Exact travelling wave solutions for some important nonlinear physical models

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1 Universiy of Wollongong Research Online Faculy of Engineering and Informaion Sciences - Papers: Par A Faculy of Engineering and Informaion Sciences 3 Eac ravelling wave soluions for some imporan nonlinear physical models Jonu Lee Universiy of Wollongong Rahinasamy Sakhivel Sungkyunkwan Universiy Publicaion Deails Lee, J. & Sakhivel, R. 3. Eac ravelling wave soluions for some imporan nonlinear physical models. Pramana: journal of physics, 8, Research Online is he open access insiuional reposiory for he Universiy of Wollongong. For furher informaion conac he UOW Library: research-pubs@uow.edu.au

2 Eac ravelling wave soluions for some imporan nonlinear physical models Absrac The wo-dimensional nonlinear physical models and coupled nonlinear sysems such as Maccari equaions, Higgs equaions and Schrodinger-KdV equaions have been widely applied in many branches of physics. So, finding eac ravelling wave soluions of such equaions are very helpful in he heories and numerical sudies. In his paper, he Kudryashov mehod is used o seek eac ravelling wave soluions of such physical models. Furher, hree-dimensional plos of some of he soluions are also given o visualize he dynamics of he equaions. The resuls reveal ha he mehod is a very effecive and powerful ool for solving nonlinear parial differenial equaions arising in mahemaical physics. Keywords imporan, soluions, wave, ravelling, models, eac, physical, nonlinear Disciplines Engineering Science and Technology Sudies Publicaion Deails Lee, J. & Sakhivel, R. 3. Eac ravelling wave soluions for some imporan nonlinear physical models. Pramana: journal of physics, 8, This journal aricle is available a Research Online: hp://ro.uow.edu.au/eispapers/43

3 PRAMANA c Indian Academy of Sciences Vol. 8, No. journal of May 3 physics pp Eac ravelling wave soluions for some imporan nonlinear physical models JONU LEE and RATHINASAMY SAKTHIVEL, School of Mahemaics and Applied Saisics, Universiy of Wollongong, Wollongong, NSW, Ausralia Deparmen of Mahemaics, Sungkyunkwan Universiy, Suwon , Souh Korea Corresponding auhor. krsakhivel@yahoo.com MS received 7 April ; revised 7 Ocober ; acceped 6 November Absrac. The wo-dimensional nonlinear physical models and coupled nonlinear sysems such as Maccari equaions, Higgs equaions and Schrödinger KdV equaions have been widely applied in many branches of physics. So, finding eac ravelling wave soluions of such equaions are very helpful in he heories and numerical sudies. In his paper, he Kudryashov mehod is used o seek eac ravelling wave soluions of such physical models. Furher, hree-dimensional plos of some of he soluions are also given o visualize he dynamics of he equaions. The resuls reveal ha he mehod is a very effecive and powerful ool for solving nonlinear parial differenial equaions arising in mahemaical physics. Keywords. Eac ravelling wave soluions; nonlinear physical models; Kudryashov mehod. PACS Nos.3.Jr;.7.Wz; 4..Jb. Inroducion The sudy of nonlinear parial differenial equaions is an acive area of research in applied mahemaics, heoreical physics and engineering fields. In paricular, here has been considerable ineres in seeking eac ravelling wave soluions of nonlinear evoluion equaions ha describe some imporan physical and dynamic processes. In he pas several decades, many powerful mehods such as variaional ieraion mehod [], homoopy analysis mehod [], homoopy perurbaion echnique [3], modified anh coh mehod [4], he Jacobi ellipic funcion mehod [], G /G-epansion mehod [6 8], he epfuncion mehod [9 ], rial equaion mehod [3], specral collocaion mehod [4] and many oher echniques were used o obain eac ravelling wave soluions of nonlinear problems. More precisely, here is no unified mehod ha can be used o handle all ypes of nonlinear problems. A powerful and effecive mehod for finding eac soluions of nonlinear differenial equaions was proposed in []. In paricular, his mehod allows DOI:.7/s ; epublicaion: 6 April 3 77

