Extended tan-cot method for the solitons solutions to the (3+1)-dimensional Kadomtsev-Petviashvili equation

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1 Interntionl Jornl of Mthemticl Anlysis nd Applictions ; (): 9-9 Plished online Mrch, ( Extended tn-cot method for the solitons soltions to the (+)-dimensionl Kdomtsev-Petvishvili eqtion Anwr J'fr Mohmd Jwd Keywords Exct Soltion, Kdomtsev-Petvishvili Eqtion, Extended n-cot Method, Nonliner Prtil Differentil Eqtions Received: Ferry, Revised: Ferry, Accepted: Ferry, Al-Rfidin University College, Bghdd, Irq Emil ddress nwr_jwd@yhoo.com Cittion Anwr J'fr Mohmd Jwd. Extended tn-cot Method for the Solitons Soltions to the (+)-Dimensionl Kdomtsev-Petvishvili Eqtion. Interntionl Jornl of Mthemticl Anlysis nd Applictions. Vol., No.,, pp Astrct he proposed extended tn-cot method is pplied to otin new exct trvelling wve soltions to the (+)-dimensionl Kdomtsev-Petvishvili eqtion nd (+)- dimensionl eqtion. he method is pplicle to lrge vriety of nonliner prtil differentil eqtions. he tn-cot method seems to e powerfl tool in deling with nonliner physicl models.. Introdction Solitons re fond in mny physicl phenomen. Solitons rise s the soltions of widespred clss of wely nonliner dispersive prtil differentil eqtions descriing physicl systems. Solitons re solitry wves with elstic scttering property. De to dynmicl lnce etween the nonliner nd dispersive effects these wves retin their shpes nd speed to stle wveform fter colliding with ech other. Onesic expression of solitry wve soltion is of the form[]: ( x, t ) f ( x λt ) () where λ is the speed of wve propgtion. For λ >, the wve moves in the positive x direction, wheres the wve moves in the negtive x direction for λ <. rvelling wves, whether their soltion expressions re in explicit or implicit forms re very interesting from the point of view of pplictions. hese types of wves will not chnge their shpes dring propgtion nd re ths esy to detect. Of prticlr interest re three types of trvelling wves: the solitry wves, which re loclized trvelling wves, symptoticlly zero t lrge distnces, the periodic wves, which rise or descend from one symptotic stte to nother. Recentl lgeric method, clled the mpping method [], is proposed to otin exct trvelling wve soltions for lrge vriety of nonliner prtil differentil eqtions (PDEs). Other methods re proposed to otin exct trvelling wve soltions sch s sine-cosine-fnction method[], tnh-coth method[-5], tn-cotfnction method [6-7], sech method [8].

