Soliton Solutions of the Time Fractional Generalized Hirota-satsuma Coupled KdV System

Transcription

1 Appl. Math. Inf. Sci. 9, No., (015) 17 Applied Mathematics & Information Sciences An International Journal Soliton Solutions of the Time Fractional Generalized Hirota-satsuma Coupled KdV System Ahmad Neirameh Department of Mathematics, Faculty of sciences, Gonbad Kavous University, Gonbad, Iran Received: 3 Oct. 01, Revised: 3 Jan. 015, Accepted: Jan. 015 Published online: 1 Jul. 015 Abstract: In this present study, the exact traveling wave solutions to the time fractional generalized Hirota-Satsuma coupled KdV system are studied by using the direct algebraic method. The exact and complex solutions obtained during the present investigation are new, whereas literature survey has revealed generalizations of solutions. The solutions obtained during the present work demonstrate the fact that solutions to the time fractional Generalized Hirota-Satsuma coupled KdV system can exhibit a variety of behaviors. It is also exhibited that the proposed method is more effective and general in nature. Keywords: Hirota-Satsuma coupled KdV system, direct algebraic method, exact wave solutions. 1 Introduction As is well known, solitons are universal phenomenon, appearing in a great array of contexts such as, for example, nonlinear optics, plasma physics, fluid dynamics, semiconductors and many other systems [1,, 3]. Studying of nonlinear evolution equations (NLEEs) modeling various physical phenomena has played a significant role in many scientific applications such as water waves, nonlinear optics, plasma physics and solid state physics [, 5, 6, 7]. Many powerful methods for finding exact solutions of NLEEs have been proposed, such as ansatz method and topological solitons [, 9, 10, 11], tanh method [1, 13], multiple exp-function method [1], simplest equation method [15, 16, 17, 1], Hirotas direct method [19, 0], transformed rational function method [1] and so on. This work aims to find out the solitary and complex wave solutions to the time fractional Generalized Hirota Satsuma coupled KdV system [] with following form: D α t u= 1 u xxx+ 3u x u+3( v + w) x, D α t v= 1 v xxx 3u x, 0<α 1, D α t w= 1 w xxx 3uw x. The Generalized Hirota-Satsuma coupled KdV system is one of the essential nonlinear equations in mathematics (1) and physics. Therefore, it is important to find solutions for this equation. This equation arises as a special case of the Toda lattice equation, a well- known soliton equation in one space and one time dimension, which can be used to model the interaction of neighboring particles of equal mass in a lattice formation with a crystal. The Generalized Hirota-Satsuma coupled KdV system has many applications in many branches of nonlinear science. One application of the Generalized Hirota-Satsuma coupled KdV system is that it can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. Another application of the sinh-gordon equation is in the field of thermodynamics, where it can be applied to exactly calculate partition and correlation functions. The paper is arranged as follows. In Section, we describe briefly the Modified Riemann Liouville derivative with properties and simplest equation method. In Section 3, we apply this method to the time fractional Generalized Hirota-Satsuma coupled KdV system. Modified Riemann-Liouville derivative and direct algebraic method In this section, we first give some definitions and properties of the modified Riemann Liouville derivative which are used further in this paper. [3,, 5] Corresponding author a.neirameh@gonbad.ac.ir