4 Jonu Lee and Rahinasamy Sakhivel us o obain all soliary wave soluions and all one-periodic soluions when we ge he epansion of he general soluion of nonlinear differenial equaion in he Lauren series. Moreover, he main advanage of his mehod is ha we can consruc eac soluions of high order nonlinear evoluion equaions more effecively in comparison wih oher mehods. In [6 8], he auhor applied his mehod o consruc he eac soluions of some nonlinear non-inegrable equaions. Ryabov [9] obained eac soluions of he Kudryashov Sinelshchikov equaion by using he Kudryashov mehod. Kabir [] used he modified Kudryashov mehod o consruc he soliary ravelling wave soluions of he Kuramoo Sivashinsky and sevenh-order Sawada Koera equaions. Eplici soluions o nonlinear problems are of fundamenal imporance. The ravelling wave soluions may be useful in he heoreical and numerical sudies of he model sysems. Therefore, finding ravelling wave soluions of nonlinear equaions is of fundamenal ineres o undersand he equaions fully. In his paper, we shall find he eac ravelling wave soluions for some nonlinear physical models and coupled equaions such as Higgs equaion, Maccari sysem and Schrödinger KdV equaion [6,7] by using he Kudryashov mehod. The compuer symbolic sysems such as Maple and Mahemaica allow us o perform complicaed and edious calculaions.. Algorihm of Kudryashov mehod Le us presen he algorihm of he Kudryashov mehod for finding eac soluions of nonlinear parial differenial equaions PDE []. We consider he nonlinear PDE in he following form: Eu, u, u, u, u,...=. Using he ravelling wave soluions u, = yη, η = k w, eq. can be convered o nonlinear ordinary differenial equaions ODE E y,wy η, ky η,w y ηη, k y ηη,...=. To find he dominan erms we subsiue yη = η p, p > 3 ino all erms of eq.. Then we compare degrees of all erms in eq. and choose wo or more erms wih he smalles degree. The maimum value of p is he pole of eq. and we denoe i as N. I should be menioned ha he mehod can be applied when N is an ineger. If he value N is nonineger, we have o use he ransformaion of soluion yη. We look for eac soluion of in he following form: yη = a + a Qη + a Qη + +a N Qη N, 4 where a i, i =,,...,N are unknown consans and Qη is he following funcion: Qη = + e. η 78 Pramana J. Phys., Vol. 8, No., May 3

5 Travelling wave soluions This funcion saisfies o he firs-order ordinary differenial equaion Q η = Q Q. 6 Equaion 6 is necessary o calculae he derivaive of funcion yη. We can calculae he necessary number of derivaives of funcion y. For insance, we consider he general case when N is arbirary. Differeniaing 4 wih respec o η and aking ino accoun 6, we have N y η = a i iq Q i, y ηη = i= N a i ii + Q i + Q + iq i. 7 i= The highes order derivaive of yη can be found in [6,7]. Ne, subsiue epressions 4, and 6 in. Then we collec all erms wih he same powers of funcion Qz and equae he resuling epression o zero. Finally, we obain algebraic sysems of equaions. Solving his sysem, we ge values for he unknown parameers. 3. Applicaions o physical models Eample 3.. Two-dimensional sine-gordon: Le us demonsrae he applicaion of Kudryashov mehod for finding he eac ravelling wave soluions of he wo-dimensional sine-gordon equaion u u u yy + m sin u =, 8 and Dodd Bullough Mikhailov equaion u + pe u + qe u =. 9 The wo-dimensional sine-gordon equaion 8 and Dodd Bullough Mikhailov equaion 9 have been widely applied in many fields such as solid-sae physics, nonlinear opics, fluid dynamics, fluid flow, quanum field heory, elecromagneic waves and so on [7]. To look for he ravelling wave soluions of eq. 8, we make ransformaion v = e iu, v, y, = V η, η = μ + αy + β, and generae he reduced nonlinear ODE in he form μ β α VV V + m V 3 V =, where he prime denoes he differenial wih respec o η. The pole of eq. is equal o N =, hen we look for he eac ravelling wave soluions in he following form: V η = a + a Q + a Q, where a, a and a are unknown consans. Subsiuing ino and aking ino accoun relaions 7, we can obain a sysem of algebraic equaions. Solving he resuling Pramana J. Phys., Vol. 8, No., May 3 79