2 Anwr J'fr Mohmd Jwd: Extended tn-cot Method for the Solitons Soltions to the (+)-Dimensionl Kdomtsev-Petvishvili Eqtion. Description of Extended n-cot Fnction Method For given nonliner evoltion eqtion, s in for vriles (+) - dimensionl, α β ω θ P (,,,,,... ) () t x y We see trvelling wve soltion of the form: U ( ξ ), nd ξ + αy + βz + ωt + θ Where,,, re considered constnts. he following chin rle z xx x () U t ω du, d ξ U x du d ξ, U α y du, d ξ U z β du d ξ U x d U d ξ, converted the PDE Eq.(), to n ordinry differentil eqtion ODE / // /// Q ( U, U, U, U,....) () withq eing nother polynomil form of their rgment, which will e clled the redced ordinry differentil eqtions of Eq.(). Integrting Eq.() s long s ll terms contin derivtives, the integrtion constnts re considered to e zeros in view of the loclized soltions. However, the nonzero constnts cn e sed nd hndled s well. Now finding the trveling wve soltions to Eq.() is eqivlent to otining the soltion to the redced ordinry differentil eqtion Eq.(). introdce the nst the new independent vrile tht leds to the chnge of vriles: du du ( + ) d ξ d d U du d U Y ( + ) + ( + ) d d ξ d tn(ξ ) (5) d U du d U d U ( + )( + ) + 6 Y ( + ) + ( + ) d d ξ d d he next step is tht the soltion is expressed in the form U (6) m m i i (ξ ) i + i (7) i i where the prmeter m cn e fond y lncing the highest-order liner term with the nonliner terms in Eq.(), nd, α, β, ω,,,..., m,,..., m re to e determined. Sstitting Eq.(7) into Eq.() will yield set of lgeric eqtions for, α, β, ω,,,...,,,..., ecse ll m coefficients of hve to vnish. Hving determined these prmeters, nowing tht m is positive integer in most cses, nd sing Eq.(7) we otin nlytic soltions t ),in closed form. he trigonometric fnctions cn e extended to m hyperolic fnctions y sing the complex form. So tht tnh-fnction expnsion soltion genertes from tn fnction expnsion soltion for tn( iξ ) i tnh( ξ), nd cot-fnction expnsion soltion genertes from coth fnction expnsion soltion for cot( iξ ) i coth( ξ ).. Applictions In this section, we will ring to er the new tn- cot method discssed in Section to the (+)-dimensionl KP eqtion nd the (+)-dimensionl eqtion which re very importnt in the field of nonliner mthemticl physics... Exct Soltions to the (+)- Dimensionl KP Eqtion he ( + )-dimensionl KP-I eqtion is given y [9]: ( 6 + ) ( + ) (8) t + x xxx x yy zz his explins wve propgtion in the field of plsm physics, flid dynmics, etc. Soliton simltion stdies for Eq.(8) hve een done y Hsin et l [-], Sentorsi et l. [],nd Anwr et l. []. o stdy the trvelling wve soltions to Eq.(8), sstitte U ( ξ ), nd ξ x + α y + β z + ωt + θ into Eq.(8) nd integrting twice with zero constnts, we hve: U // + [ ω ( α ] U + U (9) we postlte tn series, nd the trnsformtion given in Eq.(5), so tht Eq.(9) redces to: du d U [ ( + ) + ( + ) ] + AU + U () d d where:

3 Interntionl Jornl of Mthemticl Anlysis nd Applictions ; (): 9-9 [ ω ( α ] A () Now, to determine the prmeterm, we lnce the liner term of highest-order with the highest order nonliner terms. // So, in Eq.() we lnce U with U, to otin:m+ m, then m. he tn-cot method dmits the se of the finite expnsion for : U nd nd U / + // U () // Sstitting U /,U from Eq.(7) in Eq.(5), + ( ( + + AU + ( + ( + + ( + ( ) + ( + ( ) + + ( + ) ) + ( )) ( + ) )) + ( ) + ) + i then eqting the coefficient of, i,,,,, -, -, -, - leds to the following nonliner system of lgeric eqtions: ) + ( + 6 ) + A( ) + ( ) ( ) + A + ( ) + + A + ( + + ) + ( + ) A + ( ) + + A + ( + + ) + ( + ) 6 + () Solving the nonliner systems of eqtions () we cn get the following soltions, ζ + tn ( ) + cot ( ζ ) j j,,..., ξ + αy + βz + ωt + θ x (), here re two fmilies of soltions: Fmily Where: With the following cses, A + ( α, ω

4 Anwr J'fr Mohmd Jwd: Extended tn-cot Method for the Solitons Soltions to the (+)-Dimensionl Kdomtsev-Petvishvili Eqtion Cse + ( α ω, ( ) +, {( + ) + (tn ( ζ ) + cot ( ζ ))} (5) For α β, θ, ω + 6 ( {( + ) + (tn ( ζ ) + cot ( ζ ))} (6) ξ x + y + z + ( + 6) t Fig.(). Represents the solitry soliton soltion ( in Eq.(6) for 55, 5, (,, t x Fig().Solitry soliton soltion ( in Eq.(6) for 55, 5, Cse ω + ( α ( ) {( ) + (tn ( ζ ) + cot ( ζ ))} (7) Cse + ( α ω + ( ) ( {( + ) + tn ( ζ ) + 6cot ( ζ )} (8) Cse ω + ( α ( ) {( ) + tn ( ζ ) + 6cot ( ζ )} (9)

