Analytic Solution of Nonlinear Schrödinger Equation by Means of A New Approach

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1 Anlytic Solution of Nonlinr Schrödingr Eqution y Mns of A Nw Approch Ali Yunus Rohdi Optoinformtics Lortory, Dprtmnt of Physics, Spuluh Nopmr Institut of Tchnology (ITS), Sukolilo, Sury, 6111 Tlp/Fx : , Astrct Th nonlinr Schrödingr qution (NSE) hs srvd s th govrning qution of opticl soliton in th study of its pplictions to opticl communiction nd opticl switching. Vrious schms hv n mployd for th solution of this nonlinr qution s wll s its vrints. W rport in this ppr rltivly simplr nw pproch for th nlytic solution of NSE. In this schm th qution ws first trnsformd into n rctngnt diffrntil qution, which ws thn sprtd into th linr nd nonlinr prts, with th linr prt solvd in stright forwrd mnnr. Th solution of th nonlinr qution ws writtn in th form of modultion function chrctrizd y its mplitud function A nd phs function F(A). Sustituting th linr solution for A, th rctngnt diffrntil qution ws solvd for crtin initil vlu of A. It is shown tht this mthod is pplicl to othr first-ordr nonlinr diffrntil qution such s th Kortwg d Vris qution, which cn trnsformd into n rctngnt diffrntil qution. I. Introduction Th phnomnon of th solitry wv propgtion ws osrvd for th first tim y th Scottish scintist John Scott Russll in 1844, whn on dy h ws wtching wtr wvs of crtin shp kpt on trvling without chnging thir shp for distnc s fr s his y could s. To xplin th hvior of such unsul wv, Kortwg nd d Vris govrnd modl for th wv propgtion in shllow wtr in form prtil diffrntil qution clld s KdV diffrntil qution, which its solution pproprits to th fturs of th solitry wv clld s soliton [1]. Th xistnc of solitons in opticl fir ws prdictd y Zkrov nd Zt (197) ftr thy drivd diffrntil qution for th light propgting in n opticl fir, tht dmonstrtd ltr y Hzgw in 1973 t Bll Lortory. Nxt, Mollnur nd Stoln mployd th solitons in opticl fir for gnrting supicoscond pulss. Th diffrntil qution of th soliton pulss propgtion in losslss opticl fir hs n ordinrily givn in normlizd from : U 1 U j + + N U Z T U whr U, Z, nd T r rspctivly th normlizd prmtrs of th nvlop function, th distnc nd th tim propgtion, whil N is n ordr of soliton, nd (1)

2 j 1. Th ov of th nonlinr diffrntil qution with th Z nd T xchngd is known s th nonlinr Schrödingr qution (NSE). Aftr th lightwv soliton cn gnrtd in opticl fir, th NSE hs n com th primry modling which hs lso srvd s th govrning qution of opticl soliton in th study of its pplictions to opticl communiction nd opticl switching [1],[]. Thr r vrious schm tht commonly mployd for solving th NSE, mong ths r th split Fourir trnsform [3], finit diffrnc [4], vritionl pproch [5], nd prturtion mthod [6]. Du to for N 1 th NSE rducs to n intgrl form, hnc it cn solvd xctly in nlytic solution, whr its solution corrsponds to th fundmntl soliton. Sprting oth vrils T nd Z hs n don y ssuming th NSE solution in form [1],[] : jφ( z) U( T, Z) V( T), () whr oth V nd φ nd r th rl functions. Applying th oundry conditions dv T, V nd givs φ κz whr κ is constnt, nd hnc dt th NSE rducs to th following first ordr diffrntil qution : dw dt κw w, V w (3) κ It is wll-known tht upon intgrting th oth sids of Eqs.(3) givs th solution of w in hyprolic scnt function (sch-function). But w rport in this ppr nw pproch clld Short Stl Modultion Tchniqu (S-SMT) for otining th solution of w y trnsforming Eqs.(3) into n rctngnt diffrntil qution, which ws thn sprtd into th linr nd nonlinr prts, with th linr prt solvd in stright forwrd mnnr. Th solution of th nonlinr qution ws writtn in th form of modultion function chrctrizd y its mplitud function A nd phs function F(A). Sustituting th linr solution for A, th rctngnt diffrntil qution ws solvd for crtin initil vlu of A. II. Solving n rctngnt diffrntil qution y using th common procdur In clssifying th first ordr nonlinr ordinry diffrntil qution, n rctngnt diffrntil qution is dfind s : dy y dx +, y ( x ) y (4) for oth ritrry of dn constnt cofficints nd strtd from th ordinry point (x). Th common procdur of solving Eqs.(4) ing y trnsforming into rctngnt. Rcntly, w hv rportd nothr wy for solving th rctngnt diffrntil qution using Brnoulli intgrl [7], nd y pplying th stl modultion tchniqu [8]. Solving of Eqs.(4) for oth nd r positiv constnts y trnsforming into rctngnt prformd y mking chng vril u y givs th xct solution in th form :

