THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A OF THE ROMANIAN ACADEMY Volume 4 Number /03 pp. 8 34 CNOIDAL WAVES SOLITARY WAVES AND PAINLEVE ANALYSIS OF THE 5TH ORDER KDV EQUATION WITH DUAL-POWER LAW NONLINEARITY Alvaro SALAS Sachi KUMAR Ahmet YILDIRIM 3 Aja BISWAS 4 Uiversidad de Caldas Departmet of Mathematics Uiversidad Nacioal de Colombia Departmet of Mathematics Maizales Colombıa Thapar Uiversity School of Mathematics ad Computer Applicatios Patiala-47004 (Pujab) Idıa 3 Ege Uiversity Departmet of Mathematics 3500 Borova Izmir Turkey Uiversity of South Florida Departmet of Mathematics ad Statistics Tampa FL 3360-5700 USA 4 Delaware State Uiversity Departmet of Mathematical Scieces Dover DE 990-77 USA E-mail: biswas.aja@gmail.com This paper studies the 5 th order KdV equatio with dual-power law oliearity. The o-topological as well as the sigular solito solutio is obtaied. Additioally the coidal wave solutio is obtaied for two values of the power law parameter. Fially the Paileve aalysis of the equatio is also discussed. Key words: 5 th order KdV equatio; coidal wave solutio; Paileve aalysis.. INTRODUCTION There has bee a overwhelmig amout of research that has bee goig o i theory of oliear evolutio equatios (NLEEs). These NLEEs appear i various areas of scietific research amely i Applied Mathematics Theoretical ad Applied Physics Noliear Dyamics Mathematical Biology Mathematical Egieerig as well as several other areas. Oe of the essetial targets i this area of research is to carry out a itegratio of these NLEEs to extract several iterestig solutios. These solutios will be very useful i various practial applicatios. There are several tools that have bee developed i the past couple of decades to itegrate these NLEEs [-0]. Some of these tools are the Adomia decompositio method semi-iverse variatioal priciple homotopy perturbatio method exp-fuctio method simplest equatio method G / G-expasio method ad several others. These techiques of itegratio reveals several iterestig ad importat solutios to various NLEEs. Some of these solutios are coidal waves soidal waves shock waves solitos ad solitary waves compactos covatos pekos cuspos stumpos just to ame a few. This paper will focus o a particular NLEE amely the 5th order Korteweg-de Vries (KdV) equatio with dual-power law oliearity. The coidal waves ad o-topological solitary waves solutios will be extracted for this NLEE amely the 5th order KdV equatio with dual-power law olieartity.. MATHEMATICAL ANALYSIS Let us cosider the equatio [7] t x xxx xxxxx q + ( aq + bq ) q + cq + kq =0 () where q= q( x t ) is the ukow fuctio ad a b c k ad >0 are costats with ab > 0. Without loss of geerality we may suppose that a= b =. Ideed the scalig / a b qxt ( ) = Q( ξτ ) where ξ= x ad τ= t. b a ()
