Research Article On Perturbative Cubic Nonlinear Schrodinger Equations under Complex Nonhomogeneities and Complex Initial Conditions

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1 Hindawi Publishing Corporaion Differenial Equaions and Nonlinear Mechanics Volume 9, Aricle ID 959, 9 pages doi:.55/9/959 Research Aricle On Perurbaive Cubic Nonlinear Schrodinger Equaions under Complex Nonhomogeneiies and Complex Iniial Condiions Magdy A. El-Tawil and Maha A. El-Hamy Deparmen of Engineering Mahemaics, Faculy of Engineering, Cairo Universiy, Gia, Egyp Deparmen of Mahemaics, Girls College, Medina, Saudi Arabia Correspondence should be addressed o Magdy A. El-Tawil, magdyelawil@yahoo.com Received February 9; Revised May 9; Acceped July 9 Recommended by Roger Grimshaw A perurbing nonlinear Schrodinger equaion is sudied under general complex nonhomogeneiies and complex iniial condiions for ero boundary condiions. The perurbaion mehod ogeher wih he eigenfuncion expansion and variaional parameers mehods are used o inroduce an approximae soluion for he perurbaive nonlinear case for which a power series soluion is proved o exis. Using Mahemaica, he symbolic soluion algorihm is esed hrough compuing he possible approximaions under runcaion procedures. The mehod of soluion is illusraed hrough case sudies and figures. Copyrigh q 9 M. A. El-Tawil and M. A. El-Hamy. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied.. Inroducion The nonlinear Schrodinger NLS equaion is he principal equaion o be analyed and solved in many fields, see 5, for examples. In he las wo decades, here are a lo of NLS problems depending on addiive or muliplicaive noise in he random case, 7 or a lo of soluion mehodologies in he deerminisic case. Wang e al. obained he exac soluions o NLS using wha hey called he subequaion mehod. They go four kinds of exac soluions of i u u x α u p u β u p u,. for which no sign o he iniial or boundary condiions ype is made. Xu and Zhang 9

2 Differenial Equaions and Nonlinear Mechanics followed he same previous echnique in solving he higher-order NLS: i u x u α β u u iε u u iδ u u iγu.. Sweilam solved i u u x q u u, >, L <x<l,. wih iniial condiion u x, g x and boundary condiions u x L, u x L,, which gives rise o soliary soluions using variaional ieraion mehod. Zhu used he exended hyperbolic auxiliary equaion mehod in geing he exac explici soluions o he higherorder NLS: iq β q γ q q i β q β q γ q q,. wihou any condiions. Sun e al. solved he NLS: i ψ ψ x a ψ ψ,.5 wih he iniial condiion ψ x, ψ x using Lie group mehod. By using coupled ampliude phase formulaion, Porseian and Kalihasan consruced he quaric anharmonic oscillaor equaion from he coupled higher-order NLS. Two-dimensional grey solions o he NLS were numerically analyed by Sakaguchi and Higashiuchi. The generalied derivaive NLS was sudied by Huang e al. 5 inroducing a new auxiliary equaion expansion mehod. Abou Salem and Sulem sudied he effecive dynamics of solions for he generalied Schrodinger equaion in a random poenial. El-Tawil 7 considered a nonlinear Schrodinger equaion wih random complex inpu and complex iniial condiions. Colin e al. considered hree componens of nonlinear Schrodinger equaions relaed o he Raman amplificaion in a plasma. In 9, Jia-Min and Yu-Lu consruced an appropriae ransformaions and an exended ellipic subequaion approach o find some exac soluions for variable coefficien cubic-quinic nonlinear Schrodinger equaion wih an exernal poenial. In his paper, a sraigh forward soluion algorihm is inroduced using he ransformaion from a complex soluion o a coupled equaions in wo real soluions, eliminaing one of he soluions o ge separae independen and higher-order equaions, and finally inroducing a perurbaive approximae soluion o he sysem.. The Linear Case Consider he nonhomogeneous linear Schrodinger equaion: u, i α u, F, if,,,,t,,.

