Soliton Solutions of the Perturbed Resonant Nonlinear Schrödinger's Equation with Full Nonlinearity by Semi-inverse Variational Principle

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1 Qu. Phys. Le. Vol. No Quu Physis Leers Ieriol NSP Nurl Siees Pulishi Cor. Solio Soluios of he Perured Reso Nolier Shrödier's Euio wih Full Nolieriy y Sei-iverse Vriiol Priiple j isws Depre of Mheil Siees, Delwre Se Uiversiy, Dover, DE US sr: This pper rries ou he ierio of he reso olier Shrödier's euio i presee of perurio ers h re osidered wih full olieriy. The hree ypes of olier edi re sudied. They re he ui olieriy, power lw d lo lw olieriy. The sei-iverse vriiol priiple is pplied o er he lyil solio soluio. Keywords: Nolier Shrödier's euio, solio soluio,sei-iverse vriiol priiple. Iroduio The olier Shrödier's euio NLSE plys vil role i vrious res of STEM disiplies [-]. I ppers i he sudy of olier opis, pls physis, heil iosiees, uu ehis, fluid dyis d severl oher disiplies. The i feure of he NLSE is h i suppors solio soluio whih kes i very widely pplile. Solios re sle olier wves or pulses d is he ouoe of delie le ewee dispersio d olieriy. Therefore, hese solios re he esseil fris h die our dily lives. For eple, hese wves re sle pulses h rspor iforio hrouh opil _ers over rs-oiel d rs-oei dises i er of few feo-seods. Oher eples of solios i our dily lives re i ose-eisei odeses, α-heli proeis i liil siees, uler physis d severl ohers. Therefore i is iperive o fous deeply io he ieriliy spes of he NLSE h will revel solio soluios. This pper will sudy he eeded s well s he eerlized versio of he NLSE wih few Hiloi perurio ers h re oi o e ke io osiderio wih full olieriy. There re hree ypes of olieriy h will e ddressed. They re he ui NLSE, power lw olieriy d he lo lw olieriy. While i he firs wo ypes of olieriy, i is he solio soluio h will e oied, he hird lw olieriy will ive Gusso soluios. The sei-iverse vriiol priiple SVP will e ipleeed o er he solio d Gusso soluios o he perured reso NLSE. This priiple is silly iverse prole pproh. II. Goveri Euio The perured reso NLSE wih full olieriy h is oi o sudied i his pper y he SVP is ive y i F * i α i iv i

2 8 j isws: Solio Soluios of he Perured... Here i, o he lef hd side, he firs er represes he lier evoluio of he solio pulse. The oeffiie of is he roup veloiy dispersio while he oeffiie of is he olier er. The oeffiie of is he uu or oh poeil h ppers i he oe of hirl solios i uu Hll effe [5, 7,, ]. I is lso see i he oe of Mdelu fluid i uu ehis [7]. For he perurio ers o he rih hd side, he oeffiie of α is he ier-odl dispersio h shows up i olier opis. The, he oeffiie of is he selfseepei er h is lso sudied i olier opis i order o void he forio of shok wves duri solio rsissio hrouh opil fiers. The oeffiies of υ d re due o olier dispersios. Filly, he -er is fro pls physis for solios i relivisi plss [, 9, ]. The ide represes he full olieriy preer. The idepede vriles re d h represe spil d eporl vriles respeively. The depede vrile is, h is he ople vlued wve profile for he perured reso NLSE. The fuiol F represes heeerl for of olier edi d F is k ies oiuously differeile, so h k F C,, ;R U,prole hve ee sudied i he ps. I priulr, he speil se wih, for ui d power lw olieriy, ws overed i 9 [, 6]. ddiiolly, he se of lo-lw olieriy wih he se speil vlues of he preers ws ddressed i []. The se wih rih hd side euio se o zero u wih hirl olieriy ws ddressed o severl osios, ely duri d [5, 7, ]. The e rih d drk solio soluios y he sz ehod were lso oied i [7].Thus, his pper is hus oi o ddress euio o he os eerlized sei so fr, d he ool of ierio is oi o e he SVP. The sri hypohesis is ive y iφ, se where s represes he shpe of he wve profile d s v wih v ei he veloiy of he wve. The phse opoe φ, isdefied s φ 5 where represes he solio freuey d is he solio wveuer while represe he phse os. Therefore, susiui his hypohesis io d deoposi io rel d iiry prs yield he followi wo euios // α F 6 d v α { v } 7 respeively. Sei he oeffiies of he lierly idepede fuios i 7 o zero yields he solio veloiy s v α 8 d he osri odiios ewee he preers is ive y

3 j isws: Solio Soluios of he Perured... 8 v 9 I eeds o e oed h he veloiy of he solio s well s he osri odiio holds irrespeive of he ype of olieriy iuesio. Now, uliplyi oh sides of he rel pr euio 6 y / d ieri leds o / α F d K where K represes he ierio os. The, he siory ierl J is defied o e J Kds whih herefore is / J α F dds Filly, he -solio soluio hypohesis is ke o e or [ seh s ] s f s s e where is he pliude d is he iverse widh of he solioor Gusso. The fuiol f i depeds o ui or power lw olieriy. The SVP ses h he pliude e rerieved fro he oupled syse of euios ive y J 5 d J 6 III. ppliios The SVP h ws developed i he previous seio will ow e pplied o he hree ypes of olieriy h will e deiled i he followi hree suseios. The eplii vlue of he solio pliude d he iverse widh re oi o e deeried for eh of hese olier ses.. Cui Nolieriy I his se, Fu u d his is ooly referred o s he ui Shrodier's euio d i he oe of olier opis his isreferred o Kerr lw olieriy. This ype of ui NLSE is very ooly sudied i pls physis, solios due o α-heli proeisi heil iosiees, deep wer wves i fluid dyis s well s olier opis. Wih ui olieriy, he overi euio redues o * 7 i so h he orrespodi siory ierl is iv i iα i

