Analytic Solutions for A New Kind. of Auto-Coupled KdV Equation. with Variable Coefficients

Transcription

1 Theoretical Mathematics & Applications, vol.3, no., 03, ISSN: (print), (online) Scienpress Ltd, 03 Analytic Solutions for A New Kind of Auto-Coupled KdV Equation with Variable Coefficients Dongpo Hu and Cuncai Hua Abstract In this paper, we apply the extended variable-coefficient mapping method to discuss a new kind of auto-coupled KdV equation with variable coefficients. By solving nonlinear differential algebraic equations which are derived from nonlinear evolution equations, many Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric periodic solutions for the auto-coupled KdV equation with variable coefficients are derived. By selecting the appropriate parameter values, some exact solutions of the other forms are also obtained. Mathematics Subject Classification: 35M0, 35Q5, 35Q53 Keywords:the extended variable-coefficient mapping method; auto-coupled KdV equation with variable coefficients; elliptic function solutions School of Mathematics, Yunnan Normal University, KunMing, Yunnan,650500, P.R. China, hudongpo006@6.com School of Mathematics, Yunnan NormalUniversity, KunMing, Yunnan, , P.R.China, cuncai-hua@39.com Corresponding author. Article Info: Received : December, 0. Revised : February 0, 03 Published online : April 5, 03

2 70 Auto-Coupled KdV Equation Introduction Nonlinear evolution equations play an important role in description of natural phenomena. As the soliton phenomena were first observed by Scott Russell in 83[], researchers began to study the explicit solutions of nonlinear evolution equations, many powerful methods to seek explicit exact solitary wave solutions of nonlinear evolution equations have been established and developed. Since constructive methods transform the problem of solving nonlinear evolution equations into the problem of solving the corresponding systems of algebraic equations, the problem of solving nonlinear evolution equations can be simplified, and it can also reveal many of the essential attribute of equations. Some methods have been widely applied and extended, such as Bäcklund transformation[,3], sine-cosine method[], homogeneous balance method[5], tanh-function expansion method[6,7], the extended tanh-function expansion method[8,9], the Jacobi elliptic function expansion method[0,], the Riccati expansion method[]. At present, people have paid more attention to the KdV equation with variable coefficients. Using different constructive methods, some researchers have obtained exact solutions of the KdV equation with variable coefficients. For example Liu Shi-Kuo et al. applied extended Jacobi elliptic function expansion method to construct the exact solutions of variable coefficients KdV equation, getting the solitary wave solutions and soliton solutions easier[3]. Based on the idea of the homogeneous balance method, Fan En gui obtained the Bäcklund transformation and similarity reductions of general variable coefficient KdV equation[]. Woopyo Hong et al. found analytic solutions for general variable coefficient KdV equation and made use of both the truncate Painleve expansion and symbolic computation to obtain an auto-bäcklund transformation and certain soliton-typed analytic solutions[5]. Li Desheng and Zhang Hongqing obtained exact soliton-like, rational formal and trigonometric function solutions of the general variable coefficient KdV and MKdV equations by using the extended tanh-function method[6]. Based on the reduced method, the variable separation method of extended mapping method has been widely applied to solve exact solutions of nonlinear evolution equations[7 9], Zhang Sheng and Xia Tiecheng proposed a variable coefficient extended mapping method and applied this method to

3 Dongpo Hu and Cuncai Hua 7 the MKdV equaiton with variable coefficients and (+)dimensional Nizhnik- Novikov-Vesolov equations. Many new and more general exact solutions including Jacobi elliptic function solutions,hyperbolic function solutions and trigonometric function solutions are obtained. However, at present, there are less of papers which applied the extended variable coefficient mapping method to the variable coefficient coupled equations. This present work is motivated to apply the method from[9] to a new kind of auto-coupled KdV equations, and some exact solutions are derived. Particularly, by selecting some different parameter values,we obtain the corresponding other forms solutions, including the Jacobi elliptic function solutions, hyperbolic function solutions and tribonometric function solutions. The rest of this paper is arranged as follows: In Section, we shall describe the variable-coefficient extended mapping method for searching solutions of nonlinear evolution equations with variable coefficients and give the main steps of the method. In Section 3, we shall apply this method to a new kind of autocoupled KdV equation with variable coefficients and obtain some solutions. In Section, some conclusions are given. Description of method For a given partial differential, say, in two variables x and t F (u, u t, u x, u xx, u xt, ) = 0, () where u x = u, u x xx = u, u x xt = u,, the same hereafter. x t We seek solutions of Eq.() in following form[9]: u = a 0 + n a i f i () + i= n i= b i f i () + n c i f i ()f () + i= n i= d i f i ()f (), () where a i = a i (X), b i = b i (X), c i = c i (X), d i = d i (X), = (X) and X = X(x, t) are all functions to be determined later. f() satisfies the following auxiliary differential equation: f () = pf () + qf () + r, (3)

