THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 13, Number 1/2012, pp

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volue, Nuber /0, pp 4 SOLITON PERTURBATION THEORY FOR THE GENERALIZED KLEIN-GORDON EQUATION WITH FULL NONLINEARITY Anjan BISWAS, Ahe YILDIRIM, T HAYAT, Oar M ALDOSSARY 4, Ryan SASSAMAN 5 Deparen of Maheaical Sciences, Delaware Sae Universiy, Dover, DE , USA Deparen of Maheaics, Ege Universiy, 500 Bornova, Izir, TURKEY, Deparen of Maheaics Saisics, Universiy of Souh Florida, Tapa, FL , USA Deparen of Maheaics, Qaid-i-Aza Universiy, Islaabad-44000, PAKISTAN 4 Deparen of Physics, King Saud Universiy, P O Box 455, Riyadh-45, SAUDI ARABIA 5 Deparen of Maheaical Sciences, Delaware Sae Universiy, Dover, DE , USA Corresponding auhor Anjan BISWAS, E-ail: biswasanjan@gailco This paper sudies he perurbaion heory of he generalized Klein-Gordon equaion in presence of perurbaion ers ha appear wih full nonlineariy There are five fors of nonlineariy ha are considered for he Klein-Gordon equaion I is observed ha all five fors of nonlineariy lead o he sae srucure of he adiabaic dynaics of he solion velociy Key words: solions, perurbaion, adiabaiciy INTRODUCTION The Klein-Gordon equaion (KGE) appears in Theoreical Physics This equaion has been in exisence for he pas few decades [ 5] In fac, in Quanu Mechanics here are a leas a couple of versions of his 4 6 equaion ha has been sudied They are Φ odel he Φ odel [7] The generalized for of he KGE wih full nonlineariy has been recenly inegraed [0] This paper is going o carry ou he sudy of adiabaic dynaics of he velociy of he solion in presence of fully nonlinear perurbaion ers The solion perurbaion heory will be adoped o carry ou his sudy MATHEMATICAL ANALYSIS The generalized KGE (gkge) is odeled by he equaion q k q + F ( q )=0, () where he dependen variable qx (, ) represens he wave profile Also, k is a consan is a posiive ineger wih In fac, if =, equaion () reduces o he regular KGE In his paper, he following five fors of he funcion F( q ) will be considered F( q)= aq bq, () F( q)= aq bq, () n F( q)= aq bq, (4)

2 Solion perurbaion heory for he generalized Klein-Gordon equaion wih full nonlineariy n n F( q)= aq bq + cq, (5) n n+ F( q)= aq bq + cq (6) These five cases will be respecively labeled as Fors I-V In all of hese five fors, a, b c are real valued consans The KGE, given by (), has a leas wo conserved quaniies They are he oenu ( P ) he energy ( E ) ha are respecively given by P = q q d x (7) x E = ( ) d q q f q + + x x, (8) where f ( q ) is he ani-derivaive of F( q ) ha is given by q f ( q)= F( s)d s (9) Perurbaion Ters The perurbed generalized KGE ha is going o be sudied in his paper is given by q k q + F ( q )= ε R, (0) where ε is a perurbaion paraeer R represens he perurbaion ers In presence of perurbaion ers, he conserved quaniies do no say conserved Insead hey vary adiabaically Thus, he adiabaic variaion of he oenu ( P ) energy ( E ) are given by d P = ε ( q ) R d x d () x d E = ε ( q ) R d x d () The perurbaiion ers ha are going o be considered in his paper are given by R= α q + q + γ q +δ q +λ q +σ q +ν q () x x These perurbaion ers arise in he sudy of long Josephson juncions Here, in (), α is he loss er ha apperas in field heory, while accouns for dissipaive losses in Josephson juncion heory due o unneling of noral elecrons across he dielecric barrier, while σ accouns for losses due o a curren along he barrier Also, he perurbaion er due o γ is generaed by a sall inhoogeneous par of he local inducance while λ he capaciy inhoogeneiy The higher order spaial dispersion er is given by ν Here < 0 while σ >0 Thus, he perurbed KGE ha is going o be sudied is given by { } ( q ) k ( q ) + F q ε α q + ( q ) +γ ( q ) +δ ( q ) +λ ( q ) +σ ( q ) +ν ( q ) ( )= x x

