On Bilinear Equations, Bell Polynomials and Linear Superposition Principle

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1 Ameca Joua of Computatoa ad Apped Mathematcs 4, 4(5): 55-6 DOI:.59/.acam.445. O Bea Equatos, Be Poyomas ad Lea Supeposto Pcpe M. Y. Adamu *, E. Suema Mathematca Sceces Pogamme, Abubaka Tafawa Baewa, Uvesty, Bauch, gea Abstact A cass of bea dffeeta opeatos s dscussed. Some bea dffeeta equatos whch geeazed Hota bea equatos ae used though assgg appopate sgs. We fomuate a moe geea bea equato wth m D Ƥ opeatos usg dffeet atua umbes P 5. The esutg bea dffeeta equatos ae detfed by a speca kd of Be poyomas, ad aso the ea supeposto pcpe s apped to the costucto of the ea subspaces of souto. We have aso gve amoe eampes by agothm usg weghts of depedet vaabe. Keywods Bea Equatos, Be Poyoma, Supeposto Pcpe, -wave Soutos. Itoducto I ecet yeas, thee has bee much teest vestgatg dffeet kd of eact soutos of oea evouto equatos such as soto, posto, competo ad atoa soutos etc. Eact soutos pay a mpotat oe the study of oea physca pheomea. Fo eampe the wave pheomea obseved fud dyamcs, sotos the study of waves ad so o []. The eact soutos f avaabe, of those oea equatos ca factate the vefcato of umeca soves ad ad the stabty aayss of soutos. Howeve, vestgatg o estabshg eatos amog dffeet eact soutos s aso a vey mpotat topc. Sce these eatos ot oy povde a appoach to defomg eact soutos, but aso hep us to study the stuctues ad popetes of some compcated foms such as sotos [, ]. Hota peseted a dect method to sove a kd of specfc bea dffeeta equatos [4]; ad soto soutos ae despte the dvesty, a uvesa pheomeo that Hota bea equato descbe [4-6]. It s we kow that ude the Coe-Hope tasfomato u (og f) the Koteweg de-ves equato u + 6uu + u (.) t ca be tasfomed to F F 4F F + F + FF F F (.) whch eads t t * Coespodg autho: magaadamu78@yahoo.com (M. Y. Adamu) Pubshed oe at Copyght 4 Scetfc & Academc Pubshg. A Rghts Reseved (.) usg the Hota D-opeato. Usg the bea fom, the Woska soutos, cudg sotos, egatospostos ad competo, ae peseted fo some oea evouto equatos [7, 8]. The Hota D-opeato ae defed as m D D ( fg. ) 4 ( t ) m t t D + DD FF. (.4) f( t, ) g(, t ). t, t Fo eampe we have D f. g fg fg D f. g fg fg + fg (.5) D f. g fg fg + fg fg Though Hotab ea equatos ae speca, thee ae may othe dffeeta equatos whch caot be wtte the Hota bea fom. I ths pape, eghteed by the dea poposed [9], we ae gog to vestgate oe of the sevea questos posed the [9] I. e mg the D Ƥ -opeatos wth dffeet atua umbe Ƥ 5 ode to fomuate a moe geea bea equatos so as to shed moe ght o the study of some kd of geeazed bea dffeeta opeatos ad the coespodg bea equatos, whch have some ce mathematca popetes. Aothe mpotat ssue we ae aso gog to asceta s the ks betwee the bea equatos ad mutvaate Be epoeta poyoma ad the ea subspaces of soutos togethe wth the ea supeposto pcpe aso as dscussed [9].