4 7 Auto-Coupled KdV Equation and hence holds for f() and f (): f () = pf 3 () + qf(), f () = (6pf () + q)f (), f () () = p f 5 () + 0pqf 3 () + (q + pr)f(), () f (5) () = (0p f () + 60pqf () + q + pr)f (),. where = d, p, q and r are all real parameters. d To determine u = u(x, x, x 3,, t) explicitly, we take the following four steps: Step. Determining the integer n by considering the homogeneous balancing between the highest order nonlinear term(s) and the highest order partial derivative of u in Eq.(). Step. With the aid of symbolic computation of the software Maple, substituting Eq.() along with (3) and () into Eq.(), and collecting all the terms with the same order of f l ()f j ()(l = 0, ; j = 0, ±, ±,...) together, then the left- hand side of Eq.() is converted into a polynomial in f l ()f j ()(l = 0, ; j = 0, ±, ±,...). Setting each coefficient to zero yields a set of over-determined differential equations for a 0, a i, b i, c i, d i (i =,,...) and. Step 3. Solving the system of over-determined differential equations obtained in Step for a 0, a i, b i, c i, d i (i =,,...) and by use of Maple. Step. Using the results obtained in above steps to derive a series of fundamental solutions of Eq.() which depend on the solution f() of Eq.(3). For given different values of p, q, r, Eq.(3) has many kinds of Jacobi elliptic solutions, which are listed as follows: f() p q r, cd m ( + m ) cn m m m dn m m

5 Dongpo Hu and Cuncai Hua 73 f() p q r ns, dc ( + m ) m nc m m m nd m m sc m m sd m (m ) m cs m m continued ds m m (m ) mcn ± dn ns ± cs, cn m ±dn m nc ± sc ns ± ds ± icn, m ± idn, ±dn dn ±m cn ± cn±dn cn m ±dn dn m ±cn ±cn m m m m ( m ) m +m ( m ) m +m m m m m m m m +m m +m m +m m Where i =. Selecting proper values of p, q, r and corresponding f(), then substituting them along with a i, b i, c i, d i and into Eq.(), we can obtain exact solutions of Eq.(), from which hyperbolic solutions and trigonometric function solutions can be obtained in the limit cases when m and m 0. Here = sn(, m), cn = cn(, m) and dn = dn(, m) are Jacobi elliptic sine function, Jacobi elliptic cosine and Jacobi elliptic function of the third kind respectively, m denotes the modulus of Jacobi elliptic functions. Other functions are derived from these three kinds of functions[0]: ns =, nc = cn, nd = dn, sc = cn, cs = cn, sd = dn, dn ds =, cn cd = dn, dc = dn cn.

6 7 Auto-Coupled KdV Equation The Jacobi elliptic functions degenerate into hyperbolic functions when m []: lim sn(, m) = tanh, lim m lim ns(, m) = coth, lim m lim sc(, m) = sinh, lim m cn(, m) = sech, lim m nc(, m) = cosh, lim m sd(, m) = sinh, lim m dn(, m) = sech, m nd(, m) = cosh, m cd(, m) =, m lim cs(, m) = csch, lim ds(, m) = csch, lim dc(, m) =. m m m The Jacobi elliptic functions degenerate into trigonometric functions when m 0: lim sn(, m) = sin, lim lim ns(, m) = csc, lim lim sc(, m) = tan, lim cn(, m) = cos, lim nc(, m) = sec, lim sd(, m) = sin, lim dn(, m) =, nd(, m) =, cd(, m) = cos, lim cs(, m) = cot, lim ds(, m) = csc, lim dc(, m) = sec. In this paper, we consider the following coupled KdV equation with variable coefficients F (t, u, v, u t, v t, u x, v x, u xx, v xx, u xxx,...) = 0, F (t, u, v, u t, v t, u x, v x, u xx, v xx, v xxx,...) = 0. In order to search for explicit solutions of Eq.(5), we suppose that the solutions of Eqs.(5) can be expressed as n n n u = a 0 + a i f i () + b i f i () + c i f i ()f () + n n n v = A 0 + A i f i () + B i f i () + C i f i ()f () + n i= i= i= D i f i ()f (). i= i= i= i= n i= (5) d i f i ()f (), where a i = a i (X), b i = b i (X), c i = c i (X), d i = d i (X), A i = A i (X), B i = B i (X), C i = C i (X), D i = D i (X), = (X), X = X(x, t) are all functions to be determined later. By balancing the highest order nonlinear term(s) and the highest order partial derivative in Eqs.(6), we can get the values of n, n. (6)

