Computational Methods for Three Coupled Nonlinear Schrödinger Equations

Transcription

1 Applied Mathematics ISSN Olie: ISSN Prit: Computatioal Methods for Three Coupled Noliear Schrödiger Equatios M. S. Ismail S. H. Alaseri Departmet of Math Kig Abdulaziz Uiversity Jeddah Kigdom of Saudi Arabia How to cite this paper: Ismail M.S. ad Alaseri S.H. (06) Computatioal Methods for Three Coupled Noliear Schrödiger Equatios. Applied Mathematics Received: September 8 06 Accepted: November 06 Published: November 5 06 Copyright 06 by authors ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal icese (CC BY 4.0). Ope Access Abstract I this work we will derive umerical schemes for solvig 3-coupled oliear Schrödiger equatios usig fiite differece method ad time splittig method combied with fiite differece method. The resultig schemes are highly accurate ucoditioally stable. We use the exact sigle solito solutio ad the coserved quatities to check the accuracy ad the efficiecy of the proposed schemes. Also we use these methods to study the iteractio dyamics of two solitos. It is foud that both elastic ad ielastic collisio ca take place uder suitable parametric coditios. We have oticed that the ielastic collisio of sigle solitos occurs i two differet maers: ehacemet or suppressio of the amplitude. Keywords Three Coupled Noliear Schrodiger Equatios Fiite Differece Method Time Splittig Method Iteractio of Solitos. Itroductio I recet years the cocept of solito has bee receivig cosiderable attetio i optical commuicatios sice solito is capable of propagatig over log distaces without chage of shape ad velocity. It has bee foud that the solito propagatig through optical fiber arrays is govered by a set of equatios related to the coupled oliear Schrödiger equatio [] []. N iq jt + q jxx + qp q j = 0 j = N () p= where i = q j is the evelope or the amplitude of the jth wave packets. Equatio () reduces to the stadard oliear Schrödiger equatio for N = to Maakov itegrable system for N = ad recetly for this case the exact two solito solutio DOI: 0.436/am November 5 06

2 obtaied ad ovel shape chagig i elastic collisio property has bee brought out. The system for N = 3 is of physical iterest i optical commuicatio ad i biophysics this system ca be used to study the luchig ad propagatio of solitos alog the three spies of a alpha-helix shape chagig i protei [] [] [3] [4]. I this work we are goig to derive some umerical methods for solvig the three coupled oliear Schrödiger equatios (3-CNS) with iitial coditios ( ) ( ) ( ) iq + q + q + q + q q = 0 () t xx 3 iq + q + q + q + q q = 0 (3) t xx 3 iq + q + q + q + q q = 0 (4) 3t 3xx 3 3 ( ) ( ) q x 0 = g x x x x j = 3 (5) j j R ad the homogeous boudary coditios ( ) ( ) q x t = q x t = 0 j = 3 j j R The exact solutio of the 3-coupled oliear Schrödiger equatio [] [] is give by iλi R qj ( xt ) = Ak j Re sech λr + (6) α 3 j R Aj = = α j e = λ = k ( x + ikt ) j = 3 j= k + k ( j) ( ) where α k j = 3 are four arbitrary complex parameters. Further kra j gives the amplitude of the jth mode ad k I the solito velocity. May umerical methods for solvig the coupled oliear Schrödiger equatio are derived i the last two decades. Fiite differece ad fiite elemet methods are used to solve this system by Ismail [5]-[0]. A coservative compact fiite differece schemes are give i [] []. I [3] [4] Xig ü studied the bright solito collisios with shape chage by itesity for the coupled Sasa-Satsuma system i the optical fiber commuicatios. A higher order expoetial time differecig scheme for system of coupled oliear Schrödiger equatio is give i [3]. A semi-explicit multi-sypmlectic splittig scheme for 3-coupled oliear Schrödiger equatio is give i [4]. May researchers are used time splittig method for solvig the coupled o-liear schrödiger equatio the basic idea i this method is to split the origial system ito a liear subsystem ad oliear subsystem The splittig simplify the problem sice the liear problem is ucouplig relatively easy to solve ad the oliear problem ca be solved exactly due to their poit-wise coservatio law for more details see [5]-[3]. To avoid complex computatios we assume q = u + iu q = u + iu q = u + iu (7) where ( ) u xt i= 6 are real fuctios the systems ()-(4) ca be writte as i

