Application of Multiple Exp-Function Method to Obtain Multi-Soliton Solutions of (2 + 1)- and (3 + 1)-Dimensional Breaking Soliton Equations

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1 Amerc Jourl of Compuol Appled Mhemcs: ; (: 4-47 DOI:.593/j.jcm..8 Applco of Mulple Exp-Fuco Mehod o Ob Mul-Solo Soluos of ( + - (3 + -Dmesol Breg Solo Equos M. T. Drvsh,*, Mlheh Njf, Mohmmd Njf Deprme of Mhemcs, Rz Uversy, Kermshh, 6749, Ir Youg Reserchers Club, Kermshh Brch, Islmc Azd Uversy, Kermshh, Ir Absrc The mulple exp-fuco mehod s ew pproch o ob mulple wve soluos of oler prl dfferel equos (NLPDEs. By hs mehod oe c ob mul-solo soluos of NLPDEs. I hs pper, usg compuer lgebr sysems, we pply he mulple exp-fuco mehod o cosruc he exc mulple wve soluos of he ( + - he (3 + -dmesol breg solo equos. By hs pplco, we ob oe-wve, wo-wve hree-wve soluos for hese equos. Keywords Mulple Exp-Fuco Mehod, Breg Solo Equo, Exc Soluo, Mul-Solo Soluo. Iroduco The sudy of exc soluos of oler prl dfferel equos plys mpor role solo heory explc formuls of oler prl dfferel equos ply essel role he oler scece. Also, he explc formuls my provde physcl formo help us o uders he mechsm of reled physcl models. Recely, my ds of powerful mehods hve bee proposed o fd exc soluos of oler prl dfferel equos, e.g., he h-mehod[], he homogeeous blce mehod[], homoopy lyss mehod[3-8], he F expso mehod [9], hree-wve mehod[-3], exeded homoclc es pproch[4-6], he G ( expso mehod[7] he G exp-fuco mehod[8-3]. By hese lyc mehods, oe oly c ob rvelg wve soluos for NLPDEs. However, s ow h here re mulple wve soluos o some NLPDEs, for exmple, mul-solo soluos o KdV Tod lce equos[4] or mulple perodc wve soluos o he Hro bler equos[5,6]. Recely, M F[7] explored ey feure of he ler superposo prcple h ler equos possess, for Hro bler equos, whle mg o cosruc specfc sub-clss of N -solo soluos formed by ler combos of expoel rvelg wves. They proved h, ler superposo * Correspodg uhor: drvshm@yhoo.com (Mohmmd Tgh Drvsh Publshed ole hp://jourl.spub.org/jcm Copyrgh Scefc & Acdemc Publshg. All Rghs Reserved prcple c pply o expoel rvelg wves of Hro bler equos. Applcos mde o show h he preseed ler superposo prcple s helpful geerg N -wve soluos o solo equos, prculrly hose hgher dmesos. Also, M e l.[8] cosruced he mulple exp-fuco mehod o ob mulple wve soluos, cludg oe-solo, wo-solo hree-solo ype soluos o (3+-dmesol YTSF equo. Oe my fd oher wors o fd exc soluos of solo equos [9-3]. I hs pper, we pply he mulple exp-fuco mehod o ob some exc mulple wve soluos for he ( + - (3 + -dmesol breg solo equos. The (+-dmesol breg solo equo s u 4u u u u u ( x xy x xx y xxxy hs equo descrbes he (+-dmesol erco of he Rem wve propged log he y -xs wh log wve propged log he x -xs[3]. Wzwz[33] roduced exeso o equo ( by ddg he ls hree erms wh y replced by z. Hs wor, ebles us o esblsh he followg (3+-dmesol breg solo equo ux 4 ux ( uxy + uxz uxx ( uy + uz ( ( uxxxy + uxxxz where u uxyz (,,, : x y z Ths pper s orgzed s follows: he followg seco we hve bref revew o mulple exp-fuco mehod. I Secos 3 4 we pply he exp-fuco mehod o he ( + -dmesol he (3 + -dmesol breg solo equos, respecvely. I