7 Dongpo Hu and Cuncai Hua 75 3 Application General variable coefficient KdV equation is given as u t + α(t)uu x + β(t)u xxx = 0, (7) where α(t) and β(t) are arbitrary functions of t,the equation first was introduced by Grimshaw. In this paper,a new auto-coupled KdV equations with variable coefficients is as follows: u t + α(t)uv x + β(t)u xxx = γ(t)v x, (8) v t + α(t)vu x + β(t)v xxx = γ(t)u x. where β(t) = δα(t), δ is a constant. By balancing u xxx and uv x, v xxx and vu x, we get n =, n =. In order to search for explicit solutions, we assume that Eqs.(8) has the following solutions of the form: u() = a 0 + a f() + a f () + b + b f() + c f () f f () + d () + d f() f (), f () v() = A 0 + A f() + A f () + B + B f() + C f () f f () + D () f() f +D (). f () (9) where a 0 = a 0 (t), a = a (t), a = a (t), b = b (t), b = b (t), c = c (t), d = d (t), d = d (t), A 0 = A 0 (t), A = A (t), A = A (t), B = B (t), B = B (t), C = C (t), D = D (t), D = D (t), = κx + η, κ = κ(t), η = η(t). Substituting Eqs.(9),(3) and () into Eqs.(8), collecting the coefficients with the same power x µ f l ()f j ()(µ = 0, ; l = 0, ; j = 0, ±, ±, ) and setting each of the obtained coefficients to be zero, we get a set of over-determined nonlinear differential algebraic equations, which is omitted here. Solving the set of over-determined nonlinear differential algebraic equations by using symbolic computation of software Maple, we have results in the following cases: Case a 0 = c, a = 0, a = 0, b = 0, b = δrk, c = 0, d = 0, d = 0, A 0 = c, A = 0, A = 0, B = 0, B = δrk, C = 0, D = 0, D = 0, κ = k, η = (γ(τ)k α(τ)kc β(τ)qk 3 )dt. (0)

8 76 Auto-Coupled KdV Equation where c and k are arbitrary constants. Therefore, the first solution of Eqs.(8) are given as follows: u = v = c δrk f (), () We can obtain many new and more types of exact solutions of Eqs.(8). For example, selecting p = m, q = m, r = m and f() = cn yields Jacobi elliptic function solutions as u = v = c δ( m )k nc, () In the limit cases when m 0, from Eq.() we can obtain trigonometric function solutions as u = v = c δk sec, (3) Selecting p =, q = m, r = and f() = cn m ±dn yields u = v = c 3δk ( m ± dn), () cn when m 0, from Eq.()we can obtain trigonometric function solutions as Selecting p =, q = m, r = (sin ± ) u = v = c 3δk, (5) cos and f() = ±cn yields ( ± cn) u = v = c 3δk, (6) sn When m from Eq.(6) we can obtain hyperbolic function solutions ( ± sech ) u = v = c 3δk tanh, (7) When m 0 from Eq.(6) we can obtain trigonometric function solutions ( ± cos ) u = v = c 3δk sin, (8) where = kx (γ(τ)k α(τ)kc β(τ)qk 3 )dt.

9 Dongpo Hu and Cuncai Hua 77 Case a 0 = c, a = 0, a = δpk, b = 0, b = 0, c = 0, d = 0, d = 0, A 0 = c, A = 0, A = δpk, B = 0, B = 0, C = 0, D = 0, D = 0, κ = k, η = (γ(τ)k α(τ)kc β(τ)qk 3 )dt. where c and k are arbitrary constants. Therefore, the second solution of Eqs.(8) are given as follows: (9) u = v = c δpk f (), (0) Selecting p =, q = ( + m ), r = m and f = dc yields u = v = c δk dc, () When m 0, from Eq.() we can obtain trigonometric function solutions as Selecting p =, q = m, r = u = v = c δk sec, () and f = +cn yields u = v = c 3δk sn ( + cn), (3) When m, from Eq.(3) we can obtain hyperbolic function solutions as u = v = c 3δk tanh ( + sech ), () When m 0, from Eq.(3) we can obtain trigonometric function solutions as u = v = c 3δk sin ( + cos ), (5) where = kx (γ(τ)k α(τ)kc β(τ)qk 3 )dt. Case 3 a 0 = c, a = 0, a = δpk, b = 0, b = δrk, c = 0, d = 0, d = 0, A 0 = c, A = 0, A = δpk, B = 0, B = δrk, C = 0, D = 0, D = 0, κ = k, η = (γ(τ)k α(τ)kc β(τ)qk 3 )dt. (6) where c and k are arbitrary constants.