3 Differenial Equaions and Nonlinear Mechanics where u, is a complex valued funcion which is subjeced o I.C. : u, f if, a complex valued funcion, B.C. : u,, u T,.. Le u, ψ, iφ,, ψ, φ: real valued funcions. Subsiuing. in., he following coupled equaions are go as follows: φ, ψ, α ψ, α φ, G,, G,,. where ψ, f, φ, f, G, F,, G, F,, and all corresponding oher I.C. and B.C. are eros. Eliminaing one of he variables in., one can ge he following independen equaions: ψ, φ, α ψ, α φ, α ψ,, α ψ,,. where ψ, G α G, ψ, α G G..5 Using he eigenfuncion expansion echnique, he following soluions for. are obained: ( nπ ψ, T n sin T n ( nπ φ, τ n sin T n,,. where T n and τ n can be go hrough he applicaions of iniial condiions and hen solving he resulan second-order differenial equaions using he mehod of he variaional

4 Differenial Equaions and Nonlinear Mechanics parameer. The final expressions can be go as follows T n C A sin β n C B cos β n, τ n C A sin β n C B cos β n,.7 where ( nπ, β n α T A ψ n ; n cos ( β n d, β n B β n A β n ψ n ; n sin ( β n d, ψ n ; n cos ( β n d,. B β n ψ n ; n sin ( β n d, in which ψ n ; n T ψ n ; n T T T ( nπ ψ, sin T d, ( nπ ψ, sin T d..9 The following condiions should also be saisfied: C T C T T T ( nπ f sin T ( nπ f sin T d B, d B.. Finally, he following soluion is obained: u, ( ψ, iφ,,. or u, ( ψ, φ,..

5 Differenial Equaions and Nonlinear Mechanics 5 5 Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M.. The Nonlinear Case Consider he homogeneous nonlinear Schrodinger equaion: u, i α u, ε u, u, iγu, F, if,,,,t,,. where u, is a complex-valued funcion which is subjeced o he iniial and boundary condiions..

6 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M. Lemma.. The soluion of. wih he consrains. is a power series in ε if he soluion exiss. Proof. A ε, he following linear equaion is go: u, i α u, iγu, F,,,,T,,.

7 Differenial Equaions and Nonlinear Mechanics Figure 5: The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. 5 which has he soluion, see he previous secion, u, e γ w, iv,..

8 Differenial Equaions and Nonlinear Mechanics Figure 7: The ero-order approximaion of a α, T, γ wih considering only one erm in he series M. 5 5 Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. Following Pickard approximaion,. can be rewrien as i u n, α u n, iγu n, F, ε u n, u n,, n.. A n, he ieraive equaion akes he following form: i u, α u, iγu, F, ε u, u, F, εh,,.5

9 Differenial Equaions and Nonlinear Mechanics Figure 9: The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. 5 which can be solved as a linear case wih ero iniial and boundary condiions. The following general soluion can be obained: ( nπ w, T n εt n sin T n ( nπ v, τ n ετ n sin T n,,. u, e γ w, iv, u εu. A n, he following equaion is obained: i u, α u, iγu, F, ε u, u, F, εh,,.7 which can be solved as a linear case wih ero iniial and boundary condiions. The following general soluion can be obained: Coninuing like his, one can ge u, u εu ε u ε u ε u.. u n, u n εu n ε u n ε u n ε n m u n m n..9

10 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M. As n,hesoluion if exiss can be reached as u, lim n u n,. Accordingly, he soluion is a power series in ε. According o he previous lemma, one can assume he soluion of. as he following: u, ε n u n. n. Le u, ψ, iφ,, ψ, φ: real valued funcions. The following coupled equaions are go: φ, ψ, α ψ, α φ, ( ε ψ φ ψ γφ F, ( ε ψ φ φ γψ F,. where ψ, f, φ, f, and all corresponding oher I.C. and B.C. are eros. As a perurbaion soluion, one can assume ha ψ, ψ εψ ε ψ, φ, φ εφ ε φ,. where ψ, f, φ, f, k and all corresponding oher I.C. and B.C. are eros.