2 1 A. Neirameh: Soliton Solutions of the Time Fractional Generalized... Assume that f : R R,x f(x) denote a continuous (but not necessarily differentiable) function. The Jumarie modified Riemann Liouville derivative of order α is defined by the expression 1 x Γ( α) (x ξ) α 1 [ f(ξ) f(0)]dξ, α 0 0 x Dx α d f(x)= dx Γ(1 α) (x ξ) α [ f(ξ) f(0)]dξ, 0 α 1 0 ( f (n) (x)) (α n) n α n+1,n 1 () Some properties of the fractional modified Riemann- Liouville derivative were summarized and three useful formulas of them are D α x x y = Γ(1+γ) Γ(1+γ α) xγ α,γ 0 (3) D α x(u(x)v(x))=v(x)d α x u(x)+u(x)d α x v(x), () { btanh( bξ), b 0 F(ξ)= bcoth( { bξ), b 0 btan( bξ), b 0 F(ξ)= bcot( bξ), b 0 F(ξ)= 1 ξ, b=0 (11) Integer n in (9) can be determined by considering homogeneous balance [] between the nonlinear terms and the highest derivatives of u(ξ)in Eq. (). Substituting (9) into () with (10), then the left hand side of Eq. () is converted into a polynomial in F(ξ), equating each coefficient of the polynomial to zero yields a set of algebraic equations for a i,k,c. Solving the algebraic equations obtained and substituting the results into (9), then we obtain the exact traveling wave solutions for Eq. (1). D α x [ f(u(x))]= f (u)d α x u(x)=dα x f(u)(u x )α, (5) Which are direct consequences of the equality d α x(t) = Γ(1+α)dx(t). Next, let us consider the time fractional differential equation with independent variables x = (x 1,x,...,x m,t) and a dependent variable u, F(u, D α t u,u x 1,u x,u x3..., D α t u,u x1 x 1,u x x,u x3 x 3,...)=0. (6) Using the variable transformation u(x 1,x,...,x m,t)= U(ξ), ξ = x 1 + l 1 x +...+l m 1 x m + λtα β, (7) Γ(1+α) where k,l i and λ are constants to be determined later; the fractional differential equation (6) is reduced to a nonlinear ordinary differential equation H =(U(ξ),U (ξ),u (ξ),...), () Where = d dξ. We assume that Eq. () has a solution in the form u(ξ)= n i=0 a i F i (ξ), (9) Where a i (i=1,,..,n) are real constants to be determined later. F(ξ) expresses the solution of the auxiliary ordinary differential equation F (ξ)=b+f (ξ), (10) Eq. (10) admits the following solutions: 3 Application to the time fractional Generalized Hirota Satsuma coupled KdV system: Next, we study Eq. (1). Considering the following complex transformation: and ξ = x λtα Γ(1+α). u(x,t)= 1 λ U (ξ), v(x,t)= λ +U(ξ), w(x,t)=λ λu(ξ), (1) Substituting Eq. (1) into Eq. (1), we can know that Eq. (1) is reduced into ordinary differential equations: λ(u ξ ) + λuu ξ ξ + 3U λ U + 6λ + λ R=0, (13) and λu ξ ξ + U 3 λ U = 0, (1) where Ris an integration constant to be determined later. Case 1: Balancing UU ξ ξ with U in Eq. (13) gives n=1. Therefore, we may choose U = a 1 F+ a 0, (15) Substituting Eq. (15) along with Eq. (9) in Eq. (13) and setting all the coefficients of powers Fto be zero, then we obtain a system of nonlinear algebraic equations and by

3 Appl. Math. Inf. Sci. 9, No., (015) / 19 solving it, we obtain a 1 =± b+3, a 0 =± 9 16b, λ = b+3, ( R= b (b+3) + (b+3) 115 (9 16b ) (9 16b ) + b+3 16 (b+3)b) (16) From (11),(15) and (16), we obtain the solitary wave solutions of (13) as follows b+3t α U 1 = ± b+3[ btanh b(x+, (17) Where b 0and k is an arbitrary real constant. From (1) we have u 1 (x,t) = b+3 ( b (b+3) tanh b(x+ (b+3)tα Γ(1+α) ± 1 (b+3)(9 16b ) [ btanh b(x+ (b+3)tα + 1 (9 16b )) v 1 (x,t) = b+3 ± b+3[ b tanh b(x+ (b+3)tα, w 1 (x,t) = ( b+3 ) ( b+3 )[± b+3[ b tanh b(x+ (b+3)tα And (b+3)t α U = ± b+3[ bcoth b(x+, So u (x,t) = b+3 ( b (b+3) coth b(x+ (b+3)tα Γ(1+α) ) ± 1 (b+3)(9 16b ) [ bcoth b(x+ (b+3)tα + 1 (9 16b )). v (x,t) = b+3 ± b+3[ b coth b(x+ (b+3)tα, w (x,t) = ( b+3 ) ( b+3 ) [± b+3[ b coth b(x+ (b+3)tα ], Where b 0and k is an arbitrary real constant. (b+3)t α U 3 = ± b+3[ btan b(x+, (1) Therefore u 3 (x,t)= b+3( b (b+3)tan ) b (x+ (b+3)tα [ Γ(1+α) btan ± 1 ] (b+3)(9 16b ) b( x+ (b+3)t α ( Γ(1+α) ) 9 16b )), + 1 v 3 (x,t) = b+3 ± b+3[ b tan b(x+ (b+3)tα,

