BRIGHT AND DARK SOLITONS OF THE MODIFIED COMPLEX GINZBURG LANDAU EQUATION WITH PARABOLIC AND DUAL-POWER LAW NONLINEARITY

Transcription

1 Romanian Reorts in Physics, Vol. 6, No., P , 0 BRIGHT AND DARK SOLITONS OF THE MODIFIED COMPLEX GINZBURG LANDAU EQUATION WITH PARABOLIC AND DUAL-POWER LAW NONLINEARITY HOURIA TRIKI, SIHON CRUTCHER, AHMET YILDIRIM 3, T. HAYAT 5,6, OMAR M. ALDOSSARY 6, ANJAN BISWAS 7 Badji Mokhtar University, Deartment of Physics, 300 Anaba, Algeria US Army Research, Develoment and Engineering Command, Building 500, Redstone Arsenal, AL , USA 3 Ege University, Deartment of Mathematics, 3500 Bornova, Izmir, TURKEY, ahmetyildirim80@gmail.com University of South Florida, Deartment of Mathematics and Statistics Tama, FL , USA 5 Qaid-i-Azam University, Deartment of Mathematics, Islamabad-000, Pakistan 6 King Saud University, Deartment of Physics, P. O. Box 55, Riyadh-5, Saudi Arabıa 7 Delaware State University, Deartment of Mathematical Sciences, Dover, DE , USA, biswas.anjan@gmail.com Received November 9, 0 Abstract. This aer obtains the exact bright and dark soliton solutions of the comlex Ginzburg-Landau equation with arabolic and dual-ower law nonlinearities, governing the roagation of solitons through nonlinear otical fibers. The solitary wave ansatz is used to carry out the integration of the considered models. Parametric conditions for the existence of the exact solutions are given. Key words: exact bright and dark soliton solutions, comlex Ginzburg-Landau equation.. INTRODUCTION The study of the modified comlex Ginzburg-Landau (mcgl) equation has been going on for the ast few years [ 0]. There has been quite a bit of focus on the study of this equation with Kerr and ower laws of nonlinearity. One of the most imortant asects of this equation is in its integrability. Previously some authors have addressed the integrability of this mcgl equation in Kerr and ower law nonlinearity. However, in those aers the semi-inverse variational rincile was alied to carry out the integration of this equation [].

2 368 Houria Triki et al. Several methods of integrating such tyes of equations have been develoed ' in the ast few years. They are the G / G method, ex-function method, Riccati equation method, simlest equation method, Fan's F-exansion method, Lie symmetry analysis, collective variable aroach, and many others. In this aer, there will be one such method that will be addressed to solve the mcgl equation, that is known as the ansatz method. It is only the arabolic law as well as the dualower laws of nonlinearity that will be addressed.. GOVERNING EQUATION The dimensionless form of the mcgl equation that will be studied in this aer is given by [] { } qx β i qt aqxx bf q q= α q q q. () q q q xx x function Equation () is the mcgl equation with nonlinearity given by the function F. This equation for Kerr law nonlinearity was studied using the Hirota bilinear method. The modulational stability of this equation was also addressed []. In () the deendent variable qxt (, ) reresents the wave rofile that arises in various hysical systems, including nonlinear otics, lasma hysics and others. The indeendent variables are x and t that reresents the satial and temoral variables. The real valued constants are a, b, α and β where a and b reresents the coefficients of disersion and nonlinearity. The function F is a real-valued algebraic function and it is necessary to have the smoothness of the comlex F q q: C C. Considering the comlex lane C as a twodimensional linear sace differentiable, so that [] R, the function ( ) k ( ) mn, = F q q is k times continuously F q q C ( n, n) ( m, m); R. () In this aer the two forms of nonlinearity of the function F that will be considered are the arabolic law and the dual-ower law. The next two subsections are going to focus on the ansatz method to extract the -soliton solutions for these two tyes of nonlinear media.