3 where u + u + ωu = 0 u u ωu = 0 (8) t xx t xx u + u + ωu = 0 u u ωu = 0 (9) 3t 4xx 4 4t 3xx 3 u + u + ωu = 0 u u ωu = 0 (0) 5t 6xx 6 6t 5xx 5 ( u ) ( ) ( u u3 u4 u5 u6 ) ω = () The resultig systems (8)-(0) ca be writte i a matrix-vector form as where vj vj + A + Aω v 3 j = 0 j = () t x u u u 0 v = v = v = A = u u 4 u 6 Propositio The three coupled oliear Schöridiger equatios have the co- served quatities xr ( ) ( ) xr = = = x x I q x t d x q x 0 d x (3) xr ( ) ( ) xr = = = x x I q x t d x q x 0 d x (4) xr ( ) ( ) xr 3 = 3 = = x 3 x I q x t d x q x 0 d x (5) Proof : we cosider the first coserved quatity (3) from (8) we have ( ) ( ) u xt u xt + + ωu ( xt ) = 0 t x ( ) ( ) u xt u xt ωu ( xt ) = 0 t x by multiplyig (6) by u( xt ) ad (7) by u ( ) atios to obtai (6) (7) xt ad by addig the resultig equ- ( ) ( ) u xt u xt u ( xt ) + u ( xt ) + u( xt ) u( xt ) = 0. t x x x By itegratig Equatio (8) with respect to x from vaishig boudary coditios to obtai t xr x ( ) ( ) u xt + u xt dx = 0 x to (8) x R ad usig the ad this is the proof of the first coserved quatity (3). The other two coserved quatities (4) ad (5) ca be proved i the same way. The exact values of the coserved quatities usig the exact solito solutio (6) are give by the followig formula I j = kr α j j 3. = α + α + α 3 (9)

4 The paper is orgaized as follows. I Sectio we derived a secod order Crak- Nicolso scheme for solvig the proposed system. The fourth order compact differece scheme is derived i Sectio 3. I Sectio 4 we preset two fixed poit schemes to solve the block oliear tridiagoal systems obtaied i Sectios ad 3. To avoid the oliearity obtaied i the previous sectios we preset time splittig method to solve the 3-CNS i Sectio 5. The umerical compariso of the derived methods are reported i Sectio 6. Fially we draw some coclusios i Sectio 7.. Secod Order Crak-Nicolso Scheme I order to develop a umerical method for solvig the system give i () the regio R = [ x < x< xr] [ t > 0] will be covered with a rectagular mesh of poits with coordiates x = x + mh m = 0 M m t = t = k = 0 where h ad k are the space ad time icremets respectively. We deote the exact ad + umerical solutio at the grid poit ( xm t ) by v jm ad V jm respectively. We approximate the space derivative by the secod order cetral differece formula ( t) u xm = ( u( x m t) u( xm t) u( xm t) ) xu( xm t) O( h ) x h h δ + + = + where δ x is the secod order cetral differece operator. The Crak-Nicolso scheme for the 3-CNS equatio is give by V + jm V jm + raδ + x jm + jm + kw + m A( + jm + jm ) = V V 4 V V 0 (0) where k wm = ( qjm + qjm ) r = j = 3. h j= The scheme i (0) is of secod order accuracy i both directios space ad time ad it is ucoditioally stable usig vo-neuma stability aalysis see Ismail [8]. A oliear block tridiagoal system must be solved at each time step. Fixed poit method is used to do this job ad this will be discussed later. 3. Fourth Order Compact Differece Scheme A highly accurate fiite differece scheme ca be obtaied by usig the fourth order Padè compact differece approximatio for the spatial discretizatio ( t) u xm 4 = + δ x δxu xm t + O h ( ) ( ) x h together with the Crak-Nicolso scheme this will lead us to the compact fiite differece scheme jm jm ra δx δ x jm jm kwm A jm jm ( ) V V = V V V V 0 () 3