2 4 M. T. Drvsh e l.: Applco of Mulple Exp-Fuco Mehod o Ob Mul-Solo Soluos of ( + - (3 + -Dmesol Breg Solo Equos hose secos we ob oe-wve, wo-wve hree-wve soluos for our equos. The pper s cocluded Seco 4.. Mehodology M e l.[8] llusred he mulple exp-fuco mehod hree seps. I summry, her seps re: Defg solvble dfferel equos, rsformg oler PDEs solvg lgebrc sysems. The ey po of her pproch s o see rol soluos se of ew vrbles defg dvdul wves. The pplco of mulple exp-fuco mehod yelds specfc oe-wve, wo-wve hree-wve soluos o NLPDEs. To expl hese fudmel seps mulple exp-fuco mehod, cosder PDE ( + dmesos s F( xu,, x, u, (3 I he frs sep, we mus roduce sequece of ew vrbles η η( x,,,,, by solvble prl dfferel equos, for exmple, he ler PDEs, η, x η, η, ωη,,,, (4 where,,,, re he gulr wve umbers ω,,,, re he wve frequeces. Ths sep s srg po o cosruc exc soluos of oler equos. Solvg hese ler equos leds o he expoel fuco soluos, η ce ξ, ξ x ω,,,, (5 where c,,, re rbrry coss. Ay fuco η descrbg sgle wve mulple wve soluo wll be combo of ll of hose sgle wves. I he secod sep, we proceed by cosderg rol soluos he ew vrbles η,,, : M p( η, η,, η j ux (,, p prs, j r s, q ( η, η,, η rs,, j (6 N j qrs, j r s rs,, j q, where p l, j q l, j re coss o be deermed from (3. The we use he dfferel relos (4 o express ll prl dervves of u wh x erms of η,,,. Subsug hese obed expressos for prl dervves o he equo (3 geeres rol fuco equo he ew vrbles η,,, : Gxη (,,,, η (7 Equo (7 s clled he rsformed equo of he orgl equo (3. Ths sep mes possble o compue soluos o dfferel equos drecly by compuer lgebr sysems he hrd sep. Flly, he hrd sep we se he umeror of he resulg rol fuco Gxη (,,,, η o zero. Ths yelds sysem of lgebrc equos o ll vrbles, ω, pl, j, pl, j. By solvg hs sysem, wh he d of mhemcl sofwre such s Mple, we deerme polyomls p q he wve expoes ξ,,. Afer hs, he mulple wve soluo u s compued gve by x ω x ω p( ce,, ce ux (, (8 x ( ω x,, ω q ce ce for more dels o hese seps; cf.[8]. I he followg secos we pply mulple exp-fuco mehod o he ( + - he (3 + -dmesol breg solo equos o ob some mul-wve soluos. I mus be oed h ll obed soluos re ew oes for some specl vlues of prmeers hese ew soluos we c ob he soluos whch hve obed by oher mehods. 3. Oe-Wve, Two-Wve Three- Wve Soluos o he ( + -Dmesol Breg Solo Equo I hs pr, we pply he mulple exp-fuco mehod o cosruc he exc mulple wve soluos of he ( + -dmesol breg solo equo ux 4uxyux uxxuy uxxxy (9 We geere oe-wve, wo-wve hree-wve soluos of wo polyoml fucos p q s follows: Cse : Oe-wve soluos To ob oe-wve soluos, we requre he ler codos η, x η, η, y lη, η, wη ( where, l w re coss whch wll be deermed. The we ry pr of polyomls of degree oe, p( η + η, q( η b+ bη ( where,, b b re coss o be deermed. By he mulple exp-fuco mehod usg he dfferel relos (, we ob he followg soluo o he resulg lgebrc sysem wh he d of Mple, b( + b, w l ( b,, b re rbrry. Thus we c hve expoel fuco soluo o (, x l y w η e + (3 whch ges he followg oe-wve soluos y w p + e u uxy (,, (4 y w q b + be where, b re rbrry coss. I mus be