6 Jonu Lee and Rahinasamy Sakhivel sysem by using Maple, we find ha soluion of eiss only in he following wo cases: a =, a = 4, a = 4,μ= μ, α =± β + m μ,β = β, a =, a = 4, a = 4,μ= μ, α =± β m μ,β = β. 3 Thus, soluion of eq. 8 which corresponds o is given by + v u, y, = arccos, y, v, y, = arccos + csch μ ± β + m y + β, μ 4 where 4 v, y, = { + ep μ ± } β + m /μ y + β 4 + [ { + ep μ ± }] β + m /μ y + β = anh μ ± β + m y + β μ. Ne, soluion of eq. 8 which corresponds o 3 is given by + v u, y, = arccos, y, v, y, = arccos csch μ ± β m y + β, μ where 4 v, y, = + { + ep μ ± } β m /μ y + β 4 [ { + ep μ ± }] β m /μ y + β = anh μ ± β m y + β μ. 76 Pramana J. Phys., Vol. 8, No., May 3

7 Travelling wave soluions More precisely, if we ake μ = k and β = λ hen our soluions of 4 and urn ou o he soluions as epressed in []. Ne, o find a ravelling wave soluion of eq. 9, we use v = e u, v, = V η, η = μ + c. 6 Subsiuing 6 ino eq. 9, we ge μ cv V μ cv + pv 3 + q =. 7 The pole of eq. 7 is equal o N =. Therefore, we have V η = a + a Q + a Q. 8 Subsiuing 8 ino7, we can obain a sysem of algebraic equaions. Solving he resuling sysem by using Maple, he following se of soluions is obained: { a = q /3, a = 6 q /3, a = 6 q /3, p p p μ = μ, c = 3p q } /3. 9 μ p This in urn gives he following eac ravelling wave soluion of eq. 9: [ u, =ln q { }] /3 3 p +cosh { μ 3p/μ q/p /3 }. Eample 3.. Coupled Schrödinger KdV equaion: A second insrucive eample is he coupled Schrödinger KdV equaion iu = u + uv, v + 6vv + v = u, which describes various processes in dusy plasma, such as Langmuir, dus-acousic wave and elecromagneic waves [7]. We suppose ha eq. has he ravelling wave soluion of he form u, = e iθ Uη, v, = V η, θ = α + β, η = + c, where α, β and c are consans. Subsiuing eq. ino eq., we find ha c = α and U, V saisfy he following coupled nonlinear ordinary differenial sysem: U + β α U + UV =, αv + 6VV + V U =. 3 The pole of he coupled equaions 3 aren =, M =. As a resul, Kudryashov mehod admis he soluion of 3 in he following form: U = a + a Q + a Q, V = b + b Q + b Q, 4 Pramana J. Phys., Vol. 8, No., May 3 76

8 Jonu Lee and Rahinasamy Sakhivel where a, a, a, b, b and b are unknown consans o be deermined. Subsiuing 4 in he reduced ODE 3 and collecing he coefficiens of Q yields a sysem of algebraic equaions. Solving he resuling sysem by using Maple, he following ses of soluion are obained: { a = a, a = a, a =, b = b, b =, b =, α = a 3b,β = a4 + a + 9b + b + 6b a + } 4, c = α, { a =, a = 6, a =±6, b = b, b = 6, b = 6, α = 3b,β = 9b + b 3 } 4, c = α, 6 { a =±, a = 6, a =±6, b = b, b = 6, b = 6, α = 3b,β = 9b + 4b + 4, c = α }. 7 The firs wo ses and 6 give he eac wave soluions of eq. in he following form: [ { u, = a ep i a 3b }] + D, + e +α where and v, = b + + e +α D = a 4 + a + 9b + b + 6b a + 4, u, =±6 [ ep i [ v, = b + + e +α, 8 { 3b + ] + e +α + + e +α 9b + b 3 4, }] e +α + e +α. 9 Finally, he hird se 7 gives he eac wave soluions as u 3, =±6 [ { ep i 3b + 9b + 4b + [ ] e +α, + e +α v 3, = b + }] e +α + e +α Pramana J. Phys., Vol. 8, No., May 3