5 where ( ) 3 + m = + jm + jm j= w q q The scheme () ca be writte i a block tridiagoal form as ( Vjm Vjm ) + ( Vjm Vjm ) + ( Vjm + Vjm + ) ra jm jm jm ra jm jm jm V V + V V V + V k jm + 0 jm + jm + = G G G 0 where G + jm = ω + m A( V + jm + V jm ) j = 3. 4 The method is of secod order accuracy i time ad fourth order i space it is implicit ucoditioally stable see Ismail [9]. The resultig system is a block oliear tridiagoal system ad ca be solved by fixed poit method ad this will be discussed ext. From the previous methods we ca derive the geeralized fiite differece scheme ( Vjm Vjm ) + ( )( Vjm Vjm ) + ( Vjm + Vjm + ) ra ( Vjm Vjm ) ( Vjm Vjm ) ( Vjm + Vjm + ) σ σ σ k σgjm + ( σ) Gjm + σg jm + = 0 where Gjm = ω + m A( V + jm + V jm ) j = 3 4 for arbitrary parameter σ. The scheme is secod order i time ad space for σ. It is very easy to see that the previous methods ca be recovered by selectig σ = 0 ad σ = respectively. The resultig system is agai a block oliear tridiagoal + system which ca be solved for the ukow vector V jm by ay iterative solver the fixed poit method is adopted i this work. 4. Fixed Poit Method Sice the geeralized compact fiite differece scheme (3) is oliear ad implicit a iterative method is eeded to solve it. Two iterative algorithms are implemeted to perform this job []. Algorithm + ( s+ ) + ( s+ ) + ( s+ ) + ( s+ ) + ( s+ ) + ( s+ ) σvjm + ( σ) Vjm + σvjm + + ra Vjm Vjm + V jm+ = σvjm + ( σ) Vjm + σvjm + + ra jm jm jm V V + V + (4) ( s) ( s) ( s) + k σgjm + ( σ) Gjm + σg jm +. () (3) 4

6 where 3 ( s) + ( s) + ( s) Gjm = qjm q + jm A Vjm + V jm j = 3 4 j= where the superscript s deotes the sth iterate for solvig the oliear system of equatios for each iteratio. The system i (4) ca be solved by Crout s method where we eed oly oe U factorizatio for the block tridiagoal matrix at the begiig ad the solutios of lower ad upper bi-diagoal block systems at each iteratio are required oly. Algorithm σv ( σ) V σv V V V = σvjm + ( σ) Vjm + σvjm + + ra jm jm jm V V + V + ( s) ( s) ( s) + k σgjm + ( σ) Gjm + σg jm + + ( s+ ) + ( s+ ) + ( s+ ) + ( s+ ) + ( s+ ) + ( s+ ) jm + jm + jm + + ra jm jm + jm + (5) where G = q q + A V + V j = 3 3 ( s) + ( s) + ( s+ ) jm jm jm jm jm 4 j= where the superscript s deotes the sth iterate for solvig the oliear system of equatios for each time. The block tridiagoal system (5) ca be solved by Crout's method. Note that i this method we eed to do factorizatio at each iteratio. The iitial iterate + ( 0) m till the followig coditio U ca be chose as + ( 0) m V = V. We apply the iterative schemes ( s + ) ( s) 6 m + m 0 V V is satisfied. The covergece of the iterative schemes Algorithm ad Algorithm is give i []. 5. Time Splittig Method I this work we are goig to preset the time splittig method for solvig the 3-coupled oliear Schrödiger Equatio (). The basic idea i the time splittig method is to decompose the origial problem ito subproblems which are simpler tha the origial problem ad the to compose the approximate solutio of the origial problem by usig the exact or approximate solutios of the subproblems i a give sequetial order. To display this method for our system with iitial coditios m ( ) ( ) ( ) iq + q + q + q + q q = 0 t xx 3 iq + q + q + q + q q = 0 t xx 3 iq + q + q + q + q q = 0 3t 3xx 3 3 (6) 5