3 Amerc Jourl of Compuol Appled Mhemcs: ; (: oed h, f we se, b, b l r, he soluo (9 of[33] wll ob. Cse : Two-wve soluos I hs cse, we requre he ler codos, η η, η lη, η wη,,, (5, x, y, where, l, w,,, re coss hus he soluos η η re defed by x l y w η e +,,, (6 where c c re rbrry coss. We suppose h p( η, η [ η+ η + ( + ], (7 q( η, η+ η + η + where s cos o be deermed. By he mulple exp-fuco mehod usg he dfferel relos (5, we ob he followg soluo o he resulg lgebrc sysem wh he d of Mple, w l,,, (8 ( (9 ( + The, we ob he followg wo-wve soluos p( η, η u uxy (,, q ( η, η ( [ η+ η + ( + ] η η g,, l, l, c c ( revels solry wves o Eq. (9. Fgure shows he plo of soluo ( for hese specl vlues of s prmeers. Cse : Three-wve soluos Smlrly, o ob he hree-wve soluos for equo (9, we requre he ler codos, η, x η, η, y lη, η, w η,,,3, ( where, l, w,,,3, re coss hus he soluos η, η η c be defed by 3 y w η e,,,3, ( where c, c c3 re rbrry coss.we suppose h p( η, η [ η+ η + 3η3+ ( + + 3( ( ( η 3] q( η, η+ η + η + η η 3, where, 3 3 re coss o be deermed. By he mulple exp-fuco mehod usg he dfferel relos (, we ob he followg soluo o he resulg lgebrc sysem wh Mple, w l,,,3, (4 ( j (5 j,, j,,3. ( + j The, we ob he followg hree-wve soluos p( η, η, η3 u uxy (,, (6 q ( η, η, η3 where p q re defed by (3 η,,,3 re defed by (. For l r,,,3 (6, we mee soluo (6 of [3]. Fgure shows he plo of soluo (6 for some specl vlues of s prmeers. 4. Oe-Wve, Two-Wve Three- Wve Soluos o he (3 + -Dmesol Breg Solo Equo Fgure. The wo-wve soluo For l r,,, soluo ( s s sme s soluo (4 of[33]. We obed wde clss of rvelg wve soluos o Eq. (9 by seg specl vlues o he rbrry prmeers, we c cosruc forml wo-wve soluos. For sce, I hs seco, we wll pply he mulple exp-fuco mehod o cosruc he exc mulple wve soluos for he (3 + -dmesol breg solo equo ux 4 ux ( uxy + uxz uxx ( uy + uz (7 ( u + u, we dscuss hree cses of wo polyoml fucos p q o geere oe-wve, wo-wve hree-wve soluos s follows: xxxy xxxz

4 44 M. T. Drvsh e l.: Applco of Mulple Exp-Fuco Mehod o Ob Mul-Solo Soluos of ( + - (3 + -Dmesol Breg Solo Equos Cse : Oe-wve soluos Ag, we requre he ler codos, η η, η lη, η mη, η wη (8, x, y, z, where, l, m, w re coss. We he ry pr of polyomls of degree oe, p( η + η, q( η b+ bη (9 where,, b b re coss o be deermed. By he mulple exp-fuco mehod usg he dfferel relos (8, we ob he followg soluo o he resulg lgebrc sysem wh he help of Mple, b( + b, w ( m+ l (3 b where, b re rbrry coss. Sce we c hve expoel fuco soluo o (8, x l y mz w η e + + (3 By usg (3, we ob he followg oe-wve soluos for equo (7 where, l, m, w,,, re coss hus he soluos η η c be defed by y+ mz w η e,,, (34 where c c re rbrry coss. We suppose h p( η, η [ η+ η + ( + ], q( η, η + η + η +, (35 Fgure 3. The frs wo-wve soluo Fgure. The hree-wve soluo y+ mz w + e y+ mz w + e p u uxy (,, (3 q b b where, b re rbrry coss. For, b, b, l r m s our soluo (3 s he sme soluo (49 of [33]. Cse : Two-wve soluos I hs cse, we requre he ler codos, η, x η, η, y lη, η, z mη, η wη,,., (33 where s cos o be deermed. By he mulple exp-fuco mehod usg he dfferel relos (33, we ob wo soluos o he resulg lgebrc sysem wh Mple, 3 w m,,, (36 ( (37 ( + whe l,,, 3 w l,,, (38 ( (39 ( + whe m,,, we ob he followg wo-wve soluos u uxy p( η, η (,, q ( η, η [ η+ η + ( + ] + η + η + (4

Amerc Jourl of Compuol Appled Mhemcs: ; (: 4-47 45 Two specfc soluos of hese wo-wve soluos re ploed Fgures 3 4. whe m,,,3.

