Soliton Scattering on the External Potential in Weakly Nonlocal Nonlinear Media

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1 Mlysin Journl of Mhemicl Sciences 1(S) Februry: (216) Specil Issue: The 3 rd Inernionl Conference on Mhemicl Applicions in Engineering 214 (ICMAE 14) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journl homepge: hp://einspem.upm.edu.my/journl Solion Scering on he Exernl Poenil in Wekly Nonlocl Nonliner Medi Umrov, B.A. 1 nd Busul Akln, N. A. 2 1 Deprmen of Physics, Kulliyyh of Science, Inernionl Islmic Universiy Mlysi, 252 Kunn, Phng, Mlysi 2 Deprmen of Compuionl nd Theoreicl Sciences, Kulliyyh of Science, Inernionl Islmic Universiy Mlysi, 252, Kunn, Phng, Mlysi E-mil: bkhrm@iium.edu.my ABSTRACT The Nonliner Schrödinger Equion (NLSE) is one of he universl mhemicl models nd i rises in such diverse res s plsm physics, condensed mer physics, Bose - Einsein condenses, nonliner opics, ec. In his work he scering of he solion of he generlized NLSE on he loclized exernl poenil hs been sudied, king ino ccoun he wek nonlocliy of he medi. We hve pplied he pproxime nlyicl mehod, nmely he vriionl mehod o derive he equions for solion prmeers evoluion during he scering process. The vlidiy of pproximions were checked by direc numericl simulions wih solion iniilly loced fr from poenil. I ws shown h depending on iniil velociy of he solion, he solion my be refleced by poenil or rnsmied hrough i. The criicl vlues of he velociy sepring hese wo scenrios hve been idenified. Keywords: Solion, nonliner equions, scering, vriionl mehods.

2 Umrov, B.A. nd Busul Akln, N. A. 1. Inroducion The invesigion of nonliner wve processes become one of he mos ineresing problems of he modern physics nd mhemics, wih numerous pplicions in differen res of he physics nd engineering (Sco, 25); (Remoissene, 23). The loclized wves propging in nonliner medi, or solions re he objecs rcing he enion of he reserches, nd hve been heoreiclly nd experimenlly sudied firs in he conex of wer wves (Wihm, 1974). Then i become cler h solion is he universl concep nd i hs been discovered in he solid se physics, nonliner opics, Bose-Einsein condenses, plsm physics, ec (Bullough nd Cudrey, 198); (Agrwl, 1995); (Pievski nd Sringri, 23). The heoreicl sudies of solions bsed on few models, reflecing he bsic principles of he physics of nonliner wves, nd lso on he generlizions of hese models, expending he limis of heir pplicbiliy. One of his models is he nonliner Schrödinger equion (NLSE) (Ablowiz e l., 24). This is he pril differenil equion, which is he inegrble by Inverse Scering Trnsform (IST) mehod nd hs mulisolion soluions. Solions of NLSE preserve heir ideniy while propging nd inercing wih ech oher. The NLSE describes propgion of he wve pckes in wekly nonliner nd wekly dispersive medium. For pplicions i is imporn o be ble o conrol solions, which cn be chieved by inercion of solion wih noher solion or exciion of he sysem. Also he solion cn be mnged using exernl perurbions, bu in his cse i is necessry o modify he model nd include ddiionl erms. In his work we sudy he inercion of he solion of he generlized NLSE, king ino ccoun wek nonlocliy of nonlineriy, wih he loclized liner poenil. The pper is orgnized s follows. In Sec.2 he mhemicl model is formuled nd bsic equions re derived. Sec.3 is devoed o numericl simulions of he vriionl equions nd comprison of he resuls wih direc soluion of he originl wekly nonlocl NLSE. Finlly, in Sec.4 we summrize our findings, nd discuss some ineresing direcions for fuure sudies. 2. The model nd min equions We begin wih he discussion of wve propgion in nonliner nonlocl wekly dispersive medi, nd ssume h nonlineriy is of he Kerr ype. Then he following generlized NLSE equion cn be wrien s mhemicl model of he bove menioned sysem (Krolikowski nd Bng, 2). 22 Mlysin Journl of Mhemicl Sciences