5 Interntionl Jornl of Mthemticl Anlysis nd Applictions ; (): 9-9 Cse 5 ω + ( α + ( ) 5( {( + ) + (tn ( ζ ) + cot ( ζ ))} () Cse 6 ω + ( α ( ) {( ) + (tn ( ζ ) + cot ( ζ ))} () Cse 7 ω + ( α + ( ) 7 {( + ) + tn ( ζ ) + 6cot ( ζ )} () Cse 8 ω + ( α ( ) 8 {( ) + tn ( ζ ) + 6 cot ( ζ )} () Cse 9 + ( α ω (6 + ) 9 {(6 + ) + tn ( ζ ) + cot ( ζ )} () Cse + ( α ω (6 ) {(6 ) + tn ( ζ ) + cot ( ζ )} (5) Cse + ( α ω (6 + )

6 Anwr J'fr Mohmd Jwd: Extended tn-cot Method for the Solitons Soltions to the (+)-Dimensionl Kdomtsev-Petvishvili Eqtion Cse ( {(6 + ) + tn ( ζ ) + cot ( ζ )} (6) + ( α ω (6 ) Cse ω {(6 ) + tn ( ζ ) + cot ( ζ )} (7) 9 + ( α ( 9 ) + ( + 9 ) + tn ( ζ ) + cot ( )} { ζ (8) Cse Cse 5 ω 9 + ( α ( 9 ) [( 9 ) + tn ( ζ ) + cot ( )] (9) ζ 9 + ( α ω ( + 9 ) ( + 9 ) + tn ( ζ ) + cot ( )} 5 { ζ () Cse ( α ω ( 9 ) ( 9 ) + tn ( ζ ) + cot ( )} 6 { ζ () Fmily Where:, A + ( α, ω With the following cses:

7 Interntionl Jornl of Mthemticl Anlysis nd Applictions ; (): Cse 7 + ( α ω ( + ) 7 {( + ) + tn ( ζ ) + cot ( ζ )} () Cse 8 + ( α ω ( ) 8 {( ) + tn ( ζ ) + cot ( ζ )} () Cse 9 + ( α ω ( + ) 9 {( + ) + tn ( ζ ) + cot ( ζ )} () Cse + ( α ω ( ) {( ) + tn ( ζ ) + cot ( ζ )} (5) Cse ω 6 + ( α + ( 6 ) ( {( + 6) + tn ( ζ ) + cot ( ζ )} (6) Cse ω 6 + ( α ( 6 ) {( 6) + tn ( ζ ) + cot ( ζ )} (7) Cse ω 6 + ( α + ( 6 )

8 6 Anwr J'fr Mohmd Jwd: Extended tn-cot Method for the Solitons Soltions to the (+)-Dimensionl Kdomtsev-Petvishvili Eqtion Cse Cse 5 Cse 6 Cse 7 Cse 8 Cse 9 ( {( + 6) + tn ( ζ ) + cot ( ζ )} (8) ω ω ω 6 + ( α ( 6 ) {( 6) + tn ( ζ ) + cot ( ζ )} (9) + ( α ω ( + ) 5 {( + ) + tn ( ζ ) + cot ( ζ )} () + ( α ( ) 6 {( ) + tn ( ζ ) + cot ( ζ )} () + ( α ( + ) 7 {( + ) + tn ( ζ ) + cot ( ζ )} () + ( α ω ( ) 8 {( ) + tn ( ζ ) + cot ( ζ )} () + ( α ω ( 6 + ) 9 {(6 + ) + tn ( ζ ) + cot ( ζ )} ()

9 Interntionl Jornl of Mthemticl Anlysis nd Applictions ; (): Cse + ( α ω ( 6 ) {(6 ) + tn ( ζ ) + cot ( ζ )} (5) Cse + ( α ω ( 6 + ) {(6 + ) + tn ( ζ ) + cot ( ζ )} (6) Cse + ( α ω ( 6 ) {(6 ) + tn ( ζ ) + cot ( ζ )} (7) Reslts of solving (+)-dimensionl KP in this pper re comptile with tht reslts otined y Anwr et l. []... Exct Soltions to the (+)-Dimensionl Eqtion he ( + )-dimensionl eqtion []: + (8) xt x xy Sstitting t ) U ( ξ ),nd ξ x + α y + ωt + θ into Eq. (8) nd integrting once with zero constnt, we hve: y xx / + ω α xxxy /// / α (9) we postlte the following tn-cot series, nd the trnsformtion given in Eq.(),then Eq.(9) redces to: du d U d U du du α [( + )( + ) + 6 ( + ) + ( + ) ] + ω[( + ) ] α [( + ) ] (5) d d d d d Now, to determine the prmeter m, we lnce the liner term of highest-order with the highest order nonliner terms. / /// So, in Eq. (5) we lnce U with U, to otin m+ m+, then m. he tn-cot method dmits the se of the finite expnsion for : U + +, / U, / / U, /// U 6 (5) / / / / / / Sstitting U, U, U from Eq.(5) in Eq. (5),