The secod wo-wve soluo Cse : Three-wve soluos To ob he hree-wve soluos for equo (7, we requre he ler codos, η, x η, η, y lη, η, z mη, (4 η wη,,,3,, where, l, w,,,3, re coss hus he soluos η, η η c be

5 Amerc Jourl of Compuol Appled Mhemcs: ; (: Two specfc soluos of hese wo-wve soluos re ploed Fgures 3 4. whe m,,,3.the, we ob he followg hree-wve soluos for equo (7 p( η, η u uxy (,, q ( η, η (48 [ η+ η + ( + ] + η + η + Two specfc soluos of hese hree-wve soluos re ploed Fgures 5 6. Fgure 4. The secod wo-wve soluo Cse : Three-wve soluos To ob he hree-wve soluos for equo (7, we requre he ler codos, η, x η, η, y lη, η, z mη, (4 η wη,,,3,, where, l, w,,,3, re coss hus he soluos η, η η c be defed by 3 c, c 3 x + l y + mz w η e,,,3, where c re rbrry coss. We suppose h p( η, η [ η + η + η + ( ( + + ( ( + + η ] q( η, η + η + η + η (4 ( η 3, where, 3 3 re coss o be deermed. By he mulple exp-fuco mehod usg he dfferel relos (4, we ob he followg soluo o he resulg lgebrc sysem wh he help of Mple, 3 w m,,,3, (44 j ( j ( + j whe l,,,3,,, j,,3 3,,,3, (45 w l (46 j ( j ( + j,, j,,3 (47 Fgure 5. Fgure Coclusos The frs hree-wve soluo The secod hree-wve I hs pper, we hve ppled he mulple exp-fuco mehod o ob oe-wve, wo-wve hree-wve soluos of he ( + - he (3 + -dmesol breg solo equos. These soluos re ew soluos for hese equos for some specl vlues of prmeers hese ew soluos we c ob he soluos whch hve obed by oher mehods. The mulple exp-fuco mehod s oreed owrds he ese