Coidal waves solitary waves ad Paileve aalysis 9 gives us Equatio () i the form 3 5 b c b k τ ξ 6 ξξξ 0 ξξξξξ Q + ( Q + Q ) Q + Q + Q =0. (3) a a I what follows we will study followig equatio q + ( q + q ) q + cq + kq =0. (4) t x xxx xxxxx Let / / qxt ( ) = u ( Bx ( vt)) = u ( ζ) where ζ= Bx ( vt). (5) Replacig (5) ito (4) ad simplifyig we obtai followig oliear ode : '' ' '' ''' (( 5 3 ) 0 ) '' ' ''' ' ' ''' ' ' ' =0. 4 3 ' 3 '' 4 4 3 ' 5 0B k 3 + 4 ( u ) u u + 4B k 6 5 + 35 0+ 4 ( u ) 3 (4) 3 B B ku + cu u + B ku u u + + B 3+ 5 B k( u ) + 0 B ku u + c( u ) u u + + B ku + B cu vu u + u u + u u 4 4 (5) 4 4 8 4 6 (6).. Sigular Solitos We seek solutios to Equatio (6) i the form u( ζ)= Acsch( ζ ). (7) Isertig asatz (7) ito (6) ad after some algebra we obtai followig algebraic equatio i the ukow z = log( ζ ) : 4 4 3 4 6 4 4 64B k 8B c 4B c 4 v) z (( ( 8A 4A 3v) 6B k( 0 40 50 0 3) 4B c ( 6 )) z 4 4 8 4 4 4 4 3 4 4 6B k + 4 B c v z + (4A + 3B k + 60B k + 30B k + 30B k + + + + + + + 4 4 3 4 + + + + + + + + 4 4 4 4 3 4 4 4 + (4A + 3B k + 60B k + 30B k + 30B k+ 64B k + 4 3 4 4 4 + 8B c + 4B c + 4 v) z + (6B k + 4 B c v)=0. (8) Equatig to zero the coefficiets of the differet powers of z gives followig algebraic system : + 4 4 6B k 4 B c v= 0. + + + + + + + + 4 4 4 4 3 4 4 4 4 3 4 4A 3B k 60B k 30B k 30B k 64B k 8B c 4B c 4 v = 0. ( ) + ( + + + + ) ( + + ) 4 4 4 4 3 ( 8A 4A 3v 6B k 0 40 50 0 3 4B c 6 = 0. Solvig it with the aid of either Mathematica or Maple we obtai solutios. These are : 4 B (4 B k + c ) A= ± α B= ± β ad v= 4 where 4 3 Rck ( + 5 + 0 + 0 + 4) R α = ± 3 4 3 k ( + + ) ( + )( + ) (9) (0)
30 Alvaro Salas Sachi Kumar Ahmet Yildirim Aja Biswas 3 ( + 5 + 0 + 0 + 4) β = ± 4 3 R ck 4 k ( + + ) ( + )( + ) () 3 4 R= k ( + + ) 3+ + + +. (). No-Topological Solitos The same procedure is valid for the asatz u( ζ) = Asech( ζ ). (3) I this case the correspodig system is 4 4 6 4 =0 B k + B c v 4 4 4 3 8 ( 5 0 0 4) ( 3 ) =0 ( + ) + ( + + + + ) ( + + ) A + B k + + + + + B c + + 4 4 4 3 A A B k B c ad the solutios are where 4 8 45 60 30 4 3 =0 4 B ( B k + c ) ' ' A= ± α B= ± β ad v= (4) 4 β α ' ' ( 5 0 0 4) 3 ( + + ) ( + )( + ) 4 3 ± Sck + + + + + S = 3 4 k ( 5 0 0 4) k ( + + ) ( + )( + ) 4 3 ck + + + + S = 4 (5) (6) 3 4 4 3 S = k + + 6 + 5 + 35 + 0+ 4. (7) 3. CNOIDAL WAVES Equatio () admits coidal wave solutios i the special cases = = Equatio () takes the form 3.. Case I: = t x xxx xxxxx =0. = ad =. 3 q + q + q q + cq + kq (8) After the travelig wave trasformatio (5) the correspodig oliear equatio (6) coverts to 3 5 ' 4 '' ''' '' (4) ' 4 (5) ''' ' ( u u ) u 0 B ku u (3cu 5 B ku ) B u ( B ku B cu vu ) u =0. + + + + + + (9) We seek solutios to Equatio (9) i the coidal wave form u = u( ζ)= Ac( ζ m). (0) Isertig asatz (0) ito (0) gives followig algebraic equatio i the variable z= s( ζ m) :