11 Differenial Equaions and Nonlinear Mechanics Subsiuing. ino. and hen equaing he equal powers of ε, one can ge he following se of coupled equaions: φ, ψ, φ, ψ, φ, ψ, α ψ, γφ F,. α φ, γψ F,. α ψ, α φ, α ψ, α φ, ( γφ ψ ψ φ,.5 ( γψ φ φ ψ,. ( γφ ψ ψ ψ φ φ ψ φ,.7 ( γψ φ φ φ ψ ψ φ ψ,. and so on. The prooype equaions o be solved are φ i, ψ i, α ψ i, α φ i, γφ i G i, i, γψ i G i, i,.9 where ψ i, δ i, f, φ i, δ i, f and all oher corresponding condiions are eros. The nonhomogeneiy funcions G i and G i are funcions compued from previous seps. Following he soluion algorihm described in he previous secion for he linear case, he general symbolic algorihm in Figure 9 can be simulaed hrough he use of a symbolic package, mahemaica-5 is used in his paper... The Zero-Order Approximaion In his case, u, ( ψ iφ,.

12 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. where ( nπ ψ, e γ T n sin T n ( nπ φ, e γ τ n sin T n,,.

13 Differenial Equaions and Nonlinear Mechanics 5 Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. in which T n A sin β n C B cos β n, ( τ n A sin β n C B cos β n,. where he consans and variables A, C,B, A, C,andB can be go by he aid of Secion.

14 Differenial Equaions and Nonlinear Mechanics Figure 5: The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. 5 Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M. The absolue value of he ero-order approximaion is go from u, ψ φ.... The Firs-Order Approximaion u, u ε ( ψ iφ,.

15 Differenial Equaions and Nonlinear Mechanics Figure 7: The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. where ( nπ ψ, e γ T n sin T n ( nπ φ, e γ τ n sin T n,,.5 in which T n A sin β n C B cos β n, ( τ n A sin β n C B cos β n,. where he consans and variables A, B, A, andb can be evaluaed in a similar manner as he ero-order approximaion whereas C B and C B. The absolue value of he firs-order approximaion can be go using u, u, ε ( ( ψ ψ φ φ ε ψ φ..7.. The Second-Order Approximaion u, u, ε ( ψ iφ,.

16 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. Figure 9: The ero-order approximaion of a α, T, γ wih considering only one erm in he series M. where ( nπ ψ, e γ T n sin T n ( nπ φ, e γ τ n sin T n,,.9

17 Differenial Equaions and Nonlinear Mechanics 7 Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. in which T n A sin β n C B cos β n, ( τ n A sin β n C B cos β n,.

18 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M. 7 5 Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. where he consans and variables A, C,B, A, C, and B can be evaluaed similarly as he previous approximaion. The absolue value of he second-order approximaion can be go using u, u, ε ( ψ ψ φ φ ε ( ( ψ ψ φ φ ε ψ φ... Case Sudies To examine he proposed soluion algorihm, see Figure 9, some case sudies are illusraed.

19 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M..5.5 Figure 5: The ero-order approximaion of a α, T, γ wih considering only one erm in he series M... One Inpu Is On Case Sudy Taking f, f, F,, F,, and following he soluion algorihm, he selecive resuls for he ero-order approximaion are go in Figures,,and. Case Sudy Taking f, f, F,, F, and following he soluion algorihm, i has been noiced ha he same resuls for he case sudy are go.

20 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M Figure 7: The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. Case Sudy Taking f, f, F,, F, and following he soluion algorihm, he selecive resuls for he firs-ero approximaion are go in Figures, 5,and. One can noice he decrease of he soluion level and is higher variabiliy.

21 Differenial Equaions and Nonlinear Mechanics 5 Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M. Case Sudy Taking f, f, F,, F, and following he soluion algorihm, i has been noiced ha he same resuls for he case sudy are go:.. Two Inpus Are On Case Sudy 5 Taking f, f, F,, F, and following he soluion algorihm, he selecive resuls for he ero-order approximaion are go in Figures 7,,and9. One can noice ha he soluion level becomes a lile bi higher han ha of case sudy. Case Sudy Taking f, f, F,, F, and following he soluion algorihm, he selecive resuls for he ero-order approximaion are go in Figures,,and. One can noice he lile increase of he soluion level han ha of case sudies and. Case Sudy 7 Taking f, f, F,, F, and following he soluion algorihm, he selecive resuls for he ero-order approximaion are go in Figures,,and5. One can noice he small perurbaions a small values of.