3 Solion Scering on he exernl poenil iψ ψ xx + n( ψ 2 )ψ + V (x)ψ =, (1) where ψ(x, ) is he field funcion nd V (x) is he exernl poenil. In he following we consider he nrrow poenil which cn be modeled by del funcion V (x) = U δ(x), nd U is he mpliude of he poenil. The funcion n = R(x x ) ψ 2 dx is he model of nonlocl nonlineriy nd R(x) is he response funcion of nonlocl medium. In cse of he R(x) = δ(x) he response becomes singulr, nd he Eq.(1) will be reduced o NLSE equion. When he nonlocliy is wek, i.e. he response funcion is nrrow in comprison wih he loclized wve widh Eq.(1) cn be furher reduced o he following generlized NLSE (Krolikowski nd Bng, 2) iψ ψ xx + ( ψ 2 + γ 2 x( ψ 2 )ψ + V (x)ψ =, (2) where γ = 1 2 R(x)x2 dx nd i is ssumed h he response funcion is normlized R(x)dx = 1. When V (x) = Eq.(2) possesses brigh sionry solion soluion ψ(x, ) = u(x) exp (iγ), (3) nd u(x) cn be found nlyiclly in implici form (Krolikowski nd Bng, 2) ± x = 1 u nh 1 ( σ u ) + 4γ n 1 ( 4γσ), (4) where u is he mximum of u(x) nd σ 2 = (ρ ρ)/(1+4γρ), ρ = u 2, ρ = u 2. I is esy o verify h he governing Eq. (2) cn be obined from he following Lgrngin densiy (Bezuhnov e l., 28) L = i 2 (ψψ ψ ψ ) ψ x 2 + V (x) ψ 2 g 2 ψ 4 + γ 2 ( x ψ 2 ) 2, (5) by mens of he Euler-Lgrnge equion. Now one cn pply he vriionl opimizion procedure (Anderson, 1983); (Mlomed, 22) o ge he pproxime sysem of ordinry differenil equions for solion prmeers. The ril funcion which pproxime he soluion of he Eq. (2) we choose from he following considerions. When one neglecs he nonlocliy nd exernl poenil, so h V (x) = nd γ =, Eq. (2) Mlysin Journl of Mhemicl Sciences 221

4 Umrov, B.A. nd Busul Akln, N. A. will be reduced o he NLSE wih well known solion soluion. So, when he exernl poenil nd nonlocliy re wek, hey cn be considered s perurbion, nd he shpe of soluion in his cse will remin he sme s solion of NLSE, bu he prmeers, which re consns in exc soluion on NLSE, will be chnging ccording o he vriionl evoluion equions, which will be derived from verged Lgrngin. Then spil inegrion of he Lgrngin densiy L = Ldx using he ril funcion ( x ξ ψ(x, ) = Asech ) e ib(x ξ)2 +iv(x ξ)+iϕ, (6) gives rise o following verged/effecive Lgrngin [ π 2 L = N 12 2 b + π2 6 2 b ξ2 + ϕ N 6 U 2 sech2 ( ) ξ + 2γN ] 3. (7) The norm of he wve funcion N = ψ 2 dx = 2A 2 is conserved quniy. Evoluion equions for vriionl prmeers cn be derived from he Euler-Lgrnge equions d/d( L/ q i ) L/ q i =, where q i re ime dependen collecive coordines, ξ, b, ϕ. The equion for he phse ϕ reduces o dn/d = nd illusres he conservion of he norm of he wve funcion. I is decoupled from oher equions nd cn be dropped in furher nlysis. Wh remins is se of coupled equions for he widh nd cener-of-mss posiion of he solion = 4 π 2 3 2N π U ξ = U 2 sech2 ( ξ π 2 2 sech2 ( ξ ) nh ( ) [ ξ 1 2ξ nh ( )] ξ + 24γN 5π 2 4, (8) ). (9) When exernl poenil is bsen, i.e. U =, Eqs.(8) nd (9) decouple, nd from Eq.(8) one cn find he pproxime widh of he sionry( = ) solion soluion of wekly nonlocl NLSE s = 12γ/5 + 1/A 2. Perurbions my genere oscillions of he widh round his sionry poin. The velociy ξ in his cse is he consn free prmeer. Inclusion of del poenil o he sysem, couples he ime evoluion of posiion of he cener of he solion wih he evoluion of is widh. Obviously when solion is loced fr from inhomogeneiy, i does no ffeced by i, nd he solion s prmeers re consn. Some quliive resuls bou solions evoluion we cn ge, if we neglec he effec of poenil o he widh of solion. Then he equion (9) describes he scering of effecive clssicl pricle on he loclized brrier. ξ = U ( ) ξ 2 sech 2 nh s s ( ) ξ s = dv P (ξ) dξ. (1) 222 Mlysin Journl of Mhemicl Sciences