3 4 Anjan Biswas, Ahe Yildiri, T Haya, Oar M Aldossary, Ryan Sassaan ADIABATIC DYNAMICS In his secion, he adiabaic dynaics of he solion velociy will be obained for all he five fors of nonlineariy The solion perurbaion heory will be herefore applied o each of he five fors of nonlineariy This sudy will be conduced in he following five subsecions For-I In his case by virue of equaions () (), he gkge is given by q k q + aq bq =0 (4) The -solion soluion of (5) is given by A qx (, )=, (5) cosh [ B( x v)] where he apliude A he widh B are respecively given by a A = b (6) For (5), he oenu ( P ) is given by a B = 4 ( k v ) (7) 8 aa v P = (8) 5 k v The perurbed gkge is given by q k q + aq bq = ε R, (9) where R is in () The adiabaic variaion of he solion velociy, in presence of hese perurbaion ers is ( k v ) dp ε( k v ) 5 5 = = ( q ) Rd x d d x 8k aa 8k aa Thus, for he perurbaion ers given by (), equaion () reduces o (0) Now, separaing variables, () gives ε γ 5σa γκ d k 7 = v v + k v+ k = ε γ 5σa γκ v v + k v+ 7 Equaion () will now be sudied based on he srucure of he roos of he cubic in v ha is locaed in he denoinaor of he righ h side of () This is: () ()

4 4 Solion perurbaion heory for he generalized Klein-Gordon equaion wih full nonlineariy 5 γ 5σa γκ v v k v () Therefore here are four possible cases ha can arise in his siuaion They are as follows: Case Suppose he cubic given by (4) has hree real disinc roos, v, v v In his case () reduces o ε =, (4) ( )( )( ) k v v v v v v where v v v = γ + +, (5) 7σa vv vv vv k =, (6) = γk vv v (7) Thus, () inegraes o v v u v v v u v v v u v k = ( )ln + ( ) ln + ( ) ln, ε ( v v)( v v)( v v) v v v v v v where v ( = 0) = u represens he iniial velociy of he solion This is an iplici soluion i shows ha he velociy will exponenially decay depending on he sign of Case Suppose he cubic given by (4) has one real roo of ulipliciy wo In ha case, equaion () can be resrucured as ε =, k (8) ( v v ) ( v v ) where v is he real roo of ulipliciy wo v is he real roo of ulipliciy one This leads o k ( u v)( v v) ( v u)( v+ u v) = ln + ε( v v) v v ( u v)( v v) ( u v) ( v v) In his case he soluion is iplici oo (9) Case In his case, he assupion is ha here is one real roo of ulipliciy hree Hence () odifies o ε =, k (0) ( v v ) where v is he real roo of ulipliciy wo Therefore his gives v ku ( v) = v + k ε( u v ) ()

5 6 Anjan Biswas, Ahe Yildiri, T Haya, Oar M Aldossary, Ryan Sassaan 5 This is he case where he soluion is explici in v In his case, liv ()= v () which shows ha he liiing value of he solion velociy is v for large ie Case 4 For his case, he assupion is ha he cubic in v has one real roo wo iaginary roos Thus, () reduces o where v is he real roo ( ) ε =, k () ( v v ) v + c ( ) c is he produc of he iaginary roos This for inegraes o k v + c u v u + c v cu ( v) = ln ln an ε v v v + c c uv+ c Thus he adiabaic behavior of he solion velociy will be one of he foor ypes depending on he values of he perurbaion paraeers (4) For-II In his case, equaions () () ogeher gives The -solion soluion o (7) is given by q k q + aq bq =0 (5) A qx (, )=, cosh [ B( x v)] where he apliude he widh are respecively given by (6) For his for, he oenu is given by a A = b a B = ( k v ) (7) (8) A v a P = ( k v ) In his case, he perurbed gkge is herefore given by q k q + aq bq ε R (40) = so h he adiabic variaion of he solion velociy is governed by (9)