22 Differenial Equaions and Nonlinear Mechanics Figure 9: The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M. 5 5 Figure : The ero-order approximaion a α, T, γ for differen values of, considering only one erm in he series M... Three Inpus Are On Case Sudy Taking f, f, F,, F, and following he soluion algorihm, he selecive resuls for he ero-order approximaion are go in Figures, 7,and. One can noice he increase of he deph of he perurbaions.

23 Differenial Equaions and Nonlinear Mechanics.5.5 Figure : The ero-order approximaion of a α, T, γ wih considering only one erm in he series M Figure : The ero-order approximaion a α, T, γ fordifferen values of, considering only one erm in he series M. Case Sudy 9 Taking f, f, F,, F, and following he soluion algorihm, he selecive resuls for he ero-order approximaion are go in Figures 9,,and. One can noice ha he perurbaions become smaller han ha of case sudy.

24 Differenial Equaions and Nonlinear Mechanics Figure : The ero-order approximaion a α, T, γ fordifferen values of, considering only one erm in he series M. u.5.5 Figure : The firs-order approximaion of u a α, T, γ, ε wih considering only one erm in he series M... Four Inpus Are On Case Sudy Taking he case of f, f, F, F, he following final resuls for he ero-order approximaion are obained in Figures,,and. One can noice he lile increase in he soluion level.

25 Differenial Equaions and Nonlinear Mechanics 5 u 5 Figure 5: The firs-order approximaion of u a α, T, γ, ε 5 wih considering only one erm in he series M. u Figure : The firs-order approximaion of u a α, T, γ, ε wih considering only one erm in he series M..5. Exponenial Nonhomogeneiy Case Sudy Taking he case of f, f, F, e, F,, he following final resuls for he ero-order approximaion are obained in Figures 5,,and7. One can noice he low soluion level and high perurbaions.

26 Differenial Equaions and Nonlinear Mechanics Figure 7: The firs-order approximaion u a α, T, γ, ε 5 for differen values of, considering only one erm in he series M. 5 5 Figure : The firs-order approximaion u a α, T, γ, ε 5 for differen values of, considering only one erm in he series M... Exponenial Iniial Condiion Case Sudy Taking he case of f e, f, F,, F,, he following final resuls for he ero-order approximaion are obained in Figures, 9,and.

27 Differenial Equaions and Nonlinear Mechanics 7 The original equaion u, i α u, ε u, u, iγu, F, if, u, ψ, iφ, Coupled non-liner equaions φ, ψ, α ψ, ε ψ φ ψ γφ F, α φ, ε ψ φ φ γψ F, Perurbaion soluion ψ, ψ εψ ε ψ φ, φ εφ ε φ Se of coupled linear equaions φ i, ψ i, α ψ i, G i, i, ψ i, δ i, f, α φ i, G i, i, φ i, δ i, f. The ih order approximaion u i u i ε i ψ i φ i, i, u ψ φ. Figure 9: The general soluion algorihm. One can noice ha a higher soluion level is go compared wih case sudy and less perurbaions are go a small values of..7. Firs-Order Approximaion Case Sudy Taking he case of f, f, F,, F,, he following final resuls for he ero and firs-order approximaions are obained in Figures. One can noice he oscillaions of he soluion level compared wih case 7. One can noice ha he soluion level increases wih he increase of ε.