5 Solion Scering on he exernl poenil This equion cn be inegred once nd reduced o he following equion ξ = 2V P (ξ), (11) where V P (ξ) = (U /2 s )sech 2 (ξ/ s ) is he effecive poenil represening he influence of he originl loclized del poenil o he solions velociy. From he Eq. (11) i is cler h he effecive pricle cn be rnsmied hrough poenil or refleced from i depending on wheher he velociy bove or below of criicl vlue v c = U / s. 3. Numericl simulions The resuls obined bove re pproxime, nd bsed on some ssumpions, so hey should be compred wih he resuls of he direc numericl soluions of he governing equions. Numericl soluion of he generlized NLSE (2) hs been performed by he spli-sep fs Fourier rnsform mehod (Agrwl, 1995) using 248 Fourier modes wihin he inegrion domin of lengh L [ 2 2], nd he ime sep ws δ =.5. The dynmicl equions of he vriionl pproximion (8)-(9) re solved using he Runge-Ku procedure of 4-h order (Press e l., 1996). Solion ψ(x) is se in moion wih some velociy v owrds he poenil brrier V (x) iniilly loced some disnce from i. Le us firs discuss he resuls of numericl soluions of he sysem of ordinry differenil equions (8)-(9), describing he pproxime ime evoluion of he widh nd posiion of he cener of he solion. Iniil widh is chosen equl o () = s nd A = As i ws discussed before, he solion behves like clssicl pricle wih inernl degree of freedom ssocied wih he widh of solion. Depending on he mpliude of poenil, solion my be rnsmied or refleced from he poenil, nd when solion close o he poenil, no only is velociy, bu lso is widh will be ffeced by perurbion. The criicl prmeers of he sysem sepring rnsmission from reflecion quie well described by he formul, obined bove from simplified effecive pricle picure of he solion scering on he loclized wek poenil. The exmples of he resuls of numericl soluion of he Eqs. (8) & (9) shown in he Figs.(1) & (2) Also, he exmples of resuls of numericl experimen wih he governing generlized NLSE (2), on he sme rnge of prmeers nd he iniil condiion chosen in he form of exc solion from Eq.(3) wih u = A = , nd Γ = 1 re shown on he Fig.(3), which confirm quliively h solion behves like pricle nd cn be eiher rnsmied or refleced on he poenil wll. Mlysin Journl of Mhemicl Sciences 223

6 Umrov, B.A. nd Busul Akln, N. A Cener of solion posiion Figure 1: Evoluion of he widh (lef pnel) nd cener of solion posiion (righ pnel) of solion in he presence of del poenil brrier V (x) = U δ(x + 3), ccording o ODE sysems: (8)-(9). Prmeers: U =.7, γ =.2, =.98994, v =.3, ξ = Cener of solion posiion Figure 2: Evoluion of he widh (lef pnel) nd cener of solion posiion (righ pnel) of solion in he presence of del poenil brrier V (x) = U δ(x + 3), ccording o ODE sysems: (8)-(9). Prmeers: U =.1, γ =.2, =.98994, v =.3, ξ = Conclusion We hve developed vriionl pproximion o describe he scering of solions of weekly nonlocl NLSE by exernl del poenil. Dynmicl equions for he prmeers of he solion hve he form of ordinry differenil equions. Quie good greemen beween he resuls of vriionl equions nd direc numericl soluion of he originl generlized NLS equion is found for solion scering on week poenil brrier, when he pricle picure cn be effecively pplied. The fuure plnned reserch includes he considerion of solion inercion wih loclized poenil wlls s well s poenil wells of 224 Mlysin Journl of Mhemicl Sciences

7 Solion Scering on he exernl poenil x.6 x Figure 3: Evoluion of he ψ(x, ) wih solion iniil condiions nd V (x) = U δ(x+3), ccording o he Eq.(2). In he lef pnel he prmeers re he sme s in Fig.(1), nd in he righ pnel he prmeers re he sme s in Fig(2). he differen shpes. The resuls cn be useful in developmen of new mehods imed probing he exernl poenils/defecs by scering solions on hem. Acknowledgmens This work hs been suppored by he reserch grn No. FRGS of he Minisry of Higher Educion. References Ablowiz, M., Prinri, B., nd Trubch, A. (24). Discree nd Coninuous Nonliner Schrodinger Sysems. CUP, Cmbridge. Agrwl, G. (1995). Nonliner Fiber Opics. Acdemic Press, New York. Anderson, D. (1983). Vriionl pproch o nonliner pulse propgion in opicl fibers. Physicl Review A, 27(6):3135. Bezuhnov, K., Dreischuh, A., nd Krolikowski, W. (28). Brigh opicl bems in wekly nonlocl medi: Vriionl nlysis. Physicl Review A, 77(33825):1 7. Bullough, R. nd Cudrey, P. (198). Solions. Springer, Berlin. Krolikowski, W. nd Bng, O. (2). Solions in nonlocl nonliner medi: Exc soluions. Physicl Review E, 63(1661):1 6. Mlysin Journl of Mhemicl Sciences 225

8 Umrov, B.A. nd Busul Akln, N. A. Mlomed, B. (22). Vriionl mehods in nonliner fiber opics nd reled fields. Progress in Opics, 43: Pievski, L. nd Sringri, S. (23). Bose-Einsein Condension. OUP, Oxford. Press, W., Teukolsky, S., Veerling, W., nd B.P., F. (1996). Recipes. The Ar of Scienific Compuing. CUP, Cmbridge. Numericl Remoissene, M. (23). Wves Clled Solions. Conceps nd Experimens. Springer, Berlin Heidelberg. Sco, A. (25). Encyclopedi of Nonliner Science. Rouledge Tylor nd Frncis Books, New York. Wihm, G. (1974). Liner nd Nonliner Wves. York. Wiley-Inerscience, New 226 Mlysin Journl of Mhemicl Sciences