7 ( ) ( ) q x 0 = g x x x x j = 3 j j R ad the homogeous boudary coditios ( ) ( ) ( ) q xt = q xt = q xt = 0 at x= x x 3 R The system i (6) ca be writte as where q q q3 = ( + N) q = ( + N) q = ( + N) q t t t 3 ( ) 3 i = ad N = i q + q + q. x (7) (8) We solve the system (7) from t = t to t = t + i two splittig steps. We solve first the liear system equatio q j t = iq j = 3 with the homogeous Dirichlet boudary coditios jxx ( ) q xt = 0 at x= x x j R usig the fiite differece method for the time step k followed by solvig the oliear system q j t ( ) = i q + q + q3 qj j = 3 for the same time step. Equatio (30) ca be itegrated exactly i time [5] the exact solutio is ( ) ( ) = ( ( ) ( ) ( ) )( ) q xt exp i q xt q xt q xt t t q j j t t x x x j = 3 R To apply this method i systematic way we combie the splittig steps via the stadard secod order Strag splittig []. The flowchart of this method ca be described by the followig steps. Step : Solutio of the oliear problem ( ) k qjm xt = exp i q m + q m + q3 m qjm ( ) ( ) ( ) ( ) j jm q 0 = q = 0 m= M j = 3. Step : Solutio of the liear problem The solutio of the liear ca be obtaied usig the geeralized differece scheme ( ) ( ) ( ) σi + ra q jm + ( ) σ I ra q jm+ σi + ra q jm+ (3) ( ) ( ) ( ) = σi ra q jm + ( σ) I + ra q jm + σi ra q jm + (9) (30) (3) 6

8 ( ) ( ) j jm q 0 = q = 0 m= M j = 3. The solutio of this system ca be obtaied by solvig liear block tridiagoal system with costat coefficiets usig Crouts method ad this ca be executed i very efficiet way. Step 3: Solutio of the oliear problem ( ) ( ) ( ) k ( ) q + jm ( xt ) exp i q m q m q = m qjm q q m M j + + j0 = jm = 0 = = 3. The umerical scheme is of secod order accuracy i time secod ad fourth order i space for σ = 0 ad σ = respectively. It is ucoditioally stable ad co- served the coserved quatities exactly [5] [0] [3]. We deote this method by the time splittig fiite differece method by (TSFDM). 6. Numerical Results I this sectio we coduct some typical umerical examples to verify the accuracy coservatio laws computatioal efficiecy ad some physical iteractio pheomea described by 3-coupled oliear Schrödiger equatios. 6.. Sigle Solito (33) I this test we choose the iitial coditio as R q j ( x0) = AjkR exp ( ikix) sech krx + α 3 j R Aj = e = = α 3. j j = + j= ( k k ) The followig set of parameters are used h = 0. k = 0.0 x = 30 t = l (34) α = + i α = i α = ik = + 0.5i 3 The error ad the coserved quatities as well as the executio time for all methods are give i Tables -6 we have oticed that all method are coserved the coserved quatities exactly ad for accuracy the credit goes to the fourth order scheme σ =. The profile of q q ad q 3 at differet times are displayed i Figures -3. respectively. To test the coverget rate i space ad time of the proposed schemes. We defie the error orm by ( ) = E = h max u U m m m m M where u m ad U m are respectively the exact ad the umerical solutio at the grid 7

9 Table. Secod order scheme (σ = 0) (Algorithm ) cpu =.8 sec. T l( q ) l( q ) l3( q 3) Table. Fourth order scheme (σ = /) (Algorithm ) cpu =.7 sec. T l( q ) l( q ) l3( q 3) Table 3. Secod order scheme (σ = 0) (Algorithm ) cpu = 3.3 sec. T l( q ) l( q ) l3( q 3) Table 4. Fourth order scheme (σ = /) (Algorithm ) cpu = 3.4 sec. T l( q ) l( q ) l3( q 3)

