SOLITONS, KINKS AND SINGULAR SOLUTIONS OF COUPLED KORTEWEG-DE VRIES EQUATIONS

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMNIN CDEMY, Series, OF THE ROMNIN CDEMY Volue 4, Nuber /3,. SOLITONS, KINKS ND SINGULR SOLUTIONS OF COUPLED KORTEWEG-DE VRIES EQUTIONS Bouthia S. HMED, ja BISWS i Shas Uiversity, Deartet of Matheatics, College of Girls, Cairo, Egyt Delaware State Uiversity, Deartet of Matheatical Scieces, Dover, DE 99-77, US E-ail: biswas.aja@gail.co; hed_bouthia@live.co This aer studies the couled Korteweg-de Vries (KdV) equatios that describe shallow two-layered water waves i ocea shores. The asatz ethod is eloyed to retrieve the solitary wave solutios, toological ad sigular solitos of this couled syste of equatios. The costrait coditios will fall out for the existece of these tyes of solitos. The study is also geeralized to couled KdV equatios with ower law oliearity. The uerical siulatios suleet the aalytical schees. Key words: solutios, itegrability, asatz ethod, uerics.. INTRODUCTION Uique oliear heoea exist i all research fields, such as fluid echaics, lasa hysics, otical fibers, biology, solid state hysics, cheical kieatics, cheical hysics, ad so o. I the ast work, the research of travellig wave solutios of several oliear evolutio equatios layed a iortat role i the aalysis of such oliear heoea [ 5]. For the travellig wave solutios, ay ethods were atteted, such as the iverse scatterig trasfor ethod [], Hirota s biliear trasforatio [9], the tah ethod [6,,], sie-cosie ethod [5], hoogeeous balace ethod [5], ex-fuctio ethod [5] ad so o. The above ethods derived ay tyes of solutios fro ost oliear evolutio equatios. It is ow tie to start focusig o such equatios which aear i couled vector forat. They have ractical alicatios too. Quite cooly, vector couled evolutios are studied i two-layered or rather ultile-layered shallow water waves, or otical solitos i birefriget fibers ad/or desewavelegth divisio ultilexed systes [8]. The two-layered fluid waves ear ocea shores ca be odeled by couled Korteweg-de Vries (KdV) equatios or the Gear-Grishaw odels [3]. This aer will focus o couled KdV equatios; exact solitary waves, kiks (toological solitos) ad sigular solitos will be obtaied.. GOVERNING EQUTIONS The couled KdV equatios which describe the iteractio rocess of two log waves are give by Tye I q + a qq + bq + c rr =, () t x xxx x r + a qr + b r =, () t x xxx ad the couled KdV equatios with ower law oliearity are Tye II t x xxx x q + a q q + bq + c r r =, (3)

2 Bouthia S. hed, ja Biswas t x xxx r + a q r + b r =. (4) These are the two odels that rereset the couled KdV equatios. The deedet variables are q(x,t) ad r(x,t) which resectively rereset the wave rofile for the two layers of fluid. The satially ideedet variable is x while the teoral ideedet variable is t. The costat coefficiets are a j, b j, j=, ad c. I (), the first ter reresets the evolutio ter, while the secod ter alog with the fourth ter are the oliear ters. The third ter is the disersio. The for (), the first ter agai reresets evolutio, the secod ter is the cross-oliear ter while the third ter is agai the disersio. For Tye-II, the two additioal araeters are ad that rereset the ower law oliearity araeters, i.e., we get couled KdV equatios with ower law oliearity. Thus, the ower law oliearity is itroduced o a geeralized settig that could rereset the regular KdV equatio or the odified KdV (KdV) deedig o the values of ad. 3. NSTZ PPROCH Oe of the ost oular aroaches to itegratio, that has lately surfaced is the asatz ethod [3, 4, 7]. This ethod is siilar to the trial solutio ethod that is very cooly used i solvig ordiary differetial equatios. Such a trial solutio, for solitos, is assued based o the kow results of the sigle KdV equatio. This trial solutio is the substituted ito the couled syste. The liearly ideedet fuctios are the idetified, after the alicatio of the balacig ricile, ad their resective coefficiets are all set to zero. These lead to the coutatio of the solito araeters. 3.. Solitary waves I this sectio the search is goig to be for -solito solutio to the couled KdV-tye I equatios give by (), () ad tye II equatios (3) ad (4). To begi, let us assue the followig solitary wave asatz i two cases of the KdV equatio Couled KdV equatios To obtai -solito solutio of () ad () we assue that qxt (, ) =, (5) cosh rxt (, ) = cosh, (6) where ad rereset the alitudes of q ad r solitos, resectively, ad is give by = B( x vt), (7) while B is aother free araeter of the solito ad v is the velocity of the solito. Substitutig (5) ad (6) ito () ad () gives: tah tah tah ( + )( + ) tah c B b ( + ) qb a B B B tah + = cosh cosh cosh cosh cosh b ( ) + ( + ), (8) qbtah a Btah B tah ( )( ) B tah =. (9) cosh cosh cosh cosh