4 Coidal waves solitary waves ad Paileve aalysis 3 4 4 4 4 4 ( + 360 ) + ( ( 0 ( + ) ) + = 0 A B km z B m c B k m A A z C () 4 4 4 where C = A + A 4 B c( + m ) + 8 B k(m + 3m + ) v. 4 0 Equatig the coefficiets of z z ad z to zero ad solvig the obtaied algebraic system gives followig solutios : 3 3 m (5 k ( m ) + 0 k (m c) A = 0 k (m () 3 ck( m ) ± 0 k (m B = ± 0 k (m (3) 4 v= A ( A + ) + 4 B ((m + 3m + ) B k ( m + ) c). (4) We must choose the parameters c m ad k adequatelly i order to get a real valued fuctio u. For example let us cosider the choice A = m k m k m c 3 3 (5 ( ) + 0 ( ) 0 k (m ) ad 3 ck( m ) 0 k (m B =. 0 k (m (5) These umbers are real if c c> 0 ad < k < 0 ad m >. (6) 5 O the other had observe that we obtai a topological solito i the limit whe m sice limc( ζ m) = sech( ζ ). (7) m Equatio () takes the form 3.. Case II: = q + q + q q + cq + kq (8) 4 t x xxx xxxxx =0. After the travelig wave trasformatio (5) the correspodig oliear equatio (6) coverts to 4 ' ''' 4 (5) ( u u v) u B cu B u =0. Itegratig equatio (9) oce with respect to z gives + + + (9) 4 (4) '' 5 3 5B ku 5B cu 3u 5u 5 vu = 0 + + + (30) with costat of itegratio equal to zero. We seek solutios to Equatio (30) i the coidal wave form u = u( ζ)= Ac( ζ m). (3) Isertig asatz (3) ito (30) gives followig algebraic equatio i the variable z = s ( ζ m) :
3 Alvaro Salas Sachi Kumar Ahmet Yildirim Aja Biswas 5 4 4 4 4 4 3( + 0 ) + (30 ( ( + 5)) 5 6 ) + = 0 A B km z B m c B k m A A z D (3) 4 4 where D= 5A + 3A 5B c+ 5 B k(4m + ) 5 v. 4 0 Equatig the coefficiets of z z ad z to zero ad solcvig the obtaied algebraic system gives followig solutios : 3 0 k ( m ) + 30 k ( m ) c A = ± 5 k ( m ) (33) 3 3 ck( m ) ± 30 k ( m ) c B = 30 k ( m ) (34) 4 4 v= (5A + 3A + 5 B ( B k c) + 60 B km ). (35) 5 We must choose the parameters c m ad k adequatelly i order to get a real valued fuctio u. For example let us cosider the choice 3 ( ) 30 ( ) 0 ( ) + 30 ( ) A= B= ad 30 k ( m ) 5 k ( m ) 3 3 ck m k m c k m k m c (36) These umbers are real if c> 0 ad 3c < k < 0 ad 0 m <. (37) 4. PAINLEVE ANALYSIS The Paileve as a test for itegrability of partial differetial equatios (PDEs) was proposed by Weiss Tabor ad Carevale i 983 [7]. It is a geeralizatio of the sigular poit aalysis of ordiary differetial equatios (ODEs) which dates back to the work of S. Kovalevsky i 888. A PDE is said to possess the Paileve property if solutios of the PDE are sigle-valued i the eighbourhood of o-characteristic movable sigularity maifolds. Usig the stadard Kruskal s simplified method we expad the solutio q about a sigular maifold φ ( xt ) = 0 i a ifiite series α j q= φ q jφ (38) j=0 where φ ( x t) = x+ψ ( t) q 0 0 ad α is egative iteger determied by balacig the powers of φ of domiat terms i the equatio. φ is a o-characteristic maifold. Coefficiets q j are fuctios of x ad t. There are basically three steps i the Paileve aalysis viz domiat behaviour aalysis fidig the resoaces ad checkig whether arbitrary coefficiets eter at the resoace values [7]. From the domiat behaviour aalysis of equatio () we get α = where > 0. Substitutig (38) with α= ito () leads to