10 Table 5. TSFDM (σ = 0) cpu =.5 sec. T 0.0 l( q ).0580 l( q ) l3( q 3) Table 6. TSFDM (σ = /) cpu =.4 sec. T l( q ) l( q ) l3( q 3) poit ( ) Figure. Sigle solito: q. x t. I this experimet we take T =. The coverget rate order is m calculated by the formula ( h ) ( h ) l order ( rate of coverget i space) = h l h 9

11 Figure. Sigle solito: q. Figure 3. Sigle solito: q3. ( k ) ( k ) l order ( rate of coverget i time) = k l k To calculate the coverget rate i space we take the time step k sufficietly small ad the error from temporal trucatio is relatively small k 4 = 5 0. From Table 7 0

12 Table 7. Spatial order of coverget with k = at T =. h ( Re) Rate ( Im) Rate we ca easily that the rate of coverget is 4 as we claim i this work. To check the temporal coverget rate we fix the spatial step h small eough so that the trucatio from space is egligible such as h = 0.0. The results are give i Table 8 which idicate that the order is as we claim i the text. 6.. Iteractio of Two Solitos To study the iteractio of two solitos with differet parameters we choose the iitial coditio as a sum of two sigle solitos of the form where ( ) ( ) ( ) j j j ( ) ( ) q xt = 0 = q x0 + q x0 j= 3 (35) ( ) ( ) R qj ( x0) = Aj kr exp ( iki ( x+ x0) ) sech kr( x+ x0) + ( ) ( ) R qj ( x0) = Aj kr exp ( iki ( x x0) ) sech kr( x x0) +. For all examples i the case of iteractio we choose the set of parameters h = 0. k = 0.0 x = 50 x = 50 x = 5 together with differet values of dyamics of the followig cases k = ik = 0.4 i ( j) ( j) j { 3 } R 0 α α = for each test. We will study the Test : Two Solitos (with Pure Imagiary Parameters) I this test we will cosider the two set of parameters (equal ad differet ) ( ) ( ) ( ) ( ) ( 3) ( 3) α = α = i α = α = i α = α = i (36) ad ( ) ( ) ( 3) ( ) ( ) ( 3) α = i α = i α = i α = i α = i α = i (37) For the first set of parameters (36) we have oticed that the amplitudes of solito ad solito before the iteractio are S = [ ] ad S = [ ] remai uchaged after the iteractio which meas the iteractio is elastic ad this sceario is displayed i Figures 4-6. For the secod set of parameters (37) we have oticed that the amplitudes of solito ad solito before the iteractio are S = [ ] ad

13 Table 8. Temporal order of coverget with h = 0.0 at T =. h ( Re) Rate ( Im) Rate Figure 4. Elastic iteractio with equal pure imagiary parameters q. Figure 5. Elastic iteractio with equal pure imagiary parameters q.

14 Figure 6. Elastic iteractio with equal pure imagiary parameters q3. S = [ ] chage to [ ] [ ] S = ad S = after the iteractio the collisio mechaism ca be described as follows ( Suppressed ) ( Ehaced ) ( Ehaced ) ( ) ( ) ( ) S q q q 3 S q Ehaced q Suppressed q3 Suppressed ad i this case we have ielastic collisio ad we display this sceario i Figures Test : (Pure Real Parameters) I this test we choose two differet test of parameters ( ) ( ) ( 3) ( ) ( ) ( 3) α = 5 α = 3.5 α = α = 0 α = 7 α = (38) ad ( ) ( ) ( 3) ( ) ( ) ( 3) α = 5 α = 3.5 α = α = α =.5 α = 0.4 (39) For the first set of parameters (38) we have oticed that the amplitudes of solito ad solito before the iteractio are [ ] [ ] S = ad S = remai uchaged after the iteractio which meas the iteractio is elastic ad this sceario is displayed i Figures 0-. For the secod set of parameters (39) we have oticed that the amplitudes of solito ad solito before the iteractio are [ ] S = [ ] chage to [ ] [ ] S = ad S = ad S = after the iteractio this meas the collisio mechaism is ielastic ad ca be give as ( ) ( ) ( ) S q Suppressed q Ehaced q3 Suppressed 3