3 1 ( x) tn x + tn y y (5) Eqution (5) indicts tht th solution th rctngnt diffrntil qution for initil vlu y is of th form : whil th solution for ( x) tn( x) y (6) y is of th form : ( x) tn( x C) y + (7) whr C is th intgrting constnt tht must dtrmind y pplying th initil y x y. vlu ( ) II. Solving n rctngnt diffrntil qution y mns S-SMT pproch Applying th Short Stl Modultion Tchniqu (S-SMT) for solving Eqs.(4) is strtd for th cs with th initil vlu y. Th linr prt of Eqs.(4) is of form : with solution dy L dx, (8) y L x, whil th nonlinr prt is of form : dy N y N, (9) dx which its solution must writtn in th following modultion function : y y N N (1) y dx N As in th stl modultion tchniqu (SMT) sd on Brnoulli qution [8],[9], th SMT sd on rctngnt diffrntil qution is lso chrctrizd y modultion function of th nonlinr solution prt in which its initil vlu y N sids cts s mplitud A, lso modultd in th phs function F(A). Th corrsponding phs function of solving th rctngnt is of form : 1 F (A) (11) Adx Sustituting y L into y N in Eqs.(1) givs th solution of y in th form :

4 x y( x) (1) x Involving th constnt in th numrtor of Eqs.(1) is prformd y multiplying /, thn writing Eqs.(1) in form : / x y( x) (13) ( / x) Th pproch of short stl modultion is pplid to th dnomintor of Eqs.(13) y pproching th inomil into th root squr for x, thus w find : x y( x) (14) x By using n pproximtion x sin x for x, thn Eqs.(14) coms : ( ) y x tn x (15) Similr to th stps in Eqs.(6-7) for ritrry vlu of y, w tk th solution of th rctngnt diffrntil qution in th form : y (16) ( x) tn x + C whr C is dtrmind y pplying th initil vlu. Th solution of th rctngnt diffrntil qution s Rictti diffrntil of constnt cofficints [1] cn crtd sily using AF(A) digrm whn Eqs.(4) is writtn in from : Solution dy dx ( y) + (17) 1 y ( x) tn x + tn y / (18)

5 Th trm of AF(A) digrm rfrs to th xistnc of mplitud tht lso including in tngnt function of th phs function. Importnt to strssd hr tht for oth nd r positiv constnts, th trm coms, thrfor Eqs.(18) rducs to th form in Eqs.(5). Whil nlytic solution of Eqs.(18) for ritrry nd cofficints must still involving d Moivr thorm tht rltd tngnt nd rctngnt to th hyprolic of tngnt nd rctngnt [11]. III. Appliction of th S-SMT for solving th NSE of fundmntl soliton w to trnsform Eqs.(3) into nothr form of th following diffrntil qution : According to [1], w mk chng vril sin( θ) dθ dt κ sin ( θ) Th form of th rctngnt diffrntil qution is otind from Eqs.(19) ftr rprsnting th sinus function in trm of th Eulr formul, tht is of form : d dη κ ( ) 1, η T hr V to th nvlop of soliton function, th vlu of w 1 t T corrspond κ jπ / to th θ. Du to th rl of ( ), hnc th solving Eqs.() (19) () nd η r th dpndnt nd indpndnt vrils rspctivly. According π / corrsponds to th initil vlu ( ). Thrfor th procdur of solving Eqs.() is prformd schmticlly s following :