6 46 M. T. Drvsh e l.: Applco of Mulple Exp-Fuco Mehod o Ob Mul-Solo Soluos of ( + - (3 + -Dmesol Breg Solo Equos of use cpbly of compuer lgebr sysems provdes srghforwrd sysemc soluo procedure h geerlzes Hro's perurbo mehod. The mehod c be ppled o oher oler prl dfferel equos whch hve mul-solo soluos, becuse he mulple exp-fuco lgorhm s powerful geerg exc soluos of oler equos. REFERENCES [] A. M. Wzwz. (5 The h mehod: solos perodc soluos for he Dodd-Bullough-Tzhlov he Tzzec-Dodd-Bullough equos, Chos, Solos Frcls, 5, [] Z. Xqg, W. Lm, S. Weju. (6 The repeed homogeeous blce mehod s pplcos o oler prl dfferel equos, Chos, Solos Frcls, 8(, [3] S. J. Lo. (4 O he homoopy lyss mehod for oler problems, Appl. Mh. Compu., 47, [4] S. J. Lo. (5 A ew brch of soluos of boudry-lyer flows over mpermeble sreched ple, I. J. He Mss Trsfer, 48, [5] S. J. Lo. (9 A geerl pproch o ge seres soluo of o-smlry boudry-lyer flows, Commu. Noler Sc. Numer. Smul., 4(5, [6] M. T. Drvsh, F. Kh. (9 A seres soluo of he fom drge equo, Compu. Mh. Appl., 58, [7] A. Azz, F. Kh, M. T. Drvsh. ( Homoopy lyss mehod for vrble herml coducvy he flux gge wh edge coc ressce, Zeschrf fuer Nurforschug A, 65(, [8] F. Kh, M. T. Drvsh, R. S. R. Gorl ( Alycl vesgo for coolg urbe dss wh o-newo vscoelsc flud, Compu. Mh. Appl., 6(7, [9] E. G. F, Z. J. ( Applcos of he Jcob ellpc fuco mehod o specl-ype oler equos, Phys. Le. A, 35 (6, [] M. T. Drvsh, Mlheh Njf, Mohmmd Njf. ( Exc hree-wve soluos for hgh oler form of Bejm-Bo-Mhoy-Burgers equos, Ierol Jourl of Mhemcl Compuer Sceces, 6(3, 7-3 [] M. T. Drvsh, M. Njf. ( Some exc soluos of he (+-dmesol breg solo equo usg he hree-wve mehod, Ierol Jourl of Compuol Mhemcl Sceces, 6(, 3-6 [] M. T. Drvsh, Mlheh Njf, Mohmmd Njf. ( New exc soluos for he (3+-dmesol breg solo equo, Ierol Jourl of Iformo Mhemcl Sceces, 6(, [3] S. Arbb Mohmmd-Abd, M. Njf. ( Exed hree wve mehod for he (3+-dmesol solo equo, World Acdemy of Scece, Egeerg Techology, 8, [4] M. T. Drvsh, Mlheh Njf, Mohmmd Njf. ( New pplco of EHTA for he geerlzed (+- dmesol oler evoluo equos, Ierol Jourl of Mhemcl Compuer Sceces, 6(3, 3-38 [5] M. T. Drvsh, M. Njf. ( A modfco of exeded homoclc es pproch o solve he (3+-dmesol poel-ytsf equo, Ch. Phys. Le., 8(4, r. o. 4 [6] M. T. Drvsh, M. Njf. ( Some complexo ype soluos of he (3+-dmesol Jmbo-Mw equo, Ierol Jourl of Compuol Mhemcl Sceces, 6(, 5-7 [7] M. T. Drvsh, Mlheh Njf, Mohmmd Njf. ( Trvelg wve soluos for he (3+-dmesol breg solo equo by (G'/G-expso mehod modfed F-expso mehod, Ierol Jourl of Compuol Mhemcl Sceces, 6(, [8] J. H. He, M. A. Abdou. (7 New perodc soluos for oler evoluo equos usg Exp-fuco mehod, Chos, Solos Frcls, 34, 4-49 [9] F. Kh, S. Hmed-Nezhd, M. T. Drvsh, S. W. Ryu. (9 New solry wve perodc soluos of he fom drge equo usg he Exp-fuco mehod, Nol. Al.: Rel World Appl.,, 94-9 [] B.-C. Sh, M. T. Drvsh, A. Br (9 Some exc ew soluos of he Nzh-Novov-Vesselov equo usg he Exp-fuco mehod, Compu. Mh. Appl., 58(/, 47-5 [] F. Kh, M. T. Drvsh, A. Frm, L. Kvh. ( New exc soluos of coupled (+-dmesol oler sysem of Schrödger equos, ANZIAM Jourl, 5, - [] X. H. Wu, J. H. He. (8 Exp-fuco mehod s pplco o oler equos, Chos, Solos Frcls, 38(3, 93-9 [3] M. T. Drvsh, Mohmmd Njf, Mlheh Njf. ( Some ew exc soluos of he (3+-dmesol breg solo equo by he Exp-fuco mehod, Nol. Scece Le. A., (4, -3 [4] R. Hro, The drec mehod solo heory, Cmbrdge Uversy Press, Cmbrdge, 4 [5] Y. Zhg, L. Y. Ye, Y. N. Lv, H. G. Zho. (7 Perodc wve soluos of he Boussesq equo, J. Phys. A: Mh. Ge., 4, [6] W. X. M, R. G. Zhou, L. Go. (9 Exc oe-perodc wve soluos o Hro bler equos + dmesos, Mod. Phys. Le. A, 4, [7] W. X. M E. G. F. ( Ler superposo prcple pplyg o Hro bler equos, Compu. Mh. Appl., 6, [8] W. X. M, T. W. Hug, Y. Zhg. ( A mulple exp-fuco mehod for oler dfferel equos s pplco, Phys. Scr., 8, r. o. 653 [9] W. X. M. (3 Dversy of exc soluos o resrced Bo-Leo-Pempell dspersve log-wve sysem, Phys.

7 Amerc Jourl of Compuol Appled Mhemcs: ; (: Le. A, 39, [3] W. X. M, W. Srmpp. (5 Bler forms Bclud rsformos of he perurbo sysems, Phys. Le. A, 34, [3] W. X. M. (5 Iegrbly, : A. Sco (Ed., Ecycloped of Noler Scece, Tylor & Frcs, New Yor, pp [3] S. H. M, J. Peg, C. Zhg. (9 New exc soluos of he (+-dmesol breg solo sysem v exeded mppg mehod, Chos, Solos Frcls, 46, -4 [33] A. M. Wzwz. ( Iegrble (+-dmesol (3+-dmesol breg solo equos, Phys. Scr., 8, -5

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