SOLITONS AND CONSERVED QUANTITIES OF THE ITO EQUATION

Transcription

1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 3, Number 3/, pp. 5 4 SOLITONS AND CONSERVED QUANTITIES OF THE ITO EQUATION Ghodrt EBADI, A. H. KARA, Mrko D. PETKOVIC 3, Ahmet YILDIRIM 4, Anjn BISWAS 5 University of Tbriz, Tbriz, Fculty of Mthemticl Sciences, ; Irn University of the Witwtersrnd, School of Mthemtics, Wits 5, Johnnesburg, South Afrıc 3 University of Niš, Fculty of Science nd Mthemtics, Višegrdsk 33, 8 Niš, Serbı 4 Ege University, Deprtment of Mthemtics, 35 Bornov, Izmir, Turkey University of South Florid, Deprtment of Mthemtics nd Sttistics, Tmp, FL , USA 5 Delwre Stte University, Deprtment of Mthemticl Sciences, Dover, DE 99-77, USA E-mil: bisws.njn@gmil.com This pper obtins the soliton solutions of the Ito integro-differentil eqution. The GG / method will be used to crry out the solutions of this eqution nd then the solitry wve nstz method will be used to obtin -soliton solution of this eqution. Finlly, the invrince nd multiplier pproch will be pplied to recover few of the conserved quntities of this eqution. Key words: conserved quntities; solitry wve nstz method; solitons; Ito integro-differentil eqution; GG / method.. INTRODUCTION The study of nonliner evolution eqution (NLEE) hs been going on for the pst few decdes [-6]. During this time, there hs been mesurble progress tht hs been mde. There re lots of equtions tht hve been integrted. There re vrious methods of integrbility tht hs been developed so fr. In ddition to NLEEs, there hs been growing interest in the nonliner integro-differentil evolution equtions. Some of these commonly studied integro-differentil evolution equtions re the Ito eqution, the generlized shllow wter wve eqution nd mny others. There re vrious nlyticl methods of solving these NLEEs tht hs lso been developed in the pst couple of decdes. Some of these methods re ep-function method, Fn's F -method, Riccti's eqution method, Adomin decomposition method nd mny others. In this pper, the G' / G method will be developed to solve the Ito eqution. Also, the solitry wve nstz method will integrte the Ito eqution. Finlly, the conserved quntities for the Ito eqution will be computed using the invrince nd multiplier pproch bsed on the well known result tht the Euler-Lgrnge opertor nnihiltes the totl divergence.. THE G / G METHOD In this section, the G'/ G method will be described nd pplied to obtin trveling wve solution of the Ito eqution. This eqution is studied in () dimensions s well s () dimensions. therefore, this work will be done in two following subsections... () Dimensions The ()-dimensionl form of the generlized Ito integro-differentil eqution tht is going to be studied in this subsection is given by tt t ( t t ) t q q q q qq q q d = ()