3 3 Modified comlex Ginzburg Landau equation 369 For arabolic law medium, rewritten as [] 3. PARABOLIC LAW F ( s) = bs cs so that equation () can be { } qx β i qt aqxx b q c q q= α q q q, (3) q q q xx x where c is another real-valued constant. Parabolic law nonlinearity arises in the context of lasma hysics to study the nonlinear interaction between Langmuir waves and electrons. This equation () will be studied in this section and the bright and dark -soliton solution will be obtained in this context. This study will now be slit into the following two subsections where the bright and dark solitons will be studied searately. 3.. BRIGHT SOLITONS In order to obtain the bright soliton solution to (3), the starting hyothesis is taken to be [] where and the hase is given by A qxt (, )= e [ D coshτ ] iϕ, () τ= B ( x vt) (5) ϕ( xt, )= κ xω tθ. (6) Here, in ( 6), A is the amlitude of the soliton, while v is the velocity and B is the inverse width of the soliton. Also, κ is the frequency of the soliton, while ω is the wave number of the soliton and θ is the hase constant. The exonent is unknown at this oint and its value will fall out in the rocess of deriving the solution of this equation. Also D is a constant whose value in terms of the known arameters will be determined during the course of derivation of the soliton solution. Thus from ansatz (), BvAsinh τ iωa iϕ qt = e, [ D cosh τ ] [ D cosh τ] (7)

4 370 Houria Triki et al. q B A D D B A [ ] A B xx = [ D cosh τ ] [ D cosh τ ] D cosh τ iκbasinhτ Aκ iϕ e, [ D cosh τ ] [ D cosh τ] q A B x A B D A B D κ iϕ = e, q [ D cosh τ ] [ D cosh τ ] [ D cosh τ] ( ) ( ) q ( ) AB D AB D xx A B iϕ = e q [ D cosh τ ] [ D cosh τ ] [ D cosh τ] { } ( x 8 AB D ) A B iϕ [ cosh τ ] [ cosh τ ] [ cosh τ] q AB D = e. () q q D D D Substituting (7 ) into () yields ω A aaκ aa B α A B κ βa B [ D cosh τ] [ D cosh τ] α A B D β a AB D ibvasinhτ aiκbasinhτ ( ) ( ) α ( ) β ( ) ( ) 3 5 [ D cosh τ ] [ D cosh τ] [ D cosh τ] 3 5 =0. a B A D A B D AB D ba ca Now, searating the real and imaginary arts in Eq. (), we get the following air of relations: and ( a ) [ D cosh τ] BA v κ sinh τ =0 (8) (9) (0) () (3)

5 5 Modified comlex Ginzburg Landau equation 37 [ D cosh τ] [ D cosh τ] ω A aaκ aa B α A B κ βa B α ABD β a ABD ( ) ( ) α ( ) β ( ) ( ) a B A D A B D AB D ba 3 5 ca [ D cosh τ ] [ D cosh τ] [ D cosh τ] 3 5 From (3), the velocity of the soliton is given by v= κ a. (5) From the real art equation (), equating the exonents and 5 gives =5 (6) which imlies =. (7) The same value of is also obtained when the exonents and 3 are equated against each other. Now, from (), the linearly indeendent functions are [ D ] j / cosh τ for j = 0,,. Therefore, setting their resective coefficients to zero yields =0. =0, () ω A aaκ aa B α A B κ β A B (8) 3 α ABD β a ABD ba =0, (9) ( ) ( ) 5 aβ B A D αa B D ca =0. (0) Solving the above equations gives ( a ) B ( a ) αβ α κ ω =, () b c A= B, α ( βa) B 3( a β) α ()

6 37 Houria Triki et al. 6 b b c D =. ( α βa) B [ α βa] B 3( aβ) α As seen from () and (3), the soliton ulse will exist for ( a) ( a ) (3) α β 0; 3 β α 0. () Thus, the bright soliton solution to the mcgl equation with arabolic law nonlinearity (3) is given by A i( κ xωγθ t t ) qxt (, )= e, D cosh B xvt (5) where the velocity v is given by (5), the wave number ω is shown in (), the amlitude A of the soliton is given by () while the arameter D is given by (3). Note that this solution exists rovided that the constraint equations between the model coefficients α, β and a given in () are satisfied. 3.. DARK SOLITONS Now we are interested by finding the exact dark soliton solution for the considered mcgl equation with arabolic law nonlinearity (3). Dark solitons are also known as toological otical solitons in the context of Theoretical Physics. The starting ansatz is i ( A ) ϕ qxt (, )= tanhτ e, (6) where τ and ϕ are the same as in (5) and (6). Here in (6), A, and B are unknown free arameters and v is the velocity of the wave. In this case, the unknown exonent will also fall out during the course of the derivation of the soliton solution to (3). From the ansatz (6), we get {[ ] [ ] [ ] } iϕ [ A ] Bv qt = A tanh τ A Atanh τ A Atanh τ ω i tanhτ e, (7)