2 56 M. Y. Adamu et a.: O Bea Equatos, Be Poyomas ad Lea Supeposto Pcpe. Bea Dffeeta Equato Opeatos ad Bea Equato.. Bea Dp opeatos I ths secto we w summazed the esseta facts o bea dffeeta opeatos ad bea equatos assocated wth the mutvaate poyoma as gve [9]. Some deftos eated to ths may aso be foud [, ] Let P be gve a atua umbe. We toduce bea dffeeta opeatos as foows: p ; + α ( D fg. )( ) ( ) f( g ) ( ) α ( f)( )( g)( ), Whee the powes of α ae detemed by () α ( ), whee () mode wth () Ρ, Ρ (.) (.) If p the obseve the oma devatve, whe the cases of P k, k є, educe to Hota bea opeatos. (see, e. g [9] how the powes of α ca be detemed ad the dffeece betwee the Hota bea dffeeta opeato [] ad the D p - opeatos)... Bea Equatos A bea dffeeta equato assocated wth a mutvaate poyoma [9] (,... ) s defe by F F FD (,..., D ) ff. (.) p p whch educes tohota bea equato f p k, k assumg p we patcuay have the geeazed bea KdV equato 4,, t +, ( D D D ) f. f t t f f f f + 6f The geeazed bea Boussesq equato 4, t, t t ( D + D ) f. f ff f + 6f (.4) ad the geeazed bea KP equato 4 ( D, td, + D, + D, y) f. f f f f f + 6f + f f f (.5) t t y y Such geeazed bea equatos have some commo chaactestcs: Bea: the eaest eghbous to ea equatos. I ths pape, we aso woud ke to dscuss wth moe eampes (othe tha that of [9]) that ca aso povde souto o: How ca oe detfy bea equatos defed by (.)? ad: What type of eact souto ae thee to bea equato defed by (.)? though the Be epoeta poyomas ad the ea supeposto pcpe espectvey.. Reatos wth Be Epoeta Poyomas ( a, a,..., a ) be a sequece of ea o compe Let umbes. Its pata (epoeta) Be poyoma B ( a, a,...) s defe as foows [, ] k. m t, (,...) t B ( ) k k a a am (.6) k! k! m m! the eact epesso ca be wtte as B! a a (.7) k k k, ( a, a,...) ( ) ( )... ( k, ) k! k!...!! Whee the sum s ove -tupes of oegatve teges ( k,..., k ) satsfyg the costat k+ k k... Bay Be Poyomas Hee aso a summay of the ma dea of the poposed wok w be stated ad some deftos egadg the Be poyomas whee two popetes used to k bea equatos to a speca kd of Be poyomas w be empoyed (see, fo eampe [9] fo the detas). We fst epoe a eato of the Be poyomas to the D -opeatos. Fo smpcty the computatoa pocedue p we assume ξ( ) η( ) Thus usg (wok), we have p, ( fg) D f. g f e, g e (.) α ( f f )( g g) α Y ( ) ( g ) Y ( η) (.) Y( y,..., y) y ξ + αη (.) whee ξ ξ ad η η, Smay to the case of the Hota D-opeatos [], we toduce bay Be Yp; (, vw) (.4) Y ( y, y,..., y ) y ( w+ v) + α ( w v),

3 Ameca Joua of Computatoa ad Apped Mathematcs 4, 4(5): Whee v v ad w w, As a eampe we have 5 (, vw) v 5,(, vw) v + w 5,(, vw) v + vw + v (.5) 4 54 (, ) 6 4 vw v + vw + vv + w + w (, ) 5 vw v + vw + vv + vw + 5vw + w v + w 4 5 ow settg w ( ξ + η), v ( ξ η) (.6) Fom (.4), we have a combatoa fomua fo the -opeatos f g ( f g) Dp, f. g p (, ( ) v w f g Dp (.7) We ow toduce Ƥ-poyomas ode to quafy the bea equatos Ρ ( q) (, q) (.8) p; p; The fst few of the whch the case of Ƥ5 ead I tems of Ρ ( q), 5; Ρ ( q) q, 5; Ρ ( q) 5; Ρ ( q) q + q 5;4 4 Ρ ( q) q 5;5 5 (.9) f q w v g, v ( ) (.) g The combatoa fomua (.7) becomes ( fg) Dp, fg. p; (, vv q) + (.) f f g, a eato betwee bea epessos ad the Ρ -poyomas w become f Dp, f f p ; ( ) q f Theefoe the bea equato. Ρ ( ) (.) F( Dp, ) f. f wth F( ) c (.) s equvaet to a equato ad a ea combato of Ρ -poyomas q f : cρ p; ( q f) (.4) Ths