2 6 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws Here, in (), q is the dependent vrible while nd t re the independent vribles. The coefficient is constnt. The eq. () cn reduced to with using the potentil or equivlently q = v v v v v v v v =, () tt t t t t v.the eq. () is converted to the ODE = (3) 4 ( v) ceu ceu ceu u ceuu ceu u cu e u e u 3 ( v) =, (4) by the wve vribles v= u( ξ), ξ= e ct, where primes denote the derivtives with respect to ξ, nd e, c re rel constnts to be determined lter. The eq. (4) is then integrted twice. This converts it to In cu e u e u 3 ' =, (5) G'/ G method, the solution u( ξ ) of eqution (5) is considered in the finite series form N G ( ξ) i u( ξ)= Ai, (6) G ( ξ) i= where A i re positive integers with AN tht will be determined. N is positive integer tht cn be ccomplished by blncing the liner term of highest order derivtive with the highest order nonliner term in eqution (5). Here blncing u with ( u ) gives N =. Therefore, we cn write the solution of eqution (5) in the form of G ( ξ) u( ξ )= A A, A. G( ξ) The function G( ξ ) in (6) is the solution of the uiliry liner ordinry differentil eqution G ( ξ ) λg ( ξ ) µ G( ξ )=, (8) where λ nd µ re rel constnts to be determined. By using (8) nd the generl solution of (6), we cn find u, u nd ( u ) s polynomils of G'/ G. Substituting the generl solution of (8) together with u, u nd ( u ) s polynomils of G' / G into eqution (5) yield lgebric equtions involving powers of G' / G. Equting the coefficients of ech power of G' / G to zero gives system of lgebric equtions for A, λ, µ, e nd c. Then, solving this system by Mple gives i 3 A =, c= e ( λ 4 µ ), (9) where e nd µ re the rbitrry constnts. Net, depending on the sign of the discriminnt = λ 4µ, the three types of the following trveling wve solutions of the eqution (4) re obtined. When = λ 4 µ >, the hyperbolic function trveling wve solution is (7) or equivlently u( ξ ):= A ( γh( ξ )) ()

3 3 Solitons nd conserved quntities of the Ito eqution 7 λ 4µ 3 =, = e e ( 4 ) t, γ q( ξ) := h( ξ)( h( ξ )), () csinh( γξ ) c cosh( γξ) where γ ξ λ µ h( ξ)=, nd A is n rbitrry c sinh( γξ ) ccosh( γξ) constnt. In () c, c re rbitrry constnts. If them re tken s specil vlues, the vrious known results 6λ in the literture cn be rediscovered, for instnce, setting c =, c =, e=, = nd α = A, c then the solution of () cn be written s v := c tnh ξ α ( ct) tht it is the solution (4) in [6]. 6λ Setting c =, c =, e=, = nd α = A, then the solution of () cn be written s c v := ccoth ξ α ( ct). From (5) we cn etrct other ect solutions of Eq. (5) reported erlier. When = λ 4 µ =, the rtionl function trveling wve solution is or equivlently where ξ = e, h( ξ)=, c cξ c u( ξ ):= A ( h( ξ )) () q( ξ):= h ( ξ ), (3) nd A is n rbitrry constnt. Finlly, when = λ 4 µ <, the trigonometric function trveling wve solution is or equivlently u( ξ ):= A ( γh( ξ )) (4) γ q( ξ ):= ( h ( ξ )), (5) 4µ where γ ξ µ constnt. 3 =, = e e (4 ) t, h c sin( γξ ) c cos( γξ) ( ξ)=, c sin( γξ ) c cos( γξ) nd A is n rbitrry.. () Dimensions The ()-dimensionl form of the generlized Ito integro-differentil eqution tht is going to be studied in this subsection is given by [5] qtt qt ( qqt qqt ) q qt d bqyt dqt =. (6) Here, in (6), q is the dependent vrible while, y nd t re the independent vribles. The coefficients b, nd d re constnts. The eq. (6) cn reduced to