6 ( ) 1 d dη Th linr prt Th non linr prt d ( ) L d( ) N 1 ( ) dη ( ) η dη ( ) ( ) L N 1 ( ) j ( ) L ( ) N 1 ( jη) ( jη) ( jη) η θ j jtn using d Moivr thorm [11] tnh solution d j θ 1 dη tn ( ) T 1 κ κτ ( θ / ) N dη N j θ sin j sin using d Moivr rltion [11] ( jη) ( jη) Hnc, th finl solution ( ) w sinθ dw dt κw w in Eqs.(3) is of form : ( ) ( θ/) V tnθ/ k 1+ tn κτ κτ + sch ( κt) (1) Th nlytic solution of th NSE for th fundmntl soliton is in th form : U T,Z V T jφ z κsch κt () ( ) ( ) ( ) ( ) jκκ

7 III. Appliction of AF(A) digrm for solving KdV of fundmntl soliton Th KdV diffrntil qution is of form [4],[1] : y y y y + + 6y t z z t z 3 whr z nd t r rspctivly th distnc nt th tim of th soliton propgtion. Introducing th collctiv coordints [13] ξ z vt, whr v is th vlocity of soliton, nd ssuming th solution in form : y ( x,t) f ( ξ) (4) thn Eqs.(1) rducs to th form: 3 f f f ( 1 v ) + 6 f + v (5) 3 ξ ξ ξ Upon succssiv intgrtion of Eqs.(5) rspct to d ξ nd df, lso y dfining th following two prmtrs : f u 1 v 1 v 1 α w found th corrsponding of th first ordr nonlinr diffrntil qution of Eqs.(5) in th form : du αu dξ u Using u sin θ, nd rprsnting in trm of th Eulr formul to trnsform Eqs.(6) into th rctngnt diffrntil qution : d α ( ) + ( j) (7) dξ v [ ] [ ] (3) (6) Crting th solution Eqs.(7) using AF(A) digrm is otind in th form: 1 jtn jαξ+ tn ( j ) (8) or tnh 1 [ αξ + tnh ( cosθ jsin θ )] As on th NSE cs, th KdV diffrntil qution tht givs th fundmntl soliton ing corrsponding to θ π /. Hnc, ftr nglcting th imginry trm of jsin θ, w find from Eqs.(9) th rltion tnh 1 ( ) αξ, tht vntully tn θ/. This rsult llows us to writ : u sin ( θ ) sc h ξ. v Hnc, th nlytic solution of th KdV diffrntil qution for th fundmntl soliton is of form : v 1 y ( z, t ) f ( ξ ) u ( ξ ) (3) or yilding ( ) αξ 1 (9) v 1

8 v 1 1 v 1 ( z,t) sch ( z vt) y Finlly, w prsnt th visuliztion of propgtion th fundmntl soliton of solving th NSE of Eqs.() nd th KdV of Eqs.(31) in Fig.1 low. v (31) () NSE of κ 1 () KdV for v (c) KdV for v5 (d) KdV for v7.5 Fig. 1 Visuliztion of propgtion th fundmntl soliton of solving NSE nd KdV IV. Discussion W hv shown tht th solving rctngnt diffrntil dy y dx + qution for ritrry nd constnt cofficints with th initil vlu y (x ) y y mns short stl modultion tchniqu (S-SMT) givs th nlytic xct solution in AF(A) formul of modultion function, whr its mplitud in (this cs A ) modultd in tngnt function [ Ax C] tn + of th phs function, whr C is th intgrting constnt. Th AF(A) formul cn crt th nlytic xct solution of oth th NSE nd KdV for fundmntl ordr tht corrsponds to th propgtion of fundmntl soliton in losslss mdi. As ppr