3 3 Solitos, kiks ad sigular solutios of couled Korteweg-de Vries equatios 3 Now, by equatig the exoet + ad ad the exoet + ad + i (8), (9) we get By equatig the coefficiets of = =, () tah,, cosh j j = i (8) ad (9) we have + v 4b B =, () a + 4bB c =, () v 4b B =, (3) a + bb =. (4) Fro the above equatios we have b = b = b, (5) v B =, (6) b 3v =, (7) a 3v a a a a =, >. (8) a c c The followig Fig. dislays -solito solutio q(x,t), r(x,t) with the araeters a =, a =, b=, c=, v=.5, x, t. Fig.. Solito solutio: left: q(x,t), right: r(x,t) Couled KdV equatio with ower-law oliearity Startig with the sae asatz ethod as (5) ad (6), equatios (3) ad (4) give

4 4 Bouthia S. hed, ja Biswas b + ( + ) qbtah a Btah B tah ( )( ) B tah + cosh cosh cosh cosh + c cosh Btah + = (9) tah a B ( + ) qb cosh cosh tah B + + B ( + ) tah ( )( ) tah b = cosh cosh Equatig the exoet of +, + ad exoet of +, + i (9) ad () we get = () =. () Fro (9) ad (), by equatig the coefficiets of cosh + j, j =, we get 4 v b B =, () + ( + )( + ) + a bb c =, (3) 4 v b B =, (4) ( + )(+ ) a b B =. (5) Fro the above equatios we get =, (6) b = b = b, (7) v B =, (8) b v ( + )(+ ) = 4a v ( + )(+ ) bb + + = a, (9). (3)

5 5 Solitos, kiks ad sigular solutios of couled Korteweg-de Vries equatios 5 I Fig. we dislay -solito solutio q(x,t), r(x,t) with the araeters : =, a =, a =, b = b =, c =, v =.5, x, t 3.. Toological solitos Toological solitos are also kow as kiks or shock waves [3, 4]. toological solito is a oliear wave where we have a trasitio fro oe stable state to aother. Thus, these are tyically odeled by the tah fuctios. Hece, the followig asatz will be utilized to get a solutio of the couled KdV equatios. qxt (, ) tah =, (3) rxt (, ) tah =, (3) where ad are free araeters of q ad r solito resectively ad is give by (7). Fig. Solito solutio for = : left: q(x,t), right: r(x,t) Couled KdV equatios Substitutig (3) ad (3) ito () ad () resectively yields (tah tah ) + (tah tah ) + vb a B B {( )( ) tah ( + )( + )} tah { + ( )( )} c tah { ( )( )} tah } (tah tah ) =, ab vb(tah tah ) (tah tah ) {( )( ) tah ( )( ) tah + b B { + ( )( )} tah + { + ( + )( + )} tah } =. (33) (34) Now, by equatig the exoet + ad + 3 i (33) equatig the exoet + ad + 3 i (34)give = =. (35)

6 6 Bouthia S. hed, ja Biswas 6 By equatig the coefficiets of tah + j, j = ±, ± 3 i (33), by equatig the coefficiets of tah + j, j =±, ± 3 i (34) ad ut = = we have v+ 8b B =, (36) v a b B c =, (37) a bb c + + =, (38) v+ 8b B =, (39) v a bb + + =, (4) a bb Usig Male rogra to solve the above equatios we obtai the solutios of the for + =. (4) b = b = b, (4) v B =, (43) 8b 3v =, (44) a 3v a a a a =, >. (45) a c c Figure 3 dislays toological -solito solutio q(x,t), r(x,t), resectively, with the araeters a =, a =, b =, c =, v =.5, x, t Fig. 3. Toological -solito solutio: left: q(x,t), right: r(x,t) Couled KdV equatio with ower-law oliearity For the couled KdV equatio with ower law oliearity, the startig hyothesis stays the sae as i (33), (34). Thus, (3) ad (4) give

7 7 Solitos, kiks ad sigular solutios of couled Korteweg-de Vries equatios vb(tah tah ) a B(tah tah ) {( )( ) tah ( )( ) tah { ( )( )} tah + B (46) { + ( + )( + )}tah } + c (tah tah ) =, a B vb(tah tah ) (tah tah ) {( )( ) tah ( )( ) tah + b B + + (47) + { + ( )( )} tah + { + ( + )( + )} tah } =. Now, by the aid of balacig ricile, equatig the exoet + + ad + 3 ad equatig the exoet + + ad + 3 i (46) gives = /. (48) By settig the coefficiets of tah + j, tah + j, j = ±, ± 3 i (46) ad (47) to zero, sice these are liearly ideedet fuctios, we have v + ( + ( )( )) b B =, (49) + + v + a + ( + ( + )( + )) b B + c =, (5) + + a + ( + )( + ) bb + c =, (5) 3 ( )( ) b B =, (5) v + ( + ( )( )) b B =, (53) v + a + ( + ( + )( + )) b B =, (54) a + ( + )( + ) bb =. (55) fid It follows fro (5) that = or =, i.e., the case 3... Now, substitutig = i (49), (53) we Now, we ut = = ad = = i (49 55) ad we get = =. (56) B = v/b, (57) = v/ a, (58) = ( ( )/ ) /3. (59) a a a c Figure 4 dislays the kik solito solutio with the followig araeters: a =, a =, b =, c =, v =.5, x, t.