10 78 Auto-Coupled KdV Equation Therefore, the third solutions of Eqs.(8) are given as follows: u = v = c δpk f () δrk f (), (7) Selecting p =, q = m, r = and f() = cn yields m ±dn u = v = c 3δk cn ( m ± dn) ( m ± dn) 3δk, (8) cn When m 0, from Eq.(8) we can obtain trigonometric function solutions as u = v = c 3δk cos (sin ± ) 3δk, (9) (sin ± ) cos Selecting p =, q = m, r = and f() = ±cn yields u = v = c 3δpk sn ( ± cn) 3δk, (30) ( ± cn) sn When m, from Eq.(30) we can obtain hyperbolic function solutions as u = v = c 3δpk tanh ( ± sech ) 3δk ( ± sech ) tanh, (3) When m 0, from Eq.(30) we can obtain trigonometric function solutions as u = v = c 3δpk sin ( ± cos ) 3δk ( ± cos ) sin, (3) Selecting p =, q = m, r =, f() = ns ± cs yields u = v = c 3δk (ns ± cs) 3δk (ns ± cs), (33) When m, from Eq.(33) we can obtain hyperbolic function solutions as u = v = c 3δk (coth ± csch ) 3δk (coth ± csch ), (3) When m 0, from Eq.(33) we can obtain trigonometric function solutions as u = v = c 3δk (csc ± cot ) 3δk (csc ± cot ). (35) where = kx (γ(τ)k α(τ)kc β(τ)qk 3 )dt.

11 Dongpo Hu and Cuncai Hua 79 Case a 0 = c, a = 0, a = 6δpk, b = 0, b = 0, c = ±6δk p, d = 0, d = 0, A 0 = c, A = 0, A = 6δpk, B = 0, B = 0, C = ±6δk p, D = 0, D = 0, κ = k, η = (γ(τ)k α(τ)kc β(τ)qk 3 )dt. (36) where c and k are arbitrary constants. Therefore, the fourth solutions of Eqs.(8) are given as follows: u = v = c δpk f () ± 6δk pf (), (37) Selecting p =, q = m, r = m and f = cs yields u = v = c δk cs ± 6δk cs = c δk cs 6δk nsds, (38) When m, from Eq.(38) we can obtain hyperbolic function solutions as u = v = c δk csch 6δk coth csch, (39) When m 0, from Eq.(38) we can obtain trigonometric function solutions as u = v = c δk cot 6δk csc, (0) where = kx (γ(τ)k α(τ)kc β(τ)qk 3 )dt. Cases 5 a 0 = c, a = 0, a = 0, b = 0, b = 6δrk, c = 0, d = 0, d = ±6δk r, A 0 = c, A = 0, A = 0, B = 0, B = 6δrk, C = 0, D = 0, D = ±6δk r, κ = k, η = (γ(τ)k α(τ)kc β(τ)qk 3 )dt. () where c and k are arbitrary constants. Therefore, the fifth solutions of Eqs.(8) are given as follows: u = v = c 6δrk f () ± 6δk r f () f (), ()

12 80 Auto-Coupled KdV Equation Selecting p = m, q = ( + m ), r =, f = yields u = v = c 6δk sn ± sn 6δk sn (3) = c 6δk cndn ± sn 6δk sn, When m, from Eq.(3) we can obtain hyperbolic function solutions as u = v = c 6δk tanh ± sech 6δk tanh = c 6δk tanh ± 6δk sinh, () When m 0, from Eq.(3) we can obtain trigonometric function solutions as u = v = c 6δk cos sin ± 6δk sin, (5) Selecting p = ( m ), q = +m, r = and f = cn±dn yields u = v = c 3 (cn ± dn) δk ± 3δk sn (cn ± dn) (cn ± dn ) sn sn = c 3 (cn ± dn) δk sn ± 3δk cn dn + sn dn + cn, sn When m, from Eq.(6) we can obtain hyperbolic function solutions as so or u = v = c 3 (sech ± sech ) δk tanh (6) ± 3δk sech3 + tanh sech + sech tanh, (7) u = v = c 3δk sech3 + tanh sech + sech tanh, (8) u = v = c 6δk sech tanh ± sech3 + tanh sech + sech 3δk tanh = c 6δk sinh ± sech3 + tanh sech + sech 3δk tanh, (9) When m 0, from Eq.(6) we can obtain trigonometric function solutions as u = v = c 3 (cos ± ) δk sin ± 3δk cos + sin + cos sin = c 3 (cos ± ) δk sin ± 3δk + cos sin. (50)