28 Differenial Equaions and Nonlinear Mechanics 5. Conclusions The perurbaion echnique inroduces an approximae soluion o he NLS equaion wih a perurbaive nonlinear erm for a finie ime inerval. Using mahemaica, he difficul and huge compuaions problems were froned o some exen for limied series erms. To ge more improved orders, i is expeced o face a problem of compuaion. Wih respec o he soluion level, he effec of he nonhomogeneiy is higher han he effec of he iniial condiion. The iniial condiions also cause perurbaions for he soluion a small values of he space variable. The soluion level increases wih he increase of ε. References M. J. Ablowi, B. M. Herbs, and C. M. Schober, The nonlinear Schrödinger equaion: asymmeric perurbaions, raveling waves and chaoic srucures, Mahemaics and Compuers in Simulaion, vol., no., pp., 997. F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, Solion perurbaions and he random Kepler problem, Physica D, vol. 5, no. -, pp. 9,. S. Fewo, J. Aangana, A. Kenfack-Jiosa, and T. C. Kofane, Dispersion-managed solions in he cubic complex Ginburg-Landau equaion as perurbaions of nonlinear Schrodinger equaion, Opics Communicaions, vol. 5, pp. 9, 5. A. Biswas and K. Porseian, Solion perurbaion heory for he modified nonlinear Schrödinger s equaion, Communicaions in Nonlinear Science and Numerical Simulaion, vol., no., pp. 9, 7. 5 T. Caenave and P.-L. Lions, Orbial sabiliy of sanding waves for some nonlinear Schrödinger equaions, Communicaions in Mahemaical Physics, vol. 5, no., pp. 59 5, 9. A. Debussche and L. Di Mena, Numerical simulaion of focusing sochasic nonlinear Schrödinger equaions, Physica D, vol., no. -, pp. 5,. 7 A. Debussche and L. Di Mena, Numerical resoluion of sochasic focusing NLS equaions, Applied Mahemaics Leers, vol. 5, no., pp. 9,. M. Wang, X. Li, and J. Zhang, Various exac soluions of nonlinear Schrödinger equaion wih wo nonlinear erms, Chaos, Solions and Fracals, vol., no., pp. 59, 7. 9 L.-P. Xu and J.-L. Zhang, Exac soluions o wo higher order nonlinear Schrödinger equaions, Chaos, Solions and Fracals, vol., no., pp. 97 9, 7. N. H. Sweilam, Variaional ieraion mehod for solving cubic nonlinear Schrödinger equaion, Journal of Compuaional and Applied Mahemaics, vol. 7, no., pp. 55, 7. S.-D. Zhu, Exac soluions for he high-order dispersive cubic-quinic nonlinear Schrödinger equaion by he exended hyperbolic auxiliary equaion mehod, Chaos, Solions and Fracals, vol., no. 5, pp., 7. J.-Q. Sun, Z.-Q. Ma, W. Hua, and M.-Z. Qin, New conservaion schemes for he nonlinear Schrödinger equaion, Applied Mahemaics and Compuaion, vol. 77, no., pp. 5,. K. Porseian and B. Kalihasan, Cnoidal and soliary wave soluions of he coupled higher order nonlinear Schrödinger equaion in nonlinear opics, Chaos, Solions and Fracals, vol.,no.,pp. 9, 7. H. Sakaguchi and T. Higashiuchi, Two-dimensional dark solion in he nonlinear Schrödinger equaion, Physics Leers A, vol. 59, no., pp. 7 5,. 5 D.-J. Huang, D.-S. Li, and H.-Q. Zhang, Explici and exac ravelling wave soluions for he generalied derivaive Schrödinger equaion, Chaos, Solions and Fracals, vol., no., pp. 5 59, 7. W. K. Abou Salem and C. Sulem, Sochasic acceleraion of solions for he nonlinear Schrödinger equaion, SIAM Journal on Mahemaical Analysis, vol., no., pp. 7 5, 9. 7 M. El-Tawil, The average soluion of a sochasic nonlinear Schrodinger equaion under sochasic complex non-homogeneiy and complex iniial condiions, in Transacions on Compuaional Science III, vol. 5 of Lecure Noes in Compuer Science, pp. 7, Springer, New York, NY, USA, 9. M. Colin, T. Colin, and Moha, Sabiliy of soliary eaves for a sysem of nonlinear Schrodinger equaions wih hree wave ineracions, o appear in Annals de I Insiu Henri Poincare (c Nonlinear Analysis.

29 Differenial Equaions and Nonlinear Mechanics 9 9 Z. Jia-Min and L. Yu-Lu, Some exac soluions of variable coefficien cubic quinic nonlinear Schrodinger equaion wih an exernal poenial, Communicaions in Theoreical Physics, vol. 5, no., p. 9, 9. S. J. Farlow, Parial Differenial Equaions for Scieniss and Engineers, John Wiley & Sons, New York, NY, USA, 9. L. Pipes and L. Harvill, Applied Mahemaics for Engineers and Physiciss, McGraw-Hill, Tokyo, Japan, 97.