7 7 Modified comlex Ginzburg Landau equation 373 B qxx = {( )( A ) [ Atanhτ] ( )[ Atanhτ] A( )[ Atanhτ ] A( )( A )[ Atanhτ] ( 3A )[ Atanh τ] } κ [ Atanh τ] iκb {( A )[ A ] A[ A ] [ A ] } qx B q 3 tanh τ tanh τ tanh τ iϕ tanh τ tanh τ tanh τ e, = {( A ) [ A tanh ] [ A tanh ] τ τ ( A )[ A ] A( A )[ A ] A[ A ] } iϕ [ A tanh ] e, κ τ ( q ) xx B = tanh tanh q 3A Atanhτ A A Atanhτ {( )( A ) [ A τ] ( )[ A τ] [ ] [ ] i [ ] ϕ } A Atanhτ e, {( q ) } x B = tanh tanh q q 3 tanhτ tanhτ {( A ) [ A τ ] [ A τ] ( A )[ A ] A( A )[ A ] i A[ A ] ϕ } tanhτ e. Substituting (7 3) into (3), and then searating the real and imaginary arts, we can obtain the following air of equations: { ab ( ) α B βb ( ) β B }( A ) [ A tanh τ] { ab A( ) α B A βb A( ) β B A}( A ) [ A tanh τ ] ab ( ) α B β B ( ) β B [ tanh ] A τ α B A β B A( ) β B A ab A( ) [ A tanh τ] { aαββb } B ( 3A ) aα κ ω A tanh τ 3 5 b Atanh τ c Atanh τ = 0 [ ] [ ] [ ] (8) (9) (30) (3) (3)

8 37 Houria Triki et al. 8 and ( a ) ( A τ A A τ A )( A τ ) B v κ tanh tanh tanh = 0. (33) From the imaginary art equation (33), the soliton velocity is the same as in the case of bright solitons that is given by (5). From the real art equation (3), equating the exonents of [ tanh ] 5 A B τ and [ A Btanh τ ] terms in Eq. (36), one again obtains the same value of as in (7). j Now, from (3), the linearly indeendent functions are [ A Btanh τ ] for j =0, ±, ±. Therefore, setting their resective coefficients to zero yields the following system of algebraic equations: { ab ( ) B B ( ) B }( A ) α β β =0, (3) { ab A B A B A B A}( A ) α β β =0 (35) ( ) ab α B β B β B c =0, (36) α B A β B A( ) β B A ab A( ) b =0, (37) { aα ββb } B ( A ) 3 ( a α) κ ω=0. (38) The dark soliton solutions study will now be slit into the following two cases: Case I: A 0. In this case, a α B =, (39) β α β B =, (0) β ( a B )( 3A ) B ( a ) α ββ α κ ω =, b A =. 5a 7β B c () ()

9 9 Modified comlex Ginzburg Landau equation 375 Equating the two values of B from (39) and (0) shows that dark solitons will exist rovided that β = a and β ( a α ) <0. Furthermore, free arameter A can be obtained in terms of the model coefficients by substituting (39) or (0) into () to yield bβ A =, cβ a β aα 5 7 (3) which exists rovided that c 5a 7 a 0. β β α () Case II: A = 0. Here, (36 38) yields A = ±, (5) b A =, 5a 7β B c ( a B ) B ( a ) α ββ α κ ω =. (7) Eq. (6) shows that the soliton will exist for cβ 5a7 β aα. (8) Hence, finally, the dark soliton solution to the mcgl equation with arabolic law nonlinearity (3), for the two cases together, is given by i( κ xωγ t tθ) qxt (, )= Atanh B xvt e (9) where the velocity v is given by (5) and the wave number ω is given by () if A ± and by (7) when A = ±. It is worth noting that the existence of the dark soliton solution (9) deends on the characteristics of the nonlinear medium, which satisfy the conditions given above. (6). DUAL-POWER LAW NONLINEARITY In this subsection, the mcgl equation with dual ower law nonlinearity will be studied. In this case, the governing equation is given by []