6 Coidal waves solitary waves ad Paileve aalysis 33 q ( + )( + )( + )( + ) 0 4 4k 3 =. Substitutig (38) ito () it is foud that resoaces occur at (39) 3 5 40 6 + ± ( + ) 4( + ) j =. There exist complex resoace poits so we coclude that equatio () is failed i the Paileve test. (40) 5. CONCLUSIONS This paper studied the 5th order KdV equatio with dual-power law oliearity. The coidal wave solutio as well as solito solutios were obtaied. The two types of solito solutios obtaied are the otopological solito ad the sigular solitos. Additioally the Paileve aalysis was carried out where the resoace values are also discussed. I future further aalysis will be carried out for this equatio. While the adiabatic parameter dyamics was already obtaied [7] the quasi-statioary solito solutios will also be obtaied by the aid of multiple scale aalysis. The semi-iverse variatioal priciple will be applied to itegrate the perturbed 5th order KdV equatio. Such results will be reported i future. REFERENCES. E. ALIBEIGI A. NEYRAME Aalytical sudy o oliear fifth order Korteweg-de Vries equatio World Applied Scieces Joural 0 4 pp. 440 44 00.. A. BISWAS A. YILDIRIM T. HAYAT O. M. ALDOSSARY R. SASSAMAN Solito perturbatio theory for thegeeralized Klei-Gordo equatio with full oliearity Proceedigs of the Romaia Academy Series A 3 pp. 3 4 0. 3. M. CHUGUNOVA D. PELINOVSY Two-pulse solutios i the fifth order KdV equatio: Rigorous theory ad umerical approximatios Discrete ad Cotiuous Dyamical System 8 4 pp. 773 800 007. 4. F. COOPER J M. MCHYMAN A. KHARE Compacto solutios i a class of fifth-order Korteweg-de Vries equatio Physical Review E 64 pp. 06608 00. 5. Q. FENG B. ZHENG Travelig wave solutios for the fifth-order KdV equatio ad the BBM equatio by G '/ G -expasio method WSEAS Trasactios o Mathematics 9 3 pp. 7 80 00. 6. I. L. FREIRE J. C. S. SAMPAIO Noliear self-adjoitess of a geeralized fifth-order KdV equatio Joural of Physics A 45 3 pp. 030 0. 7. L. GIRGIS A. BISWAS Solito perturbatio theory for oliear wave equatios Applied Mathematics ad Computatio 6 7 pp. 6 3 00. 8. D. KAYA S. M. EL-SAYED O a geeralized fifth order KdV equatio Physics Letters A 30 pp. 44 5 003. 9. E. V. KRISHNAN Q. J. A. KHAN Higher-order KdV-type equatios ad their stability Iteratioal Joural of Mathematics ad Mathematical Scieces 7 4 pp. 5 0 00. 0. A. RAMANI B GRAMMATICOS T BOUNTIS The Paileve property ad sigularity aalysis of itertable ad oitegrable systems Physics Reports 80 3 pp. 59 45 989.. A. H. SALAS C. A. GOMEZ J. E. CASTILLO H Symbolic Computatio of solutios for the geeral fifth-order KdV equatio Iteratioal Joural of Noliear Sciece 9 4 pp. 394 40 00.. K. TOMOEDA Local aalyticity i the time ad space variables ad the smoothig effect for the fifth-order KdV-type equatio Advaces i Mathematical Physics 0 pp. 3838 0. 3. H. TRIKI A. BISWAS Solito solutios for a geeralized fifth-order KdV equatio with t-depedet coefficiets Waves i Radom ad Complex Media pp. 5 60 0. 4. A. M. WAZWAZ The exteded tah method for ew solito solutios for may forms of the fifth order KdV equatio Applied Mathematics ad Computatio 84 pp. 00 04 007. 5. A. M. WAZWAZ Aalytic study o the geeralized fifth order KdV equatio: New solitos ad periodic solutios Commuicatios i Noliear Sciece ad Numerical Simulatio 7 pp. 7 80 007.
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