s a chaactezato fo ou geeazed bea equatos oe dmesoa case... Mutvaate Bay Be Poyomas Fo a c fucto y y (,..., ), we ca defe the vaabes as [9] y... y,... y t... (... ),..., ad the mutvaate Be poyomas,...,,...,,..., Y ( y) Y ( y ) y y e... e,,..., whch ca be computed though ep(... )... +,... t t +...!...!,...,... t t, +...!..!,..., Y y (.5) (.6) (.7) Thee eampes dffeeta poyomas fucto ae Y y + yy, t t t Y y + y + y y+ yy t, t, t, t t Y y + y y + y y + y y t, t, t t, t + y yy+ yy + yy t t t (.8) Based, o (.4) we ca show that the homogeous popety ,..., α,..., α,...,,..,. Y ( y ) y ( y ), (.9) ad the geea Lebtz ue ( fg)... fg... ( f... f)( g... g) Π (.) mpes the addto fomua fo the mutvaate Be poyomas:

4 58 M. Y. Adamu et a.: O Bea Equatos, Be Poyomas ad Lea Supeposto Pcpe I a smay mae [9] uses Y ( y y )... Y Y ( y),.. + Π ( ),..( ) ( y),.., (.) ξ(,.., ) η(,..., ) (.) f e, g e fo the sake of computatoa coveece ad by (.9) ad (.) we ca compute that p p ( fg) D,..., D f. g ξ ξ η... Π ( e... e )( e... e ) α... Πα Y( ),...,( ) ( ),..., ( ) ξ Y η Y( ),...,( ) ξ,..., Y t,..., α η,...,... Π,...,,..., (,...,,...,,..., ). Y y ξ + α η ( ) ( ) (.) Let us ow toduce bay mutvaabe Be poyomas dffeeta poyomas fom: Y p... ( V, w) ,.. ξ + α η Whee ( y ), (.4) w ( ξ + η), V ( ξ η) (.5) The fom (.) a combatoa fomua foows Ρ, Ρ, ( fg) D... D f. g. (.6) f Ρ;... ( V, w ( fg) g aga o empoyg the foowg mutvaabe -poyomas Ρ ( q) ( v, w q) (.7) Ρ;... Ρ; η... Fo eampe, we have Ρ ( q), 5 t, 5, ( ) t q qq t qq t Ρ + 5, ( ) 6 6 t q q qq t qqt qq t Ρ + + It ow foows that Ρ (.8) η Ρ, η Ρ, f D... D f. f Ρ ( q f ) (.9) Thus, a bea equato ( ) Ρ... ; FD (... D ) ff. wth p p F c (.) s equvaet to a equato o a ea combato of mutvaate Ƥ-poyomas... Ρ;... (.)... q f : c Ρ ( q f ) whee the coeffcets c... s ae costats. Ths s a chaactezato fo geeazed bea equatos as dscussed [9], defed though the D Ρ opeatos. 4. Lea Supeposto Pcpe 4.. Subspaces of Soutos F(,..., ) be a mutvaate poyoma. Let Cosde a bea equato FD (... D ) ff. (4.) p,, p, Defe a set of wave vaabes

5 Ameca Joua of Computatoa ad Apped Mathematcs 4, 4(5): θ k k,, (4.),, Whee the k, s ae costats, ad fom a bea combato of epoeta waves Whee a the θ k, k, ε ε, (4.) f e e ε s ae abtay costats ote that thee ae bea dettes of the fom θ FD (... D ) e. e p,, p, θ Fk ( + αk,..., k + αk ) e. e,,,,, θ θ (4.4) Whee the powes of α obey the ue (.). Foowg the cteo fo obtag the ea subspaces of soutos defed by [] (see [5] ad [6] fo detas) whch stated a theoem that suppot a abtay ea combato of epoeta waves ad a equvaet theoem o the ea subspaces of epoeta -wave soutos as [7] we ca use a type of paametezato fo wave umbes ad fequeces ad st the sequeta souto pocedues as foows: We toduce of the depedet vaabes: ( w ( ),..., w ( ) w,..., w, (4.5) whee the weght w s ca be both postve ad egatve. F(,..., ), defed Fom a homogeous poyoma by (.), some weght. Paameteze,,...,, k k usg a paamete w k : k, bk, (4.6) ad the deteme the popotoa costats the coeffcets c,..., s. b s ad 4.. Iustatve Eampes I ths secto a few cocete eampes w be gve to ustate the effectveess of the appoach. To peset ustatve eampes we cosde the (+)-dmesoa case wth as aga the (+)-dmesoa case as peseted [9] ( w ( ), w( y), wt ( )) ( w, w, w) (4.7) ad θ k + y wt, y t w (4.8) y w, t bk w bk, The upo fomg a homogeeous mutvaate