10 376 Houria Triki et al. 0 { } n n qx β iqt aqxx b q c q q = α q q q. q q q xx x (50) Notice that Eq. (50) reduces to the mcgl equation with arabolic law nonlinearity (3) if = n... BRIGHT SOLITONS For searching bright otical soliton solution, we use the same starting hyothesis as in (). This leads to the same imaginary art equation as given by (3) and thus the velocity is still as in (5). The real art equation reveals [ D cosh τ] A B D ( a) ( ) AB D [ D cosh τ] ( ) ( ) α ( ) β ( ) ( ) [ D cosh τ] ω A aaκ aa B α A B κ βa B α β a B A D A B D AB D n n ba [ ] ( n ) [ ] ( n D cosh τ D cosh τ ) ca =0. Now, from (5), matching the exonents ( n ) and gives which imlies (5) (n )= (5) = n (53) for n 0. (5) The same value of is obtained on equating the exonents ( n ) and. Similarly, from (5) as in the arabolic law case, the linearly indeendent functions j are / [ D cosh τ ] for j = 0,,. Thus, setting their resective coefficients to zero yields the following arametric equations:

11 Modified comlex Ginzburg Landau equation 377 ( ) ( ) ω A aaκ aa B α A B κ β A B =0, (55) α β (56) n ABD a ABD ba =0, n =0. aβ B A D αa B D ca (57) Solving the above equations gives ( a ) B n ( a ) αβ α κ ω =, (58) n nb b n c D =,(59) α ( β a)( n ) B α ( n )( βa) B ( aβ )( n ) α n bn nc A= Bn. α ( n )( βa) B ( aβ )( n ) α As seen from (59) and (60), the soliton ulse will exist for ( n )( a) ( a )( n ) (60) α β 0, & β α 0. (6) Finally, the bright soliton of the comlex modified mcgl equation with dual ower law nonlinearity (50) is A i( κ xωγθ t t ) qxt (, )= e, (6) D cosh B n ( xvt) where the velocity v is given by (5), the wave number ω is shown in (58), the arameter D is given by (59) while the amlitude A of the soliton is given by (60). Note that this solution exists rovided that the constraint equations between the model coefficients α, β and a given in (6) are satisfied... DARK SOLITONS For dark solitons, we use the same solitary wave ansatz as in (6). This leads to the same imaginary art equation as in (33) and thus the velocity of the soliton is the same as given by (5). Now, the real art equation is

12 378 Houria Triki et al. { ab ( ) α B βb ( ) β B }( A ) [ A tanh τ] { ab A( ) α B AβB A( ) β B A}( A ) [ A tanh τ ] ab ( ) α B β B ( ) β B [ A tanh τ] α B A β B A( ) β B A ab A( ) tanh A τ { aαββb } B ( 3A ) ( aα) κ ω [ A tanh ] τ n n b Atanh τ c Atanh τ = 0. [ ] [ ] [ ] By equating the exonents of ( n A Btanh τ ) and ( A Btanh ) (63) τ terms in Eq. (63), one obtains the same value of as in (53) along with the same restriction as (5). By collecting the coefficients of the linearly indeendent functions ( A Btanh ) zero, yields j τ for j =0, ±, ± in Eq. (63), and setting them to { ab ( ) B B ( ) B }( A ) α β β =0, (6) { ab A( ) B A B A( ) B A}( A ) α β β =0, (65) ( ) ab α B β B β B c =0, α B A β B A β B A ab A { aα ββb } B ( A ) 3 As in arabolic law, the two cases are ( a ) α κ ω b =0, =0. (66) (67) (68) Case I: A 0. Here, ( β ) aαβ n a B =, β (69)