4 150 A. Neirameh: Soliton Solutions of the Time Fractional Generalized... w 3 (x,t) = ( b+3 ) ( b+3 )[± b+3[ b tan b(x+ (b+3)tα Now, u (x,t) = b+3 [± b+3[ b cot b(x+ (b+3)tα ], v (x,t) = b+3 ± b+3[ b cot b(x+ (b+3)tα, Figure 1: Complex solitary waves described by solution; case1, v 3 (x,t) for b=1,α = 1. w (x,t) = ( b+3 ) ( b+3 ) [± b+3[ b cot b(x+ (b+3)tα ], Where b 0and k is an arbitrary real constant. For b = 0 U 5 = 6Γ(1+α) xγ(1+α)+3t α ± Substituting in (1) we have [ u 5 (x,t)= b+3 b+3 19 ( Γ(1+α) xγ(1+α)+3t α ) 6Γ(1+α) xγ(1+α)+3t α ± ] = 3Γ(1+α) xγ(1+α)+3t α + 1 3, v 5 (x,t)= λ 6Γ(1+α) xγ(1+α)+3t α ±, ( w 5 (x,t)= (b+3) 3 + b+3 6Γ(1+α) xγ(1+α)+3t α ± ), Case : Balancing U ξ ξ with U 3 in Eq. (1) gives N= 1. Therefore, we may choose Figure : solitary waves described by solution; case1, u 3 (x,t) for b=1,α = 1. Where b 0and k is an arbitrary real constant. And (b+3)t α U = ± b+3[ btan b(x+, (19) U = a 1 F+ a 0, (0) Substituting Eq. (0) along with Eq. (10) in Eq. (1) and setting all the coefficients of powers Fto be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain a 1 =± b 5, a 0 =± 3bi λ = b 5 (1)

5 Appl. Math. Inf. Sci. 9, No., (015) / From (11),(0) and (1), we obtain the complex travelling wave solutions of (13) as follows U 1 =± b 5 [ btanh ] b( x 5Γ(1+α) ) ± 3bi Substituting in (1) we have u 1 (x,t) = ( b 5 )[ btanh ± bi 3b 15 5 [ btanh v 1 (x,t) = b 5 +[± b 5 [ btanh ± 3bi 15 ], w 1 (x,t) = b 5 [± b b 5 5 [ btanh ± 3b i ], Figure : Complex solitary waves described by solution; case, w 1 (x,t) for b= 1,α = 1,. Where b 0 and k is an arbitrary real constant. And U = ± So ± 3bi b 5 [ bcoth u (x,t)=( b 5 )[ bcoth ] ± b( x btα 5Γ(1+α) ) bi 15 3b [ 5 bcoth b( x bt ] α 5Γ(1+α) ), v (x,t) = b 5 +[± b 5 [ bcoth ± 3bi 15 ], w (x,t) = b 5 [± b 5 b 5 [ bcoth ± 3b i ], Where b 0and k is an arbitrary real constant. U 3 =± b [ ] btan b( x 5 5Γ(1+α) ) ± 3bi Now u 3 (x,t)=( b 5 ) [ btan b( x 5Γ(1+α) ) ] ± bi 15 3b 5 [ btan b( x 5Γ(1+α) ) ], Figure 3: Complex solitary waves described by solution; case, u 1 (x,t) for b= 1,α = 1.