4 8 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws 4 with using the potentil. The eq. (7) is converted to the ODE or equivlently v v v v v v v v bv dv =, (7) tt t t t t yt t ( ) cbef u cde u = (8) 4 ( v) c eu ce u ce u u ce u u ce u u (9) 3 ( v) ( c bf de) u e u e ( u ) =, by the wve vribles v= u( ξ), ξ = e fy ct, where primes denote the derivtives with respect to ξ, nd e, f nd c re rel constnts to be determined lter. The eq. (9) is then integrted twice. This converts it into () 3 ' ( c bf de) u e u e ( u ) =. Here blncing u with ( u ) gives N =. Therefore, we cn write the solution of eqution () in the form of G ( ξ) u( ξ )= A A, A. G( ξ) The function G( ξ ) in () is the solution of the uiliry liner ordinry differentil eqution (). Substituting the generl solution of () together with u, u nd ( u ) s polynomils of G'/ G into eqution () yield lgebric equtions involving powers of G'/ G. Equting the coefficients of ech power of G'/ G to zero gives system of lgebric equtions for Ai, λ, µ, e, f nd c. Then, solving this system by Mple gives where 3 A =, c= bf de e ( λ 4 µ ), () e, f nd µ re the rbitrry constnts. Net, depending on the sign of the discriminnt = λ 4µ, the three types of the following trveling wve solutions of the eqution () re obtined. When = λ 4µ >, the hyperbolic function trveling wve solution is () or equivlently u( ξ ):= A ( γh( ξ )) (3) 3 where ξ ( λ µ ) = e fy bf de e ( 4 ) t, γ q( ξ) := h( ξ)( h( ξ )), (4) h csinh( γξ ) c cosh( γξ) λ µ ( ξ)=, γ =, c sinh( γξ ) ccosh( γξ) 4 nd A is n rbitrry constnt. In (3) c, c re rbitrry constnts. If them re tken s specil vlues, the vrious known results in the literture cn be rediscovered, for instnce, setting 6λ c =, c =, e=, =, f =, c= b d ( λ 4 µ ) nd α = A, then the solution (7) cn be

5 5 Solitons nd conserved quntities of the Ito eqution 9 c ( b d) written s v ( ξ):= α c ( b d) tnh ( y ct), tht it is the solution (73) in [6]. Setting 6λ c =, c =, e=, =, f =, c= b d ( λ 4 µ ) nd α = A, then the solution of (7) cn be c ( b d) written s v ( ξ):= α c ( b d)coth ( y ct). From () we cn rediscover some other ect solutions of eq. (6). When = λ 4 µ =, the rtionl function trveling wve solution is or equivlently where ξ ξ = e fy, h=, c cξ c u( ξ ):= A ( h( ξ )) (5) q( ξ):= h ( ξ ), (6) nd A is n rbitrry constnt. Finlly, when = λ 4 µ <, the trigonometric function trveling wve solution is u( ξ ):= A ( γh( ξ )) (7) or equivlently γ q( ξ ) := ( h ( ξ )), (8) 3 where ξ ( µ ) = e fy bf de e (4 ) t, nd A is n rbitrry constnt. c sin( γξ ) c cos( γξ) 4µ γ c sin( γξ ) c cos( γξ) =, =, h ξ 3. THE ANSATZ METHOD Ito eqution is generliztion of the biliner Korteweg-de Vries (KdV) eqution. It is sometimes referred to s the etension of the NLEE of KdV or modified KdV (mkdv) type to higher order. This eqution is studied in () dimensions s well s () dimensions. This study will therefore be split in the following two subsections. 3. () Dimensions The dimensionless form of the Ito eqution in dimensions is given by [5] tt t t t t d = q q q q qq bq q. (9) Here in (9) nd b re ll constnts. In this section, the serch will be for -soliton solution to (9). The technique tht will be used to crry out the clcultions is the solitry wve nstz method. Therefore, the strting nstz for the -soliton solution to (9) is tken s

6 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws 6 qt (, )= Asech p τ, () where τ = B( vt). (3) Here in (), A is the mplitude of the soliton while B is the inverse width of the soliton. Also, v is the velocity of the soliton while the vlue of the unknown eponent p will be determined while deriving the solution to the Ito eqution. Substituting () into (9) yields p p pvabsech τ pp ( ) vab sech τ 4 4 p 4 p pvab τ pp p p vab τ sech sech p p p p vab τ p p va B τ 4 p p ( 3) sech sech τ τ sech p p p va B p va B sech p p bvp A B τ bvp p A B τ sech sech =. (3) From (38) equting the eponents p nd p gives p =. (33) It needs to be noted tht the sme vlue of p is obtined when the eponents p nd p 4 re equted with ech other. Now, from (3) setting the coefficients of the linerly independent functions sech p j τ for j =,,4 to zero yields v=4 B, (34) 4B A =, b (35) 6B A =. (36) 5 3b Now, equting the two vlues of A from (35) nd (36) yields the condition b =. In such cse, both epressions (35) nd (36) reduce to B A =. (37) Therefore the -soliton solution to Ito eqution is given by qt (, )= Asech [ B ( vt)], (38) where the mplitude-width reltion is given by (37) nd the velocity of the soliton is given by (44). 3.. () Dimensions The dimensionless form of the Ito eqution in () dimensions is given by [5] tt t t t t yt t q q ( q q qq ) bq q d cq dq =. (39) Here,, b, c nd d re ll constnts. In order to seek -soliton solution to (39) the strting nstz is still given by (), where in this cse τ = B B y vt. (4)