10 8 Anwr J'fr Mohmd Jwd: Extended tn-cot Method for the Solitons Soltions to the (+)-Dimensionl Kdomtsev-Petvishvili Eqtion α α + + ( ) ( ) α ω 6 ( ) + [ + ] ( ) α ( ) ( ) ( ) α α α (5) then eqting the coefficient of i, i,,, -,- leds to the following nonliner system of lgeric eqtions: ( ) α α ω α α α + [ + ] + 6 α α ω α 8 ( ) α 6 α ω + α Solving the nonliner systems of eqtions (5) we cn get:,, ω 6 α + cot x + y 6 t + ( α α θ ) α θ For,, soltion in Eq.(5) is (5) the solitry + cot( x + y 6 (55) Figre () solitry soltion (, for y, -5 x 5, t 5. References (5) [] Wzwz A. M.,Prtil Differentil Eqtions nd Solitry Wves heor Higher Edction Press, Beijing nd Springer-Verlg Berlin Heidelerg,( 9). [] E. Yom, Chos, Solitons nd Frctls, 9,. [] Jwd, Anwr J fr Mohmd, Soliton Soltions for Nonliner Systems (+ )-Dimensionl Eqtions, IOSR Jornl of Mthemtics (IOSRJM) ISSN: vol. no. 6,(). [] Pres E. J., Oservtions on the tnh-coth expnsion method for finding soltions to nonliner evoltion eqtions, Appl. Mth. Compt. Vol. 7,(), pp [5] A. M.Wzw Mltiple-soliton soltions for the KP eqtion y Hirot s iliner method nd y the tnh coth method, Applied Mthemtics nd Compttion 9,(7), pp.6 6. (, t 5 Figre ().solitry soltion (, for -5 x 5, t 5. x 6 8 [6] Jwd, A.J.M., "New Exct Soltions of Nonliner Prtil Differentil Eqtions Using n-cot Fnction Method." Stdies in Mthemticl Sciences 5, no. (),pp.-5. [7] Anwr J'fr Mohmd Jwd, New Solitry wve Soltions of Nonliner Prtil Differentil Eqtions, interntionl jornl of scientific nd engineering reserch, Vol., (7), (), pp [8] Willy Heremn, Exct Soltions of Nonliner Prtil Differentil Eqtions he nh/sech Method, Wolfrm Reserch Inc., Chmpign, Illinois Octoer 5 Novemer, (). [9] Kznietsov E.A., Msher C.L., Exp J.. heor.phys. 6, (986) 97.. Conclsion he exct trvelling wve soltions to (+)-dimensionl KP nd (+)- dimensionl eqtions hve een stdied y mens of the extended tn-cot method. It cn e esily seen tht the implemented method sed in this pper is powerfl nd pplicle to lrge vriety of nonliner prtil differentil eqtions. [] Hsin Nher nd Frh Aini Adllh, New Approch of G /G expnsion method nd new pproch of generlized G /G expnsion method for nonliner evoltion eqtion, AIP Advnces,, 6; doi:.6/.7997,. [] Hsin Nher mil, Frh Aini Adllh, M. Ali Ar, Generlized nd Improved G /G expnsion method for (+)-dimensionl modified KdV-Zhrov-Kznetsev eqtion, Plos One, 8(5): e668. Doi:.7/Jornl. Pone.668,.

11 Interntionl Jornl of Mthemticl Anlysis nd Applictions ; (): [] Sentorsi A., Infed E., Phys. Rev. Lett. 77, (986), 855. [] Anwr J'fr Mohmd Jwd, Ali A. J. Adhm, Soliton Soltion to the (+)-dimensionl Kdomtsev-Petvishvili Eqtion y the nh-coth Method, Interntionl Jornl of Scientific & Engineering Reserch, Volme, Isse 9, (),pp. 5-.