6 15 A. Neirameh: Soliton Solutions of the Time Fractional Generalized... v 3 (x,t) = b 5 +[± b 5 [ btan b(x ± 3bi 15 ], w 3 (x,t) = b b [± b [ btan b(x ± 3b i ], Conclusion In this present work we deduce that the referred method can be extended to solve many systems of nonlinear partial differential equations which are arising in the theory of solitons and other areas such as physics, biology, and chemistry. With the help of symbolic computation (Maple), a rich variety of exact solutions are obtained by applying direct algebraic method, and the method can be applied to other nonlinear evolution equations. References Figure 5: solitary waves described by solution; case u 3 (x,t) for b=1,α = 1. Where b 0and k is an arbitrary real constant. U =± b [ bcot ] b( x 5 5Γ(1+α) ) ± 3bi and u (x,t) = ( b 5 )[ bcot b(x ± bi 3b 15 5 [ bcot b(x v (x,t) = b 5 +± b 5 [ bcot b(x ± 3bi w (x,t) = b b ± 5 5, b 5 [ bcot b(x ± 3b i, Where b 0 and k is an arbitrary real constant. For b = 0 don t have any solution. [1] H. Eleuch and R. Bennaceur, An Optical Soliton Pair Among Absorbing Three-Level Atoms, J. Opt. A : Pure Appl. Opt. 5, (003). [] E. A. Sete, A. A. Svidzinsky, Y. V. Rostovtsev, H. Eleuch, P. K. Jha, S. Suckewer and M. O. Scully, IEEE,Using quantum coherence to generate gain in the XUV and X-ray: gainswept superradiance and lasing without inversion, Journal of Selected Topics in Quantum Electronics 1, (01). [3] M. Remoissenet, Waves Called Solitons: Concepts and Experiments, Springer [] P. G. Drazin, R. S. Johnson, soliton: An Introduction, Cambridge Texts in Applied Mathematics, 196. [5] H. Jabri, H. Eleuch, T. Djerad,Lifetimes of atomic Rydberg states by autocorrelation function, Laser Phys. Lett., (005). [6] H. Eleuch, N. Ben Nessib and R. Bennaceur, Quantum Model of emission in weakly non ideal plasma, Eur. Phys. J. D 9, 391 (00). [7] E. A. Sete and H. Eleuch, Interaction of a quantum well with squeezed light: Quantum statistical properties, Phys. Rev. A, 0310 (010). [] Biswas A. Topological 1-soliton solution of the nonlinear Schrodinger s equation with Kerr law nonlinearity in (+1) dimensions. Commun. Nonlinear Sci. Numer. Simulat. 009(1):5-7. [9] Ebadi G, Biswas A. The G /G method and topological soliton solution of the K(m, n) equation. Commun. Nonlinear Sci. Numer. Simulat. 011;16:377. [10] Biswas A. 1-Soliton solution of the K(m, n) equation with generalized evolution. Phys. Lett. A 00;37(5):601-. [11] Biswas A. 1-Soliton solution of the K(m, n) equation with generalized evolution and time-dependent damping and dispersion. Comput. Math. Appl. 010;59():53-. [1] Ma WX. Travelling wave solutions to a seventh order generalized KdV equation. Phys. Lett. A 1993;10:1-. [13] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 199;60(7):650-. [1] Ma WX, Huang TW, Zhang Y. A multiple expfunction method for nonlinear differential equations and its application. Phys. Scr. 010(): [15] Kudryashov NA. Exact solitary waves of the Fisher equation. Phys. Lett. A 005;3(1): [16] Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Soliton Fract. 005;(5):

7 Appl. Math. Inf. Sci. 9, No., (015) / [17] Vitanov NK, Dimitrova ZI. Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics. Commun. Nonlinear Sci. Numer. Simul. 010;15(10):36 5. [1] Vitanov NK, Dimitrova ZI, Kantz H. Modified method of simplest equation and its application to nonlinear PDEs. Appl. Math. Comput. 010;16(9): [19] Hirota R. Exact solution of the Korteweg de Vries equation for multiple collision of solitons. Phys. Rev. Lett. 1971(7):119-. [0] Hirota R. The Direct Method in Soliton Theory. Cambridge University Press; 00. [1] Ma WX, Lee J-H. A transformed rational function method and exact solutions to the (3+1)-dimensional Jimbo Miwa equation. Chaos Solitons Fract. 009(): [] A.S.Arife, Solat Karimi Vanani, Ahmet Yildirim., Numerical Solution of Hirota-Satsuma Couple Kdv and a Coupled MKdv Equation by Means of Homotopy Analysis Method., World Applied Sciences Journal 01/011(13): [3] J.J. Truiljo, M.Rivero, B.Bonilla, On a Riemann-Liouvill Generalize Taylor s Formula, J.Math.Anal. 31, pp (1999). [] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: theory and applications, Gordon and Breach, Amsterdam, [5] J. L. Lavoie, T. J. Osler, R. Tremblay, Frational Derivatives and Special Functions, SIAM Review, V.1, Issue, pp.0-6. Ahmad Neirameh Assistant Prof in Department of Mathematics, Faculty of Science, University of Gonbad e Kavoos, Iran. His research interests are in the areas of applied mathematics and mathematical physics including the mathematical methods and models for complex systems,he has published research articles in reputed international journals of mathematical and engineering sciences. He is referee and editor of mathematical journals.