15 Figure 7. Ielastic iteractio with differet pure imagiary parameters q. Figure 8. Ielastic iteractio with differet pure imagiary parameters q. Figure 9. Ielastic iteractio with differet pure imagiary parameters q3. 4

16 Figure 0. Elastic iteractio with pure real parameters q. Figure. Elastic iteractio with pure real parameters q. Figure. Elastic iteractio with pure real parameters q3. 5

17 ( ) ( ) ( ) S q Ehaced q Suppressed q3 Ehaced we display the iteractio sceario i Figures Test 3: Solito Iteractio (Nozero Real ad Imagiary Parts) I this test we choose the two sets of parameters ( ) ( ) ( 3) = + i = + i = + i ( ) ( ) ( 3) α α α α = i α = i α = i (40) ad Figure 3. Ielastic iteractio with pure real di eret parameters q. Figure 4. Ielastic iteractio with pure real differet parameters q. 6

18 Figure 5. Ielastic iteractio with pure real differet parameters q3. ( ) ( ) ( 3) ( ) ( ) ( 3) α = + i α = + i α = + i α = 3 + i α = + i α = i (4) For the first set of parameters (40) we have oticed that the amplitudes of solito ad solito before the iteractio are [ ] [ ] S = ad S = remai uchaged after the iteractio which meas the iteractio is elastic ad this sceario is displayed i Figures 6-8. For the secod set of parameters we have oticed that the amplitudes of solito ad solito before the iteractio are [ ] S = [ ] chage to [ ] [ ] S = ad S = ad S = after the iteractio the collisio mechaism ca be give as follows ( ) ( ) S q q Suppressed q3 Ehaced ( ) ( ) S q q Ehaced q3 Suppressed ad i this case we have ielastic collisio ad we display this sceario i Figures Coclusio I this work we have derived differet methods for solvig the 3-coupled oliear Schrödiger equatio usig fiite differece method ad time splittig method with fiite differece methods. All schemes are ucoditioally stable ad highly accurate ad coserve the coserved quatities exactly. The iteractio of two solitos is discussed i details for differet parameters. We have oticed that to have elastic iteractio the followig costrait 7

19 Figure 6. Elastic iteractio with o zero real ad imagiary parameters for q. Figure 7. Elastic iteractio with o zero real ad imagiary parameters for q. Figure 8. Elastic iteractio with o zero real ad imagiary parameters for q3. 8

20 Figure 9. Ielastic iteractio with o zero real ad imagiary differet parameters for q. Figure 0. Ielastic iteractio with o zero real ad imagiary differet parameters for q. Figure. Ielastic iteractio with o zero real ad imagiary differet parameters for q3. 9