6 6 Solion perurbaion heory for he generalized Klein-Gordon equaion wih full nonlineariy 7 ( k v ) dp ( k v ) = = ( q ) Rdx d d x aa k aa k For he perurbaion ers given by (), equaion (4) reduces o (4) k = ε γ 57σa γκ v v + k v+ 5 This leads o he sae four cases of inegraion as in For-I The only difference is he coefficien of he v er in he cubic in v (4) For-III Here, equaions () (4) ogeher iply q n k q + aq bq =0 (4) This is he generalized for of he nonlinear KGE The special case = reduces o he firs ype of nonlinear KGE ha was sudied along wih is perurbaion ers [0] In paricular he case = wih 6 n = is called he Φ odel ha appears in solid sae Physics, Condensed Maer Physics as well as Quanu Field Theory [6] Also one resricion is ha n The -solion soluion o (45) is given by A qx (, )=, cosh n [ B( x v)] (44) where he apliude he inverse widh are respecively given by an ( + ) A = b n (45) an ( ) B = 4 k v ( ) (46) The oenu of he solion is given by Γ aa v Γ n P = ( n ) ( k v ) Γ + n (47) When he perurbaion ers are urned on, (45) changes o q n k q + aq bq = ε R, (48) so ha he adiabic variaion of he solion velociy is governed by

7 8 Anjan Biswas, Ahe Yildiri, T Haya, Oar M Aldossary, Ryan Sassaan 7 ( n ) ( k v Γ ) dp + ( n ) ( k v Γ ) + = n ε = n ( q ) Rd x d d x aa k aa k (49) Γ Γ Γ Γ n n For he perurbaion ers given by (), equaion (5) inegraes o k =, ε γ 4 σ a( + n)( n ) γκ (50) v v + k v+ ( n+ ) which leads o he sae analysis conclusions as in he previous wo fors of nonlineariy 4 For-IV In his case, equaions () (5) ogeher iplies q k q + aq bq + cq (5) n n =0 In his case, he -solion soluion is given by [0] qx (, )= A ( ) D+ cosh[ B( x v )] n where he apliude ( A ) he inverse widh ( B ) are respecively given by, (5) n a( n + ) D A =, b (5) while he consan D is a B=( n ), ( k v ) b n D = nb ac( + n) (54) (55) The oenu of he solion is given by A v a n D P= F,, ;, + B, n n n n n ( n ) v k (56) where in (58), F( uvwz, ; ; ) is he Gauss' hypergeoeric funcion defined as n Γ( w) Γ ( u + n) Γ ( v+ n) z Fuvwz (, ; ; )= ( u ) ( v ) (57) Γ Γ Γ ( w+ n ) n! n=0 B( uv, ) is he bea funcion Γ ( u) is he Euler's gaa funcion I is clear fro (58) ha i is necessary o have n (58)

8 8 Solion perurbaion heory for he generalized Klein-Gordon equaion wih full nonlineariy 9 for he exisence of solions Wih he perurbaion ers are urned on, (5) changes o q k q aq bq n cq n + + = ε R, (59) so ha he adiabic variaion of he solion velociy is governed by n n k v ( n ) dp k v ( n ) = = ( q ) Rd x d A k am d A k am x This leads o he adiabaic variaion of he solion velociy being given by d = k v ε γ γκ + ( ) + v v J k v (60) (6) where σa nm ( n ) M nn ( M ) 4 n( n ) M5 J = ( n ) + +, (6) M 4M M M wih n D M = F,, + ; B,, n n n n 4n 5 D M = F,, + ; B,, n n n n n n 5 D n 5 M = F,, + ; B,, n n n n n 5 D 5 M4 = F,, + ; B,, n n n n n n 5 D n M5 = F, +, ; B + +, n n n n Thus, (6) leads o he sae adiabaic dynaics of he solion velociy hrough hose sae four cases (6) (64) (65) (66) (67) 5 For-V In his case, equaions () (6) ogeher iply ha he generalized KGE is q k q aq bq n cq n+ + + =0 (68) The opological -solion soluion o (70) is given by where he free paraeers A B are given by qx (, )= Aanh n[ Bx ( v)], (69) 4b n A = ( n ) a (70)