9 Travelling wave soluions Moreover, if we ake a =, b = 8 + 9/8 and α = hen eq. 8 provides a new ravelling wave soluion of he coupled Schrödinger KdV equaion [ + 9 u, = ep i + ], e + v, = e + + e +. 3 The 3D plo of soluions 3 and 3 are shown in figures 4. The soluion 3 represens a bell-ype wave soluion which is shown graphically in figure 4. Eample 3.3. Coupled nonlinear physical models: Consider he following coupled Higgs equaion []: u u + u u uv =, v + v u =, 33 and coupled inegrable +-dimensional nonlinear sysem in he following form []: iu + u + uv =, v + v y + u =. 34 Firs, we consider he coupled Higgs equaion 33. To obain he ravelling wave soluion of 33, we consider he ransformaion u = e iθ Uη, v = V η, θ = p + r, η = + c. 3 Subsiuing 3 ino33, we have c U + p r U UV + U 3 =, c + V U UU =. 36 Re u..... Figure. 3D plo of eq. 3: Real par periodic soluion. Pramana J. Phys., Vol. 8, No., May 3 763

10 Jonu Lee and Rahinasamy Sakhivel Im u..... Figure. 3D plo of eq. 3: Imaginary par periodic soluion. Abs u... Figure 3. Kink-ype wave soluion of eq. 3. Inegraing he second equaion in 36 and neglecing he consan of inegraion we find c + V = U. 37 Subsiuing 37 ino he firs equaion of he sysem and inegraing we find c 4 U + c + p r U + c U 3 =, 38 where prime denoes differeniaion wih respec o η. The pole of eq. 38 is equal o N =. Therefore, Kudryashov mehod admis soluion in he following form: Uη = a + a Q. 39 Subsiuing 39 ino38, we can obain a sysem of algebraic equaions. Solving he resuling sysem wih he aid of Maple, we can obain four ses of soluions: {a = a, a = a, p =±r, r = r, c = }, Pramana J. Phys., Vol. 8, No., May 3

11 Travelling wave soluions v..4.6 Figure 4. Bell-ype wave soluion of eq. 3. {a = a, a = a, p =±r, r = r, c = }, 4 { } a = c +, a = c + c + r, p =±, r =r, c =c, 4 { } a = c +, a = c + c + r, p =±, r =r, c =c. 43 According o eqs 4 and 4, we obain he eac ravelling wave soluions in he following form: u, =e ±ri± a + a, v + e +, = a c + a + + e + 44 and u, =e ±ri± a + a, v + e, = a c + a. + + e 4 Due o eq. 4, we ge he following ravelling wave soluion: [ u 3, = c + c + r ep i ± + r], + e +c v 3, = e +c Pramana J. Phys., Vol. 8, No., May 3 76

12 Jonu Lee and Rahinasamy Sakhivel Finally, eq. 43 leads o he eac ravelling wave soluion in he form [ u 4, = c + c + r ep i ± + r] +, + e +c v 4, = e +c In paricular, if we ake c = and r = hen eq. 47 provides a new ravelling wave soluion of he coupled Higgs equaion 33 in he following form: u, = i ep[i + ] +, 48 v, = + + e + + e Furher, he behaviour of he obained soluions 48 and 49 are shown graphically see figures 8. Ne, we consider he Maccari sysem 34. Le us assume ha he ravelling wave soluion of 34 has he form u = e iθ Uη, v = V η, θ = p + qy + r, η = + y + c. Subsiuing ino34, we have U r + p U + UV =, c + V UU =. Inegraing he second equaion in he sysem and neglecing he consan of inegraion we find c + V = U. Re u..... Figure. Real par periodic soluion of eq Pramana J. Phys., Vol. 8, No., May 3