7 7 Solitons nd conserved quntities of the Ito eqution Here in (4), A represents the mplitude of the soliton, while B nd B represents the inverse width in the - nd y -directions respectively. Finlly, v represents the velocity of the soliton nd the eponent p, which is unknown t this point, will be determined in terms of n during the course of derivtion of the soliton solution. Substituting () into (39) gin yields pvasech pp va sech p τ τ p p pvabsech τ pp ( ) p p vab sech τ p p( p )( p )( p 3) vabsech τ p( p ) va Bsech τ p p p 4 p p p p va B sech τ p va B sech τ bvp A B sech τ bvp( p ) A B sech τ p p p p cvp AB sech τ cvp( p ) AB sech τ dvp AB sech τ dvp( p ) AB sech τ = Similirly s in the previous sub-section, the sme vlue of p s in (33) is yielded. Proceeding s in the () dimensionl cse, eqution (39) gives 3 (4) v= cb db 4 B, (4) 4B A =, b (4) 6B A =. (44) 5 3b Now, equting the two vlues of A from (43) nd (44) yields to the condition b = s in the previous subsection. In such cse, both epressions (43) nd (44) reduce to B A =. Finlly, the -soliton solution to (39) is given by qyt (,, ) = Asech B By vt (45), (46) where the mplitude A is relted to the width B of the soliton s given by (45) nd the velocity of the soliton is given by (4). 4. CONSERVATION LAWS In the following subsection, more conserved quntities re derived using the multiplier pproch. For this, we resort to the invrince nd multiplier pproch bsed on the well known result tht the Eulert Lgrnge opertor nnihiltes totl divergence. Tht is, if ( T, T ). is conserved vector corresponding to conservtion lw [] DT long the solutions of the differentil eqution in question, then t t D T =, (47) t E [ DT D T ]=, (48) q t where E q is the pproprite Euler-Lgrnge opertors. Moreover, if there eists nontrivil differentil function Q, clled multiplier, such tht for some conserved vector, then t Q.(eqution) = DT D T (49) t

8 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws 8 Eq [ Q.(eqution)] =, (5) from which knowledge of ech multiplier Q leds to conserved vector vi Homotopy opertor. The elborte nd tedious clcultions revel tht number of multipliers nd, hence, nontrivil conservtion lws (nd densities/conserved quntities) re vilble. In this cse, following (), the Euler opertor is 4.. () Dimensions E v nd multipliers obtined re Q = v, Q =, Q3 = t, Q4 = g( t ), where g( t ) is n rbitrry function including the cse Q =. We note tht there re no derivtive dependent multipliers upto third order. The corresponding conserved densities re, respectively, vi the homotopy opertor 3 Tt = ( vt v v 6vvt 9v 6 v (5vv 3 v ) v v 8 vv ), (5) 9 Tt = ( vt 5v vt 5vv 5vv 6v 5vv 6 v ), (5) 3 Tt = ( tvt v 5tv tvt 5tvv 6tv 5(4 v tv tv ) 6 tv ), (53) 4 t t T = ( g v g(v 5v v 5vv 6 v )). (54) The Lie point symmetry lgebr hs bsis X =, X =, X =, X = 3 t v. (55) Hence, the respective conserved quntities re t 3 v 4 t v 3 68AB Tt d =, (56) 5 T t d =, (57) A d =, (58) B 3 Tt T 4 t d =. (59) These conserved qutities re ll computed using the -soliton solution tht is given by (46). 4.. () Dimensions The condition on Q for the multidimensionl cse is bsed on (7) leding the zero th order multipliers Q = α( y) v b4 α ( y), Q = β( y), Q = γ( y bt), Q = δ ( t, y), (6) 3 4