9 40 Anjan Biswas, Ahe Yildiri, T Haya, Oar M Aldossary, Ryan Sassaan 9 an B = 8 v k ( ) In addiion, he consrain relaion beween he nonlinear coefficiens a, b, c he exponens n given by ( ) (7) 4 a + 4 bc = n a (7) us hold in order for he solions o exis The oenu of he solion is given by 8vA B P = (4 n)(4 + n) Equaions (7) (75) iply ha n ± 4 (74) Wih he perurbaion ers, equaion (70) odifies o q k q aq bq n cq n+ + + = ε R (75) Thus he adiabaic variaion of he solion velociy in his case is given by ( ) ( ) d v ε γ 8 6 = σan n n + γκ v v + k v + d k 6 9n Wih he perurbaion ers given by (), he law of adiabaic variaion of he solion velociy akes he for k =, ε ( ) γ 8 6 σan n n + γκ v v + k v+ (77) ( 6 9n ) which again leads o he four cases as observed in he firs for of nonlineariy I needs o be noed ha equaions (78) (79) iply ha n ± 4 (78) for he solion perurbaion heory o be valid, in his case (7) (76) 4 CONCLUSIONS This paper sudied he solion perurbaion heory of he gkge in presence of fully nonlinear perurbaion ers The key observaion is ha he srucure of he adiabaic dynaics of he solion velociy says he sae as ha of he regular KGE wih is perurbaion ers ha was sudied in 009 [9] This siuaion is siilar o he opological solions ha are observed in he case of sine-gordon equaion is generalizaion o full nonlineariy [6, 8] This is a very iporan observaion for he case of KGE ha is being ade for he firs ie in his paper In fuure, he quasi-saionary solion soluion of he gkge will be obained by he ehod of uliple scale perurbaion heory Those resuls will be repored in fuure ACKNOWLEDGMENTS The firs, hird fourh auhors (AB, TH & OMA) would like o recognize hankfully appreciae he suppor fro King Saud Universiy (KSU-VPP-7)

10 0 Solion perurbaion heory for he generalized Klein-Gordon equaion wih full nonlineariy 4 REFERENCES K C BASAK, P C RAY, R K BERA, Soluion of non-linear Klein-Gordon equaion wih a quadraic non-linear er by Adoian decoposiion ehod, Counicaions in Nonlinear Science Nuerical Siulaion, 4,, pp 78 7, 009 A BISWAS, C ZONY, E ZERRAD, Solion perurbaion heory for he quadraic nonlinear Klein-Gordon equaions, Applied Maheaics Copuaion, 0,, pp 5 56, 008 G CHEN, Soluion of he Klein-Gordon for exponenial scalar vecor poenials, Physics Leers A, 9, -5, pp 00 0, A EBAID, Exac soluions for he generalized Klein-Gordon equaion via a ransforaion Exp-funcion ehod coparison wih Adoian's ehod, Journal of Copuaional Applied Maheaics,,, pp 78 90, A ELGARAYAHI, New periodic wave soluions for he shallow waer equaions he generalized Klein-Gordon equaion, Counicaions in Nonlinear Science Nuerical Siulaion,, 5, pp , A L FABIAN, R KOHL, A BISWAS, Perurbaion of opological solions due o sine-gordon equaion is ype, Counicaions in Nonlinear Science Nuerical Siulaion, 4, 4, pp 7 44, D FENG, J LI, Exac explici ravelling wave soluions for he (n+)-diensional Φ 6 field odel, Physics Leers A, 69, 4, pp 55 6, S JOHNSON, A BISWAS, Topological solion perurbaion of sine-gordon equaion wih full nonlineariy, Physics Leers A, 74, 4, pp , 00 9 R SASSAMAN, A BISWAS, Solion perurbaion heory for phi-four odel nonlinear Klein-Gordon equaions, Counicaions in Nonlinear Science Nuerical Siulaion, 4, 8, pp 6 9, R SASSAMAN, A BISWAS, Topological non-opological solions of he generalized Klein-Gordon equaions, Applied Maheaics Copuaion, 5,, pp 0, 009 R SASSAMAN, A BISWAS, Topological non-opological solions of he Klein-Gordon equaions in + diensions, Nonlinear Dynaics, 6,, pp 8, 00 R SASSAMAN, A HEIDARI, F MAJID, A BISWAS, Topological non-opological solions of he generalized Klein- Gordon equaions in + diensions, Dynaics of Coninuous, Discree Ipulsive Syses: Series A, 7, a, pp 75 86, 00 R SASSAMAN, A HEIDARI, A BISWAS, Topological non-opological solions of he nonlinear Klein-Gordon equaion by He's sei-inverse variaional principle, Journal of Franklin Insiue, 47, 7, pp 48 57, 00 4 A M WAZWAZ, The anh sine-cosine ehods for copac noncopac soluions of he nonlinear Klein-Gordon equaion, Applied Maheaics Copuaion, 67,, pp 79 95, Y ZHENG, S LAI, A sudy of hree ypes of nonlinear Klein-Gordon equaions, Dynaics of Coninuous, Discree Ipulsive Syses: Series B, 6,, pp 7 79, 009 Received Noveber 8, 0