8 8 Bouthia S. hed, ja Biswas Sigular solitos There are several oliear evolutio equatios that suort sigular solutios ad they are alicable i various areas of atheatics ad hysics. Let us assue the followig sigular solutios: qxt (, ) = sih rxt (, ) = sih, (6), (6) where ad are costats of the q ad r solito, resectively, ad is give by (7). Fig. 4. Kik solutio: left: q(x,t), right: r(x,t). Fro (6) ad (6) we have Couled KdV equatio vbcoth a Bcoth B coth ( + )( + ) B coth b + ( + ) sih sih sih cosh c Bcoth = sih b ( ) + ( + ) vbcoth a Bcoth B coth ( )( ) B coth + =. sih cosh sih sih Fially we get (6) (63) v B =, (64) b 3v =, (65) a 3 v ( a+ a) a+ a =, >. (66) a c c

9 9 Solitos, kiks ad sigular solutios of couled Korteweg-de Vries equatios Couled KdV equatios with ower-law oliearity I this case, (3) ad (4) give b + + vbcoth a Bcoth B coth ( )( ) B coth + sih sih sih sih + c sih Bcoth + = b + + vbcoth a Bcoth Bcoth ( )( ) B coth + =. sih sih sih sih (67) (68) Doig the sae calculatio of subsectio 3.. we get B = v, (69) b = ( + )( + ) v = a + a ( + )( + ) v ( + ) v a 4, /( + ). (7) (7) 4. CONCLUSIONS This aer studied the couled KdV equatios that aear i shallow two-layered water waves i lakes, rivers ad ocea shores. Solitary waves, kiks, ad sigular solito solutios were obtaied. The asatz ethod was alied i order to obtai these secial solutios. The solito solutios also itroduced several costrait coditios that ust reai valid i order for these solutios to exist. The uerical siulatios clearly illustrate the solutios that were aalytically obtaied. Other aroaches that will be used to study these couled KdV equatios which will reveal further oliear wave solutios will be reorted elsewhere. REFERENCES. M.. BDOU, The exteded tah-ethod ad its alicatios for solvig oliear hysical odels, l. Math. Cout., 9, , 7.. M.J. BLOWITZ, P.. CLRKSON, Solitos, No-liear Evolutio Equatios ad Iverse Scatterig Trasfor, Cabridge Uiversity Press, Cabridge, BISWS, M.S. ISMIL, -solito solutio of the couled KdV equatio ad Gear-Grishaw odel, l. Math. Co., 6, ,. 4.. BISWS,. YILDIRIM, T. HYT, O. M. LDOSSRY, R. SSSMN, Solito erturbatio theory for the geeralized Klei-Gordo equatio with full oliearity, Proc. Roaia cad., 3,. 3 4,. 5. G. EBDI,. YILDIRIM,. BISWS, Chiral solitos with Boh otetial usig G'/G ethod ad exoetial fuctio ethod, Ro. Re. Phys., 64, ,. 6. E.G. FN, Exteded tah-fuctio ethod ad its alicatios to oliear equatios, Phys. Lett., 77,. 8,.

10 Bouthia S. hed, ja Biswas 7. P. GREEN, D. MILOVIC,.K. SRM, D.. LOTT,. BISWS, Dyaics of suer-sech solitos i otical fibers, J. Nol. Ot. Phys. ad Materials, 9, ,. 8. J.H. HE, X.H. WU, Ex-fuctio ethod for oliear wave equatios, Chaos. Solito. Fract., 3,. 7 78, R. HIROT, Exact eveloe solito solutios of a oliear wave equatio, J. Math. Phys., 4,. 85 8, G. JOHNPILLI,. YILDIRIM,. BISWS, Chiral solitos with Boh otetial by Lie grou aalysis ad travelig wave hyothesis, Ro. J. Phys., 57, ,.. W. MLFLIET, Solitary wave solutios of oliear wave equatios,. J. Phys., 6, , 99.. E.J. PRKES, B.R. DUFFY, autoated tah-fuctio ethod for fidig solitary wave solutios to oliear evolutio equatios, Cout. Phys. Cou. 98,. 88 3, H. TRIKI,. YILDIRIM, T. HYT, O.M. LDOSSRY,. BISWS, Toological ad o-toological solito solutios of the Bretherto equatio, Proc. Roaia cad., 3,. 3 8,. 4. H. TRIKI, S. CRUTCHER,. YILDIRIM, T. HYT, O.M. LDOSSRY,. BISWS, Bright ad dark solitos of the colex Gizburg-Ladau equatio with arabolic ad dual-ower law oliearity, Ro. Re. Phys., 64, ,. 5. M.L. WNG, Exact solutios of a cooud KdV-Burgers equatio, Phys. Lett., 3, , 996. Received July 3,