9 9 Solitons nd conserved quntities of the Ito eqution 3 with corresponding densities ( 5 '' 6 6 Tt = b α v α vtv bα vyv dα v αv 36αv 4 v α v 7α v v t y t y 3 bα ( v bv dv v v b v v dv 5vv v 6 v ) 3(5 v bα (4d v v ) '' b α α vt bvy dv vv v 4( 5 ( )))), Tt = ( 5 bβ v 5 v ( bβ β ( v v )) β( v 5bv dv 5v v bv dv 5v v 6v 6 v )), t y t y 3 (5 5 ( ) Tt = bγ v v bγ γ v v γ( vt 5bvy dv 5v vt bvy dv 5v v 6v 6 v )), 4 Tt = ( 5 bδyv δ tv δ ( vt bvy dv 5 vv 5 vv 6 v )). Prticulr cses of these, for e.g., would be from choosing with densities p p p 3 Q = yv, Q = y, Q =( y bt), ( Pt = vtv bvyv dv v 6 vvt 9 bvv dvv vv v 9v v v 8v v 8 vv ), y ( Pt = yvt byvy bv dyv yv yvt byvy dyv 5yv v 6yv 5 v( b yv yv ) 6 yv ), 3 ( 5 Pt = bt y vt b tvy 5 byvy bdtv 5bv dyv 5btv 5yv btv yv b tv t t y byv bdtv dyv 5btv v 5yv v y 6btv 6yv 5 v( b ( bt y) v ( bt y) v ) 6btv 6 yv ). 5. CONCLUSIONS In this pper, the Ito eqution, which is nonliner integro-differentil evolution eqution is studied. First the G ' / G method is used to crry out the integrtion of this eqution nd its solutions re obtined. Subsequently the -soliton solution of the Ito eqution in nd dimensions re obtined. In this contet it ws proved thtthe soliton solution will eist provided the nonlocl term collpses to zero. This -soliton solution is then used to compute the non-trivil conserved quntities where the corresponding conserved densities re computed using the multiplier pproch.

10 4 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws ACKNOWLEDGMENTS The third uthor (Mrko D. Petkovic) grtefully cknowledge the support from the reserch project 44 of the Serbin Ministry of Science. REFERENCES. A. BISWAS, A. YILDIRIM, T. HAYAT, O. M. ALDOSSARY, R. SASSAMAN, Soliton perturbtion theory for thegenerlized Klein-Gordon eqution with full nonlinerity, Proceedings of the Romnin Acdemy, Series A, 3,, pp. 3 4,.. A. CEASER, S. GOMEZ, New trveling wves solutions to generlized Kup-Kuperschmidt nd Ito equtions, Applied Mthemtics nd Computtion, 6,, pp. 4 5,. 3. C. LI, Y. ZENG, Soliton solutions to higher order Ito eqution: Pfffin technique, Physics Letters A, 363,, pp. 4, D. L. LI, J-X. ZHAO, New ect solutions to the ()-dimensionl Ito eqution; Etended homoclinic test technique, Applied Mthemtics nd Computtion, 5, 5, pp , Q. P. LIU, Hmiltonin structures for Ito's eqution, Physics Letters A, 77,, pp. 3 34,. 6. A. M. WAZWAZ, Multiple-soliton solutions for the generlized ()-dimensionl nd the generlized ()-dimensionl Ito equtions, Applied Mthemtics nd Computtion,,, pp , 8. Received Mrch 5,