21 ( ) ( ) ( 3) = = ( ) ( ) ( 3) α α α α α α must be satisfied ad for other values the iteractio is ielastic ad differet behaviors occur (ehacemetsuppressio) i the amplitude of each solito. This behavior is i agreemet with [] [] [3] [4]. The derived methods ca be used to solve similar oliear problems. Refereces [] Kaa T. ad akshmaa M. (003) Exact Solito Solutios of Coupled Noliear Schrödiger Equatio: Shape-Chagig Collisios ogic Gates ad Partially Coheret Solito. Physical Review E 67 Article ID: [] Kaa T. ad akshmaa M. (00) Exact Solito Solutios Shape Chagig Collisios ad Partially Coheret Solitos i Coupled Noliear Schrödiger Equatios. Physical Review etter [3] ü X. ad i F. (06) Solito Excitatios ad Shape-Chagig Collisios i Alpha Helical Proties with Iterspie Couplig at Higher Order. Commuicatios i Noliear Sciece ad Numerical Simulatio [4] ü X. (04) Bright-Solito Collisios with Shape Chage by Itesity Redistributio for the Coupled Sasa-Satsuma System i the Optical Fiber Commuicatios. Commuicatios i Noliear Sciece ad Numerical Simulatio [5] Ismail M.S. (008) Numerical Solutio of Coupled Noliear Schrödiger Equatio by Galerki Method. Mathematics ad Computers i Simulatio [6] Ismail M.S. Al-Basyoui K.S. ad Aydi A. (05) Coservative Fiite Differece Schemes for the Chiral Noliear Schrödiger Equatio. Boudary Value Problem [7] Ismail M.S. (008) A Fourth Order Explicit Schemes for the Coupled Noliear Schrödiger Equatio. Applied Mathematics ad Computatio [8] Ismail M.S. ad Taha T.R. (00) Numerical Simulatio of Coupled Noliear Schrödiger Equatio. Mathematics ad Computers i Simulatio [9] Ismail M.S. ad Alamri S.Z. (004) Highly Accurate Fiite Differece Method for Coupled Noliear Schrödiger Equatio. Iteratioal Joural of Computer Mathematics [0] Ismail M.S. ad Taha T.R. (007) A iearly Implicit Coservative Scheme for the Coupled Noliear Schrödiger Equatio. Mathematics ad Computers i Simulatio [] Hu X. ad Zhag. (04) Coservative Compact Differece Schemes for the Coupled Noliear Schrödiger System. Numerical Methods for Partial Differetial Equatios [] Kog. Ji. ad Zhu P. (05) Compact ad Efficiet Coservative Schemes for Coupled Noliear Schrödiger Equatios. Numerical Methods for Partial Differetial Eq- 30

22 uatios [3] Bhatt H.P. Abdul Q. ad Khaliq M. (04) Higher Order Expoetial Time Differecig Scheme for System of Coupled Noliear Schrödiger Equatios. Applied Mathematics ad Computatio [4] Xu Q. Sog S. ad Che Y. (04) A Semi-Explicit Multi-Symplectic for a 3-Coupled Noliear Schrödiger Equatios. Computer Physics Commuicatio [5] Atoie X. Bao W. ad Besse C. (03) Computatioal Methods for the Dyamics of the Noliear Schrödiger/Gross-Pitaevskii Equatios. Computer Physics Commuicatios [6] Aydi A. ad Karasoze B. (007) Symplectic ad Multisymplectic Methods for Coupled Noliear Schrödiger Equatios with Periodic Solutios. Computer Physics Commuicatios [7] Bao W. Tag Q. ad Xu Z. (03) Numerical Methods ad Compariso for Computig Dark ad Bright Solitos i the Noliear Schrödiger Equatio. Joural of Computatioal Physics [8] Che A. Zhu H. ad Sog S. (00) Multi-Symplectic Splittig Method for the Coupled Noliear Schrödiger Equatio. Computer Physics Commuicatios [9] Ma Y. Kog. ad Cao Y. (0) High Order Compact Splittig Multi-Symplectic Method for the Coupled Nolier Schrödiger Equatios. Computers ad Mathematics with Applicatios [0] Talleei A. ad Dehgha M. (04) Time-Splittig Pseudo-Spectral Domai Decompositio Method for the Solito Solutiots of the Oe-ad Multidimesioal Noliear Schrödiger Equatio. Computer Physics Commuicatios [] Wag S. ad Zhag. (0) Split-Step Orthogoal Splie Collocatio Methods for Noliear Schrödiger Equatios i Oe Two ad Three Dimesios. Applied Mathematics ad Computatio [] Wag T. Guo B. ad Zhag. (00) New Coservative Differece Schemes for a Coupled Noliear Schrödiger System. Applied Mathematics ad Computatio [3] Dehgha M. ad Taleei A. (00) A Compact Split Step Fiite Differece Method for Solvig the Noliear Schrödiger Equatios with Costat ad Variable Coefficiets. Computer Physics Commuicatios

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