13 Dongpo Hu and Cuncai Hua 8 where = kx (γ(τ)k α(τ)kc β(τ)qk 3 )dt. Conclusion In this paper, a new kind of auto-coupled KdV equation with variable coefficients is proposed. We apply the extended variable coefficient mapping method to the coupled model, and obtain many exact solutions which include Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric function solutions of auto-coupled KdV equation with variable coefficients. Acknowledgements. This work is supported by the Natural Science Foundation of China (Grant No.07758, 600) and the Planning Program of Leading Youth Scholars of Science and Technology of Yunnan Province of China (Grant No.008PY059). References [] J.S. Russel, Report on waves, Rep. th Meet, Brit. Assoc. Adv. Sci., (8), [] M. Wang, Y. Wang and Y. Zhou, An auto-bäcklund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications, Physics Letters A, 303(), (00), 5 5. [3] B. Li, Y. Chen and H. Zhang, Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order, Physics Letters A, 305(6), (00),

14 8 Auto-Coupled KdV Equation [] A. Bekir, New solitons and periodic wave solutions for some nonlinear physical models by using the sine cosine method, Physica Scripta, 77(), (008), 50. [5] M. Wang, Exact solutions for a compound KdV-Burgers equation, Physics Letters A, 3(5-6), (996), [6] W. Malfliet and W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 5, (996), 563. [7] W. Malfliet and W. Hereman, The tanh method: II. Perturbation technique for conservative systems, Physica Scripta, 5, (996), 569. [8] Z. Lü, H. Zhang, On a further extended tanh method, Physics Letters A, 307(5-6), (003), [9] A.M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Applied Mathematics and Computation, 87(), (007), 3. [0] S.Liu, Z. Fu, S. Liu and Q. Zhao, Expansion Method about the Jacobi Elliptic Function and Its Applications to Nonlinear Wave Equations, Acta Physica Sinica, 50(), (00), [] E. Fan and J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, 305(6), (00), [] Gegenna Taogetusang, Constructing New Exact Solutions to Several Nonlinear Evolution Equations by Riccati Equation, Journal of Inner Mongolia Normal University (Natural Science), 0(5), (0), [3] S. Liu, Z. Fu, S. Liu and Q. Zhao, Jacobi elliptic function expansion solution to the variable coefficient nonlinear equations, Acta Physica Sinica, 5(9), (00), [] E. Fan, Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations, Physics Letters A, 9(), (00), 6 30.

15 Dongpo Hu and Cuncai Hua 83 [5] W. Hong and Y.D. Jung, Auto-Bäcklund transformation and analytic solutions for general variable-coefficient KdV equation, Physics Letters A, 57(3-), (999), 9 5. [6] D. Li and H. Zhang, Improved tanh-function method and the new exact solutions for the general variable coefficient KdV equation and MKdV equation, Acta Physica Sinica, 5(7), (003), [7] C.L. Zheng, J.P. Fang and L.Q. Chen, New variable separation excitations of (+ )-dimensional dispersive long-water wave system obtained by an extended mapping approach, Chaos, Solitons & Fractals, 3(5), (005), [8] M.A. Abdou, Exact periodic wave solutions to some nonlinear evolution equations, International Journal of Nonlinear Science, 6(), (008), [9] S. Zhang and T. Xia, Variable-coefficient extended mapping method for nonlinear evolution equations, Physics Letters A, 37(), (008), [0] Ahmad T. Ali, New generalized Jacobi elliptic function rational expansion method, Journal of computational and applied mathematics, 35(), (0), 7 7. [] W. Hong and Y.D. Jung, A Generalized F-expansion Method and its Application to (+ )-dimensional Breaking Soliton Equations, Internatioal Journal of Nonlinear Science, 5(), (008), 5 3.