13 Travelling wave soluions Im u..... Figure 6. Imaginary par periodic soluion of eq. 48. Abs u... Figure 7. Kink-ype wave soluion of eq. 48. Subsiuing ino he firs equaion of he sysem and inegraing we find c + U c + r p U U 3 =, 3 where prime denoes differeniaion wih respec o η. The pole of eq. 3 is equal o N =. Therefore, we have U η = a + a Q. 4 Subsiuing 4 ino 3, we can obain a sysem of algebraic equaions and proceeding as before we find he following se of soluion: # $ a a a =, a = a, p = p, q = q, r = p, c =. Pramana J. Phys., Vol. 8, No., May 3 767

14 Jonu Lee and Rahinasamy Sakhivel v...4 Figure 8. Bell-ype wave soluion of eq. 49. The ravelling wave soluions for he coupled eq. 34 according o is given by [ { u, y, = a ep i p + qy + p }] + + ep [ + y + a / ], v, y, = a c ep [ + y + a / ]. 6 Remark 3.4. When q = and p =, he Dodd Bullough Mikhailov equaion 9 reduces o he Liouville equaion u + e u =. 7 By repeaing he soluion procedure as above, we can obain ravelling wave soluion of eq. 7 in he following form: [ u, = ln μ c + cosh{μ + c} ]. 8 Noe.3. I is noed ha Kudryashov mehod can be suiable o he nonlinear parial differenial equaions wih higher order nonlineariy. I should be menioned ha all he obained soluions are verified by puing hem back ino he original equaions wih he aid of Mahemaica. 768 Pramana J. Phys., Vol. 8, No., May 3

15 4. Conclusion Travelling wave soluions In his paper, we have successfully implemened he Kudryashov mehod o esablish eac soluions of he wo-dimensional sine-gordon equaion and Dodd Bullough Mikhailov equaion. Furher, we apply he mehod o solve he coupled nonlinear models such as Maccari equaions, Higgs equaions and Schrödinger KdV equaions. The resul reveals ha nonlinear evoluion equaions can be easily handled by Kudryashov mehod and ha he performance of his mehod is reliable and effecive. The mehod is sraighforward and concise, and i can also be applied o oher nonlinear problems. References [] J H He, In. J. Nonlinear Mech. 34, [] M Dehghan and R Salehi, Z. Naurforsch. A J. Phys. Sci. 66, 9 [3] J H He, Compu. Mehod. Appl. Mah. 78, [4] L Wazzan, Commun. Nonlinear Sci. Numer. Simula. 4, 64 9 [] J Lee and R Sakhivel, Mod. Phys. Le. B4, [6] A Jabbari, H Kheiri and A Bekir, Compu. Mah. Appl. 6, 77 [7] B Salim Bahrami, H Abdollahzadeh, I M Berijani, D D Ganji and M Abdollahzadeh, Pramana J. Phys. 77, 63 [8] S Zhang, L Dong, J-M Ba and Y-N Sun, Pramana J. Phys. 74, 7 [9] M Dehghan, J Manafian and A Saadamandi, In. J. Mod. Phys. B, 96 [] R Sakhivel and C Chun, Z. Naurforsch. A J. Phys. Sci. 6a, 97 [] R Sakhivel, C Chun and J Lee, Z. Naurforsch. A J. Phys. Sci. 6, 633 [] S Zhang, Phys. Le. A37, 6 7 [3] Y Gurefe, A Sonmezoglu and E Misirli, Pramana J. Phys. 77, 3 [4] M Dehghan and F Fakhar-Izadi, Mah. Compu. Model. 3, 86 [] N A Kudryashov, J. Appl. Mah. Mech., [6] N A Kudryashov, Commun. Nonlinear Sci. Numer. Simula. 7, 48 [7] N A Kudryashov, D I Sinelshchikov and M V Demina, Phys. Le. A37, 74 [8] N A Kudryashov, P N Ryabov and D I Sinelshchikov, J. Compu. Appl. Mah. 3, 43 [9] P N Ryabov, Appl. Mah. Compu. 7, 38 [] M M Kabir, A Khajeh, E Abdi Aghdam and A Yousefi Koma, Mah. Meh. Appl. Sci. 34, 3 [] E Fan and Y C Hon, Appl. Mah. Compu. 4, 3 3 [] M Tajiri, J. Phys. Soc. Jpn, Pramana J. Phys., Vol. 8, No., May 3 769