4 8 j isws: Solio Soluios of he Perured... ds J / α 8 For he ui NLSE, he -solio soluio hypohesis is [7] s seh s 9 Susiui his hypohesis io 8 d rryi ou he ierio redues he siory ierl o J α The euios 5 d 6 i his se, fer siplifiio, rerespeively ive y α α Solvi he oupled syse of euios d leds o he polyoil euio for he pliude s α Fro, i e esily see h he solio pliude e epliily oied provided, or. Oe he pliude is ville, he widh e reovered fro he relio 5 h e oied fro d. Euio iediely poses resriio > 5 Therefore he -solio soluio o 7 is ive y [ ] i e v seh, 6 where he pliude d he widh re respeively ive y d d he veloiy of he solio is ive y 8. This soluiois vlid s lo s he osri odiios ive y 9 d 5 hold.

5 j isws: Solio Soluios of he Perured Power Lw Nolieriy I his suseio, he perured reso NLSE will e sudied wih power lw olieriy. I his se, u u F. Power lw olieriyis lso sudied i he oe of olier opis where speil se of opil fiers is desied wih solio rsissio i id []. I his se, euio odifies o i iv i i i * α 7 where he resriio is < < 8 o void solio ollpse d i priulr 9 i order o eliie self-fousi siulriy i olier opis [].This leds o he siory ierl ei ds J / α For power lw olieriy, he -solio soluio sz is [7] s seh s Susiui his hypohesis io d rryi ou he ierio redues he siory ierl o J α The euios 5 d 6 i his se, fer siplifiio, rerespeively ive y α d α Euios d leds o he polyoil euio for he solio pliude s α 5 i fro d, he solio widh e reoveredfro

6 8 j isws: Solio Soluios of he Perured... Therefore, he -solio soluio o 7 is ive y e i [ ] v, seh 7 where he pliude d he widh re respeively ive y 5d 6 d he veloiy of he solio is sill ive y 8. This soluio is vlid s lo s he osri odiios ive y 9 d5 hold.. Lo Lw Nolieriy I he se of lo lw olieriy, F u l u, d hus he perured reso NLSE is ive y * 8 i iv i L iα i so h he siory ierl i his se rsfors o / J α L ds 9 Now, hoosi he Gusso sz ive y, he siory ierl9 redues o J π α π 5 π l π The, euio 5 d 6, i his se ives π α l d π α l 5 respeively. Solvi his oupled syse yields he pliude of he Gusso s ep W ep where 5 5 α 6 d W is he Ler's W-fuio h is defied o e he iverse of 6

7 j isws: Solio Soluios of he Perured f e 7 Euio irodues he osri odiio, fro he defiiio of Ler's fuio, s ep 8 Oe he pliude of he Gusso is ville, he widh e oied fro 5 9 whih e reovered fro d. Hee, filly, he Gusso soluio of 8 is ive y v i, e e 5 where he Gusso pliude d he iverse widh e oied fro d 9 respeively. esides 5, he ddiiol osri odiio i his se is ive y 8 h us lso hold i order for he Gussos o eis. The veloiy is i see i 8 lo wih he resriio 9. IV. Colusio This pper ddressed he perured reso NLSE where he perurio ers re osidered wih full olieriy. The SVP is ppliedo rry ou he ierio of he perured reso NLSE. Thus he solio soluios were oied. There re hree ypes of olieriy h re disussed i his pper. They re he ui olieriy, powerlw olieriy d filly he lo lw olieriy. This SVPis esseilly iverse prole ehis h ieres he perured NLSE. The soluios re oied d he solio preers re i ers of fuios d Ler's W-fuio. Severl osri odiios uoilly fell ou fro he heil sruure of he soluio preers. This pper eopssed severl sudies h were odued i he ps. The speil ses of he resuls of he pper were lredy oied efore s idied. Thus, he ppliio of he NLSE i olier opis, pls physis, hirl solios i uler physis d heil iosiees re ll olleively sudied i his pper. Therefore, hese eerlized resuls re oi o serve s he sri poi for furher ivesiio of he NLSE i his direio. Thus, he fuure of his re of reserh sds o sro fooi. Oe iedie epsio of his reserh is o look io severl oher fors of olier edi, suh s he proli lw olieriy, polyoil lw olieriy, dul d riple power lw olieriy s well s he surle lw olieriy, jus o e few. This is jus he ip of he ieer. Referees [] S. if, D. Milovi, E. Zerrd &. isws. "Solios i relivisiplss y He's vriiol priiple". pplied Physis Reserh.Volue, Nuer, -6.. [] M. G. ee & J. F. You. "Hih-order solio rekup dsolio self-freuey shifs i irosruured opil fier".jourl of Opil Soiey of eri. Volue, Nuer 7,

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