Solitary Solutions and Singular Periodic Solutions. of the Drinfeld-Sokolov-Wilson Equation. by Variational Approach

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1 Applied Mathematial Sienes, Vol. 5, 11, no. 38, Solitary Solutions and Singular Periodi Solutions of the Drinfeld-Sokolov-Wilson Equation by Variational Approah Wei-Min Zhang Shool of Mathematis, Jiaying University Meizhou, Guangdong, 51415, China Abstrat Via the semi-inverse method, variational priniples are established for the Drinfeld-Sokolov-Wilson (DSW) equation. Based on the obtained variational formulations, exat solitary solution and exat singular periodi wave solution an be obtained using the variational approah. Keywords: The semi-inverse method; Variational priniple; Drinfeld-Sokolov -Wilson (DSW) equation; soliton 1. Introdution The nonlinear equations of mathematial physis are major subjets in physial siene. The investigation of exat traveling wave solutions to nonlinear evolution equations plays an important role in the study of nonlinear physial phenomena. The wave phenomena are observed in fluid dynamis, plasma, elasti media, optial fibers, et. The importane of the exat solutions failitates the verifiation of numerial solvers and aids in the stability analysis of solutions. So, looking for exat traveling wave solutions,espeially exat solitary wave solutions, has long been a major role in the study of physial phenomena. Even today, the researh of seeking different

2 1888 Wei-Min Zhang types of solitary wave solutions and finding new method of solving solution has not slowed down in some areas of physis [1]. Reently, He [] proposed a new method alled He`s variational method [4-9] to seek solitary wave solutions of nonlinear evolution equations. The key idea and the standard appliation of the method an be found in []. In this paper, we onsider the following Drinfeld-Sokolov-Wilson (DSW) equation ut + αvvx = (1) vt + βvxxx + γuvx + εuxv = where α, βγεare,, nonzero parameters. The system was first proposed by Drinfeld and Sokolov [1, 11] and Wilson [1] whenα = 3, β = γ = and ε = 1.The equation an be solved by many methods, e.g. F-Expansion Method [13], Deomposition Method [14], Tanh-method [15]. In this paper, we will apply He`s variational approah to searh for its solitary solutions.. Variational formulation We introdue transformations uxt (, ) = u( ξ ), vxt (, ) = v( ξ), ξ = x t () Where is an arbitrary onstant. Substituting () into (1) we obtain u ' + αvv ' =, (3) v ' + βv ''' + γuv ' + εu ' v =. (4) where the prime expresses the derivative with respet toξ. By integrating Eq (3) we find 1 1 u = ( k+ αv ), (5) where k is a onstant of integration to be further determined. Substituting Eq.(3) and (5) into Eq.(4) yields 1 ( k ) v' + βv''' + α( γ + ε) v v' = (6) Integrating the Eq.(6) we have

3 Solitary solutions and singular periodi solutions ( k ) v+ βv'' + α( γ + ε) v = (7) 6 By the semi-inverse method [16-19] we an obtain the following variational formulations: J1() v = [ ( k ) v + α( γ + ) ε v β(')] v dξ, (8) J( v) = [ ( k ) v + α( γ + ε) v + βvv''] dξ. (9) 4 3. Solitary Solution and singular periodi wave solution The variational approah to solutions was first outlined by Prof. He in his review artile Some asymptoti methods for strongly nonlinear equations (Setion.) []. Following the basi idea, many researhers [4-9] found He`s variational approah was very effetive for searhing for various solitary wave solutions. In this setion, we apply He`s variational approah to searhing for solitary wave solution for Eq.(1). Following the standard appliation of the variational approah suggested in Refs. [-9], we searh for a solitary solution in the form v= v( ξ ) = pse h( qξ ), (1) where p, q are unknown onstants to be further determined. Substituting Eq. (1) into Eq. (8), we have k 1 J v p h qξ dξ α γ ε p h qξ dξ 4 4 1() = se ( ) ( ) se ( ) β - [( se ( ))'] p h qx dx (11) 4 ( k ) p 1 p αγ ( + ε) = p qβ +. q 6 36q where the prime expresses the derivative with respet to ξ. Making J 1 = J 1 () v, stationary with respet to p and q results in 3 J1 ( k ) p 1 p αγ ( + ε) = pqβ + =, (1) p q 3 9q 4 J1 ( k ) p 1 p αγ ( + ε) = p β =. (13) q q 6 36q

4 189 Wei-Min Zhang From Eq. (1) and (13), we get k = q β ( ), p=± q 3β. (14) α( γ + ε) Substituting Eq.(14) into Eq.(1) and (5), The solitary wave solution of the Drinfeld-Sokolov-Wilson (DSW) equation an be approximately expressed in the form 6q uxt (, ) = ( qβ ) + β se h[ qx ( t)] γ + ε, (15) 3 vxt (, ) =± q β se hqx [ ( t)] αγ ( + ε). (16) To determine onstant q, substituting Eqs.(15) and (16) into Eq.(1) yields ± β q β + γ h q x t q x t αγ + αε = 3( )( 1 ) se [ ( )]tanh[ ( )] Solving Eq.(17) we have, (17) or q =± (when β > ), (18) β q=± i (when β < ). (19) β Substituting Eq.(18) into Eqs.(15) and (16) we have 6 uxt = h x t γ + ε β (, ) se [ ( )], () vxt (, ) =± 3 se h[ ( x t)], (1) αγ ( + ε) β Whih are the exat bell-type solitary wave solutions of the Drinfeld-Sokolov-Wilson (DSW) equation (1) when β > ; Substituting Eq.(19) into Eq.(15) and (16) we have 6 uxt = x t γ + ε β (, ) se [ ( )]. ()