9 in Eqs.() for th NSE nd Eqs.(31) for KdV, th nlyticl solution of oth intgrl nonlinr prtil diffrntil qution lso form n AF(A) formul, whr ch of mplitud trm lso modultd in hyprolic scnt nd qudrtic of scnt hyprolic of th phs function rspctivly. V. Conclusions Applying th Short Stl Modultion Tchniqu (S-SMT) sd on rctngnt diffrntil qution simplifid y AF(A) digrm pplicl to solv oth of NSE nd KdV diffrntil qution nlyticlly for fundmntl soliton. AF(A) digrm of th S-SMT cn justify th following intgrl formultion 1 dθ dθ sc h ( θ) nd ln tn( θ / ) sin θ θ θ Hnc, th AF(A) digrm is rcommndd to mployd to formult nothr intgrls tht cn trnsformd into rctngnt. Th solution of n intgrl nonlinr prtil diffrntil qution such s th NSE nd KdV diffrntil qution for soliton propgtion of fundmntl soliton in losslss mdi hv th sm fturs, tht r in modultion function of form AF(A) formul, whr its mplitud trm A lso modultd in th phs function F(A). ACKNOWLEDGMENTS : I would lik to thnk Prof. M.O. Tji (Dprtmnt of Physics, ITB, Bndung) for introducing m into th intrst lnd of solitons, nd lso for his support my suggstions in dvloping smrt mthods of solving th nonlinr diffrntil qutions. Lst, ut not lst, my grtitud gos to my prnts, my wif Sri Yulini nd my dughtrs (Astrid Dnyls nd Ndy Frmg) for thir continus support during this rsrch. REFERENCES : [1] Izuk, K, Elmnts of Photonics, Vol.II, Chptr 15, pp: , John Wily nd Sons,. [] Agrwl G.P, Fir-Optic Communiction Systm, Chptr 9, pp: , Wily Sris in Microwv nd Opticl Enginring, 199. [3] Kczmrk,T, nd Kczmrk, A Nonlinr Lightguid s A Trnsmission Mdium of Ultrshort Chirpd Hyprolic Scnt Shp nd Gussin Pulss, Scintific- Procdings of RGA Tchnicl Univrsity, Sris Computr Sicnc, 1. [4] Aguyo,L, t.l, Exploring th Bhvior of Soliton on A Dsktop Prsonl Computr, Rvist Mxin D Fisic E-5(1) [5] Rpti, Z, Kvrdikis, P.G, Smrsi, A, nd Bishop, A.R, Vritionl Approch to th Modultion Instility, rxiv:cond-mt/4461, 4. [6] Jong H.L, Anlysis nd Chrctriztion of Fir Nonlinritis with Dtrministics nd Stochstic Signl Sourcs, P.hD Dissrttion in Elctricl Enginring, Univrsity of Virgini,. [7] Rohdi,A.Y, Introducing Brnoulli Intgrl for Solving Som Prolm of Physics, Procding of Syimposium of Physics nd Its Applictions, Dprtmnt of Physics, ITS, Sury, My, 7. A6.1-5.

10 [8] Rohdi,A.Y, Applying of Modultion Schm of Solving Brnoulli Diffrntil Eqution, Journl of Physics nd Its pplictions, Vol 3, No.1, pp 1:6, Dprtmnt of Physics, ITS,Sury, 7. [9] Rohdi,A.Y, Introducing Stl Modultion Tchniqu for Solving n Inhomognous Brnoulli Diffrntil Eqution, Sumit to Symposium ICOLA 7, Univrsity of Indonsi, 7. [1] Rohdi,A.Y, Anlyticl Solution of th Rictti Diffrntil Eqution for High Frquncy using Stl Modultion Tchniqu, Prsntd on ICMNS, ITB, Bndung, Nopmr, 6. [11] Spigl,M.R,(1968)., Mthmticl Hndook of Formuls nd Tls, Schum s Outlin Sris, McGRAW-Hill Book Compny. [1] Much,S, Introduction to Mthods of Applid Mthmtics Advncd Mthmticl Mthods for Sintists nd Enginrs, Chptr 7, pp.6:63, Much Pulishing Compny, 1. [13] Louis Morls-Molin t.l, Soliton Rtchts in Homognous Nonlinr Klin Gordon Systm, rxiv:cond-mt/5174 v, Nop. 5.

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