13 3 Modified comlex Ginzburg Landau equation 379 ( β ) aαβ n a B =, β aαββb 3A B aα n κ ω =, A nb =. ab 3 n β B 3 n c n (70) (7) (7) Equating the two values of B from (69) and (70) shows that dark solitons will exist rovided that β = a and β aα β n βa >0. (73) Additionally, we can determine the arameter A of the soliton in terms of the model coefficients by substituting (69) or (70) into (7) for β = a which gives nb A =. a( 3n ) β ( 3n ) ( αβ) c n The second case is given by (7) Case II: A = 0. Here, Eqs. (6)-(68) yields A = ±, (75) A nb =, ab 3 n β B 3 n c n { } aα ββb B aα κ n ω =. n Hence, finally, the dark soliton solution to the mcgl equation with dual-ower law nonlinearity (50) is given by n i( κ xωγ t tθ) qxt (, )= Atanh B xvt e (78) where the velocity v is given by (5) and the wave number ω is given by (7) if A ± and by (77) when A = ±. It is worth noting that the existence of the dark soliton solution (77) deends on the characteristics of the nonlinear medium, which satisfy the conditions given above. (76) (77)

14 380 Houria Triki et al. 5. CONCLUSIONS In this aer, the soliton ansatz method is used to integrate the mcgl equation. There are two tyes of nonlinearity that are considered in this aer. They are the arabolic law and its generalization, namely the dual-ower law nonlinearity. Both bright and dark -soliton solutions were obtained. The arameter restrictions in all four cases were laid down that fell out automatically from the soliton arameters. These results are going to be very useful in carrying out further analysis. Later these relations will be alied to soliton erturbation theory and to formulate the quasi-article theory. The quasi-stationary soliton solutions will also be studied with the erturbation terms resent. The stochasticity will also be taken into consideration. All such results will be reorted gradually in future ublications. Acknowledgments. The fourth, fifth and sixth authors (TH, OMA & AB) would like to recognize and thankfully areciate the suort from King Saud University (KSU-VPP-7). REFERENCES. A. Biswas, Progress in Electromagnetic Research Letters, 96, 7 (009).. K. W. Chung and Y. Y. Cao, Submitted. 3. W. P. Hong, Z. Naturforsch., 6A, (007).. K. Maruno, A. Ankiewicz, N. Akhmediev, Physics Letters, A 37, 3 0 (005). 5. D. Mihalache, Romanian Journal of Physics, 50, (005). 6. D. Mihalache, D. Mazilu, Romanian Reorts in Physics, 6, (009). 7. D. Mihalache, Romanian Reorts in Physics, 6, (00). 8. D. Mihalache, Proceedings of the Romanian Academy, A, 7 (00). 9. D. Mihalache, Romanian Reorts in Physics, 63, 9 (0). 0. D. Mihalache, Romanian Reorts in Physics, 63, (0).. A. Mohamdou, A. K. Jiotsa, T. C. Kofane, Chaos, Solitons & Fractals, 7, (005).. A. Mohamdou, F. Ndzana, T. C. Kofane, Physica Scrita, 73, (006). 3. F. B. Pela, A, J. Kenfack, M. M. Faye, Nonlinear Oscillations, 0, (007).. L. H. Shehata, H. M. Taha, Physica status solidi (b) Basic Solid State Physics,, (005). 5. S. Shwetanshumala, Progress in Electromagnetic Research Letters, 3, 7 (008). 6. V. Skarka, N. B. Aleksic, Physical Review Letters, 96, (006); V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, D. Mihalache, Physical Review Letters, 05, 390 (00); L.-C. Crasovan, B. A. Malomed, D. Mihalache, Physics Letters, A 89, (00); L.-C. Crasovan, B. A. Malomed, D. Mihalache, Physical Review, E 63, (00). 7. J. M. Soto-Creso, N. Akhmediev, Mathematics and Comuters in Simulation, 69, (005). 8. J. M. Soto-Creso, P. Grelu, N. Akhmediev, N. Devine, Physics Letters, A. 36, 3 36 (007). 9. E. Yomba, T. C. Kofane, Physica, D 5, 05 (999). 0. E. Yomba, T. C. Kofane, Chaos, Solitons & Fractals, 5, (003).