5 Solitary solutions and singular periodi solutions 1891 vxt (, ) =± 3 se[ ( x t)]. (3) αγ ( + ε) β Whih are the exat singular periodi wave solution of the Drinfeld-Sokolov-Wilson (DSW) equation (1) when β <. 4. Disussion of Examples In this setion, two pratial numerial simulation examples are illustrated to show the effetiveness and onveniene of the semi-inverse method for seeking exat solitary wave solution and periodi wave solution of nonlinear wave equations. Example 1. If we set α = 3, β = γ = and ε = 1, the Drinfeld-Sokolov-Wilson equation (1) is redued [1-1] ut + 3vvx =, (4) vt + vxxx + uvx + uxv = From () and (1) the exat bell-type solitary wave solution of Eqs.(4) read 3 uxt (, ) = se h[ ( x t)], (5) vxt (, ) =± se h[ ( x t)]. (6) The figure.1 shows the shape of the exat bell-type solitary wave solution of the Drinfeld-Sokolov-Wilson equation (4) (only shows the shape of Eq.(6) with = ) v t x 4 4 Fig.1.( the shape of Eq.(6) with = )

6 189 Wei-Min Zhang Example. If we set α = 3, β =, γ = and ε = 1, the Drinfeld-Sokolov-Wilson equation (1) is redued ut + 3vvx =, (7) vt vxxx + uvx + uxv = From () and (3) the exat singular periodi wave solution of Eqs.(7) read 3 uxt (, ) = se [ ( x t)], (8) vxt (, ) =± se[ ( x t)]. (9) The figure. shows the shape of the exat singular periodi wave solution of the Drinfeld-Sokolov-Wilson equation (7) (only shows the shape of Eq.(9) with = ). 1 v t 1 1 x Fig..( the shape of Eq.(9) with = ) 5. Conlusion We established a variational formulation for the Drinfeld-Sokolov-Wilson (DSW) equation by the semi-inverse method. Its exat solitary wave solution and exat singular periodi wave solution an be obtained using He`s variational method. The method provide a new mathematial tool to seek for solitary solutions and exat singular periodi wave solution and an be extended to other nonlinear equations.

7 Solitary solutions and singular periodi solutions 1893 Referenes [1] A.Biswas,1-soliton solution of the generalized Radhakrishnan,Kundu,Lakshmanan equation, Physis Letters A,373(3), (9). [] J. H. He, Some Asymptoti Methods for Strongly Nonlinear Equations, Internat J. Modern Phys. B (1), (6). [3] J. H. He, Variational approah for nonlinear osillators, Chaos Solitons and Fratals 34(1), (7). [4] L.Xu, Variational approah to solitons of nonlinear dispersive K(m,n) equations, Chaos Solitons and Fratals 37(1), (8). [5] T. Ozis, A. Yidirim, Appliation of He`s semi-inverse method to the nonlinear Shrodinger equation, Computers & Mathematis with Appliations 54(7-8), (7). [6] J. Zhang, Variational approah to solitary wave solution of the Zakharov equation, Computers & Mathematis with Appliations 54(7-8), (7). [7] Z. L. Tao, Solitary solutions of the Boiti-Leon-Manna-Pempinelli Equation using He`s Variational method, Z. Naturforshung. 63a,, (8). [8] Z. L. Tao, Variational approah to the Benjiamin Ono equation, Nonlinear Analysis: Real Word Appliations 1, (9). [9] S.S.Ganji, D.D.Ganji, Z.Z.Ganji and S.Karimpour, Periodi Solution for Strongly Nonlinear Vibration Systems by He s Energy Balane Method, Ata Appl Math 16, 79-9(9). [1] V. G. Drinfeld and V. V. Sokolov, Equations of Korteweg-de Vries type and simple Lie algebras, Sov. Math. Dokl. 3, (1981). [11] V. G. Drinfel d and V. V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 3, (1985). [1] G. Wilson, The affine Lie algebra (1) C and an equation of Hirota and Satsuma, Phys.Lett. A, 89(7),33-334(198) [13] X. Q. Zha and H. Y. Zhi, An Improved F Expansion Method and Its Appliation to Coupled Drinfeld-Sokolov-W ilson Equation, Commun.Theor.Phys,5, (8). [14] M.In,On numerial doubly periodi wave solutions of the oupled Drinfel d Sokolov Wilson equation by the deomposition method, Applied Mathematis and Computation, 17(1), 41-43(6).

8 1894 Wei-Min Zhang [15] Z. B. Li, Traveling wave solutions of nonlinear mathematial physis equations, Siene Press (Beijing), 7. [16] J.H.He, Variational priniple for two-dimensional inompressible invisid flow, Phys.Lett.A 371,39-4(7). [17] J.H.He, Variational theory for two-dimensional longitudinal beam dynamis, Phys. Lett.A 35(4-5), 76-77(6). [18] S.Adali,Variational priniples for multi-walled arbon nanotubes undergoing bukling based on nonloal elastiity theory, Phys.Lett.A 37, (8). [19] Z. L. Tao, Variational approah to the invisid ompressible fluid, Ata Appl. Math. 1, (8). Reeived: September, 1