under m =. If < m < he evoluion of he soluions is similar o Figs. and. For simpliciy we do no simulae hem in his paper.

12 77 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion. Fig.. Evoluion plos of he soluion (9) of he form φ II (ξ) wih he parameers: Γ () =. α () =cos() h = h = c = c = c = m = c = c = µ =. ( c = h =.; ( c =. h =. Family. ()= a ()(c + φii i (ξ )) + µφii i (ξ ) exp(i(c + λ ())) (9) i = ξ =c [ (c µ(c µ µ h c c c h ) c c µ c c c h µ + c µ h + c c µ) α ()d ] + c α ()= α ()c α ()= α ()c λ ()=c α ()d + c α ()= α ()+ c α ()(µ h h ) (a ()) [ h = µ h + c h (c µ + )µ a ()=c exp Γ ()d] h = h = () and h h h are also consrained by he condiions in φ II (ξ ) φ II (ξ ) and φ II (ξ ) respecively µ c c c c c c are arbirary consans and α () Γ () are arbirary funcions of. As shown in Fig. he inensiy of a dark-like soliary wave and a W-shaped soliary wave is increasing exponenially due o Γ () =. >. A he same ime due o he periodiciy of he rigonomeric periodic funcions α () he ime shif and he group velociy of he soliary wave and W-shaped soliary wave are changing while he waves keep heir shape in propagaing along he fiber. Family 9. 9 ()= h α () µ c λ () α () h +µ ( h ) ( ξ µ µ ξ µ µ ) exp(i(λ () + c λ () + λ ())) () ξ = c λ () + c c λ ()+c h = µ h h = µh h = h = λ ()= Γ ()= α ()α () α ()α () α ()α () λ ()= λ ()c µ h + c λ ()+c α ()d + c α ()λ () α ()=α ()=α ()= ()

13 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion Fig. 9. Evoluion plos of he soluion () wih he parameers: ( α ()=.sin() α ()= h = / c = c = c = c = µ = ; ( α ()=. α ()= h = / c = c = c =. c = µ =. and h c c c c c are arbirary consans and α () α () are arbirary funcions of. From Fig. 9 one can see ha he evoluion of he soluion () is similar o ha shown in Figs. b and d if he main conrol funcions α () and α () are of he same (or similar) forms. As shown in Fig. 9 under he same (or similar) conrol funcions he evoluion of he Weiersrass ellipic doubly periodic ype soluion () is similar o ha of he Jacobi ellipic funcion soluion (). Family. ()= h ) ] ( c exp[ ξ µ Γ ()d µ ) + h µ ( ξ µ µ exp(i(c + λ ())) () ) ξ =c ( (c µ h +c )α ()+c α ()d +c h = µ h h = µh h = h = λ ()= ( c µ h c )α () +( µ c c h c )α ()d + c α ()= a () [ µ h α ()c + α ()a ()c + α ()c µ c h ] α ()= (a ()α ()+α ()c µ h ) a () ] a ()=c exp[ Γ ()d. () Here h c c c c c are arbirary consans and α () α () α () are arbirary funcions of. From Fig. one can see ha he evoluion of he soluion () is similar o ha shown in Fig. if he main conrol funcions α () and α () are of he same (or similar) forms. Under he same (or similar) conrol funcions he evoluion of he Weiersrass ellipic doubly periodic ype soluion () is similar o ha of he Jacobi ellipic funcion soluion (). Remark. ) The soluions obained in he presen paper include many previous resuls by many auhors such as: if c = m = he Theorems and in [7] can be reproduced by our resuls () (); if m = he soluions () and () in [7] can be also reproduced by () (); c) if seing m = and α ()=α ()=α () = in Family he soluions () and (9) in [7] can be recovered. Bu o our knowledge he oher soluions have no been repored earlier.

14 77 B. Li e al. The Inhomogeneous Higher-Order Nonlinear Schrödinger Euaion Fig.. Evoluion plos of he soluion () wih he parameers: h = c = c =. c = c = c = µ =. ( Γ ()=. α ()=α ()=cos(); (Γ ()=. α ()=α ()=/( + ). ) The soluion presening he propery of he brigh soliary wave or dark soliary wave is mainly deermined by he funcion form of he soluion. If he soluion is only relaed o he funcion anh(ξ ) or sech(ξ ) he soluion describes only he dark soliary wave or brigh soliary wave such as () () and () each soluion presens one form. Bu if he soluion combines funcion anh(ξ ) wih funcion sech(ξ ) suchas () () () 7 () and () he soluion may presen various shapes which are deermined by whose effec is dominan. If anh(ξ )[sech(ξ )] is dominan he soluion describes he dark soliary wave (he brigh soliary wave). If boh effecs are nearly eual he soluion presens a combined brigh and dark soliary wave. ) Because he soluions obained conain some arbirary funcions and arbirary consans he soluions presen abundan srucures. As shown in he figures we found W-shaped soliary waves combined-ype soliary waves dispersion-managed soliary waves periodic propagaing Jacobi ellipic funcion waves and Weiersrass ellipic funcion waves. For simpliciy o each soluion we only considered some special parameer condiions. If aking differen parameers in each soluion abundan srucures can be expeced.. Summary and Discussion In his paper on he basis of symbolic compuaion we presened a generalied sub-euaion expansion mehod o consruc some exac analyical soluions of nonlinear parial differenial euaions. Making use of he mehod we invesigaed several exac analyical soluions of he inhomogeneous high-order nonlinear Schrödinger euaion including no only he group velociy dispersion self-phase-modulaion bu also various high-order effecs such as he hird-order dispersion self-seepening and self-freuency shif. As a resul a broad class of exac analyical soluions of he IHNLSE was obained which include brigh and dark soliary wave soluions combinedype soliary wave soluions dispersion-managed soliary wave soluions Jacobi ellipic funcion soluions and Weiersrass ellipic funcion soluions. As shown in he figures produced by compuer simulaion hese soluions possess abundan srucures. From our resuls many previous soluions of some nonlinear Schrödinger-ype euaions can be recovered by means of some suiable choices of he arbirary funcions and arbirary consans. The mehod developed provides a sysemaic way o generae various exac analyical soluions of he IHNLSE wih varying coefficiens and can be used also for oher PDEs. The developmen of a new mahemaical algorihm o discover analyical soluions especially soliary wave soluions in nonlinear dispersion sysems wih spaial parameer variaions is imporan and migh have significan impac on fuure research.

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