poyoma some weght,,, (4.9) F c y t,,, we sove the system (4.4) ad (4.6) fo the popoto costats b, b, b ad the coeffcets c,, s ode that the coespodg bea equato ad the assocated ea subspace of soutos cosstg of ea combato of epoeta waves we w be ascetaed. We ae ow gog to peset two cocete ustatve eampes by appyg ths geea dea beow ode to shed moe ght o the geea dea Eampe Eampes wth postve weghts et us set the weghts of depedet vaabes ( w ( ), w( y), wt ( )) (,, 4) (4.) Ad cosde a poyoma beg homogeous weght 4 4 F c + c y + c t (4.) Foowg the paametezato of wave umbes ad fequeces (4.8), we set the 4 θ k + bk y+ bk z+ bk t, (4.) Whee k, ae the abtay costats but the b, b ad b ae to be detemed popotoa costats ow, a dect computato show that the coespodg bea equato eads FD (, D, D ) ff. 5, 5, y 5, t cff 8cf f + 6cf + c ff c f f y y (4.) The coespodg ea subspace of -wave soutos s gve by (4.4) 4 θ k + bk y+ bk z+ bk t f εe εe (4.4) whee ε, ae abtay costats but the b ad b ad ae defed by popotoa costats ae 4c 4c b, b c c (4.5) Eampe Eampe wth postve ad egatve weghts Let us set weghts of depedet vaabes ( w ( ), wy ( ), wz ( ), wt ( )) (,,) (4.6) ad cosde a poyomas beg homogeous weght F c + c y t + c yt (4.7)

6 6 M. Y. Adamu et a.: O Bea Equatos, Be Poyomas ad Lea Supeposto Pcpe Foowg the paametezato of wave umbes ad fequeces, we get the wave vaabes θ k + bk y+ bk t (4.8), whee k, ae abtay costat but the popotoa costat b ad b ae to be detemed by (4.5) Smay, a sma dect computato shows that the coespodg bea equato eads FD (, D, D ) ff. 5, 5, y 5, t c ff f c f + c ff 4c f f yyt ty y + c f f c f f + 4c f f t yt y t ty y cff + cff cf f yyt yt y t (4.9) ad t possess the ea subspace of -wave soutos detemed by θ k + bk y+ bkt ε ε (4.) f e e Whee the ε s ad k s ae abtay but b ad b eed to satsfy 5. Cocusos c c c cc b, b (4.) We have dscussed o the kd of the geeazed bea dffeeta opeatos, the k wth Be D Ρ, poyomas ad apped the ea supeposto pcpe to the coespodg bea equatos. We med the D Ρ, -opeato wth dffeet atua umbe p 5 to fomuate a moe geea bea equato as t was posed fo futhe vestgato [9]. REFERECES [] R. Hota (4) The dect method soto theoy. Cambdge, Cambdge Uvesty, Pess. [] P.P Godste; Hts o the Hota bea method, ACTA, Physcapoca A (6),(7) -5. [] R. Hota; Eact souto of the KdV equato fo mutpe cosos of soto. PhysLett. 7,(97) [4] R. Hota; Eact -soto souto of wave equatos of og wave o shaow wate ad oea attces. J. Math. Phys. 4 (7)(97) [5] M. Y. Adamu ad E. Suema; O the Geeazed Bea Dffeeta Equatos, IOSR Joua of Mathematcs, (4) () 4-. [6] M. Y. Adamuad E. Suema; O ea Supeposto pcpe Appyg to Hota Bea Equatos, Ameca oua of computatoa ad apped mathematcs ()() 8-. [7] W.X. Ma, J.-H. Lee; A tasfomed atoa fucto method ad eact soutos to the + dmesoa Jmbo Mwa equato, Chaos Sotos Factas4 (9) [8] A.M. Wazwaz; Mutpe-soto soutos fo a (+)-dmes oa geeazed KP equato, Commu.oea Sc. ume.smu. 7 () [9] We-Xu Ma; Bea equato, Be poyomas ad ea supeposto pcpe; J. of Phy. (4). [] SadekBououb, Mocef Abbas: ew dettes fo compete Be poyomas based o fomua of Ramaua. J. of t. Seq. (); 9 #p4. [] Dae Bmae, Jua B. G Mchae D. Wee; Some Covouto dettes ad a veseeato vovg pata Be poyomasthe eectoc oua of combatocs 9(4) (). [] W.X. Ma, T.W. Huag, Y. Zhag; A mutpe ep-fucto method fo oea dffeeta equatos ad ts appcato, Phys. Sc. 8 () 65.