Applied Mathematical Sciences, Vol. 5, 2011, no. 46, 2289-2295 Exact Solutions to a Generalized Sharma Tasso Olver Equation Alvaro H. Salas Universidad de Caldas, Manizales, Colombia Universidad Nacional de Colombia asalash2002@yahoo.com FIZMAKO Research Group Abstract We apply the Cole Hopf transformation to linearize the forced Sharma Tasso Olver (STO equation in order to obtain some its exact solutions. Many kinds of solutions are formally derived for the unforced version. Mathematics Subject Classification: 35C05 Keywords: Sharma Tasso Olver equation; Cole Hopf transformation; linearized KdV equation; Mathematica 1 Introduction In this work we consider the forced Sharma Tasso Olver (STO equation u t +3αu 2 u x +3α(u 2 x + uu xx+αu xxx = F (t, (1 where u(x, t is the unknown function dependent on the variables x and t, α is an arbitrary real parameter and F (t is an external force. In the case wqhen F (t = 0 we obtain the Sharma-Tasso-Olver (STO equation [1],[2] u t +3αu 2 u x +3α(u 2 x + uu xx +αu xxx =0. (2 Some exact solutions to equation (2 were given in [5]. Our aim is to obtain exact solutions to forced version of this equation. 2 Solutions to forced STO equation.
2290 A. H. Salas We apply the Cole-Hopf transformation [3],[4] u(x, t = ξ log(v(ξ + F (tdt, where ξ = ξ(x, t =x + h(tdt, (3 being h(t an unknown function to be determined later. Substitution of (3 into (1 gives ( h(t+3α ( F (tdt 2 (v (ξ 2 v(ξv (ξ + 3α ( F (tdt 2 (v (ξv (ξ v(ξv (ξ + α(v (ξv (ξ v(ξv (ξ = 0. (4 The last term (4 equals zero if v(ξ satisfies following second order differential equation : v (ξ =kv(ξ, k 0. (5 whose general solution is given by v(ξ =c 1 exp( kξ+c 2 exp( kξ (6 If condition (5 holds, equation (4 reduces to ( ( 2 h(t+α(k +3 F (tdt (kv(ξ 2 v (ξ 2 =0. (7 From equation (7 we obtain ( ( h(t = α k +3 2 F (tdt. (8 From (3,(6 and (8 we see that following is a solution to forced STO equation (1 : u(x, t = c 1 exp( kξ+c 2 exp( kξ c 1 k exp( kξ c2 k exp( kξ + F (tdt, ξ = x αkt 3α ( F (tdt 2 dt. We always may choose the constants c 1 and c 2 to make solution (9 real valued. If k>0 we may choose c 1 and c 2 in the field of real numbers. If k<0, say k = λ 2 we take c 1 = c 2 0 and then u(x, t = cot(λξ+ F (tdt, ξ = x + αλ 2 t 3α ( (10 F (tdt 2 dt. (9
Solutions to STO 2291 3 Solutions to STO equation. In this section we obtain exact solutions to unforced version of equation (1, i.e. F (T 0.Thus, we will obtain exact solutions to equation (2. To this end, we again make use of the Cole-Hopf transformation in the form u(x, t =A x log f(x, t, A = const. (11 Substituting (11 into (2, integrating once with respect to x and taking the constant of integration equal to zero we obtain (A 1αf x ((A 2f 2 x +3ff xx +f 2 (f t + αf xxx = 0 (12 Equation (12 suggest the choice A = 1. Then, our problem reduces to solve following linear equation f t + αf xxx = 0 (13 Equation (13 is the simplest nontrivial third order partial differential equation. It is a simple model for the unidirectional propagation of linear dispersive waves. The key feature of equation (13 is that waves disperse, in the sense that those of different frequencies move at different speeds. Our goal is to understand the dispersion process. To streamline the argument, we replace real trigonometric solutions by complex exponentials, and so look at a solution whose initial profile f(0,x = exp(iλx (14 is a complex oscillatory function of wave number λ. (We will be observing oscillations in both space and time, and will reserve the term frequency for the time oscillations. Anticipating the induced wave to exhibit temporal oscillations, let us try an exponential solution ansatz f(x, t = exp(i(λx ωt (15 representing a complex oscillatory wave of temporal frequency ω and wave number (spatial frequency λ. Since f t = f t = iω exp(i(λx ωt and f xxx = 3 f x 3 = iλ3 exp(i(λx ωt, satisfies the partial differential equation (13 if and only if its frequency and wave number satisfy the dispersion relation ω = αλ 3 (16
2292 A. H. Salas Therefore, the exponential solution (15 of wave number λ takes the form f(x, t = exp(iλ(x + αλ 2 t. (17 It is easy to see that function It is easy to see that f(x, t = exp(k(x αk 2 t. (18 is also a solution to equation (13. By virtue of superposition principle, any linear combination of functions of the form (17 or (18 is also a solution to equation (13. This in turns gives following solutions to STO equation (1 : u 1 (x, t = x log c j sin(λ j (x + αλ 2 j t for n =1, 2, 3,... (19 u 2 (x, t = x log c j cos(λ j (x + αλ 2 jt u 3 (x, t = x log c j exp( k j (x αkj 2 t u 4 (x, t = x log c j sinh( k j (x αkj 2 t u 5 (x, t = x log c j cosh( k j (x αkj 2 t for n =1, 2, 3,... (20 for n =1, 2, 3,... (21 for n =1, 2, 3,... (22 for n =1, 2, 3,... (23 m u 6 (x, t = x log c j exp( k j (x αkj 2 t + d l cosh( k l (x αkl 2 t for m, n =1, 2, 3,... l=1 (24 and so on. We may obtain many other solutions STO equation (2 by using following property of its solutions : If u = u(x, t is a solution to equation (2, then function v = v(x, t defined by v = u x + u 2 u = u + x log u (25
Solutions to STO 2293 is also a solution to equation (2. This may be checked by direct computation. Formula (25 allows us to generate infinitely many solutions starting from a known solution. For example, it is easy to see that f(x, t =x + sin(λ(x + αλ 2 t is a solution to equation (13 and then, from (11 (with A = 1 function u = x log(x + sin(λ(x + αλ 2 t = 1 λ sin(λ(x + αλ2 t x + cos(λ(x + αλ 2 t (26 is solution to equation (2. Consequently, by formula (25 we obtain following solution to STO equation (2 : v = v(x, t = is also a solution to STO equation (2. λ2 cos(λ(x + αλ 2 t λ sin(λ(x + αλ 2 t 1. (27 4 Other solutions to linear dispersive equation f t + αf xxx =0. We may try different ansatze to solve equation (13. solutions : They give following Ansatz : f(x, t = sinh(k 1 x w 1 t sin(k 2 x w 2 t Solution to (13 : f(x, t = sinh(k 1 x αk 1 (k 2 1 3k2 2 t sin(k 2x αk 2 (3k 2 1 k2 2 t Solution to (2 : u(x, t =k 1 coth(k 1 x αk 1 (k 2 1 3k 2 2t+k 2 cot(k 2 x αk 2 (3k 2 1 k 2 2t (28 Ansatz : f(x, t = sinh(k 1 x w 1 t cosh(k 2 x w 2 t Solution : f(x, t = sinh(k 1 x αk 1 (k 2 1 +3k2 2 t cosh(k 2x αk 2 (3k 2 1 + k2 2 t Solution to (2 : u(x, t =k 1 coth(k 1 x αk 1 (k 2 1 +3k 2 2t+k 2 tanh(k 2 x αk 2 (3k 2 1 + k 2 2t (29 Solutions to STO equation (2 given in (28 and (29 are interesting, since they are linear combinations of two functions. Another solution to (13 in the form of a product of an exp function by a polynomial function is : f(x, t =(A + Bx + Cx 2 6αCk 2 tx +9α 2 Ck 4 t 2 3αk(2C + Bkt exp(kx αk 3 t. (30 Other techniques to solve nonlinear pde s may be found in [6]-[17].
2294 A. H. Salas 5 Conclusions We have obtained exact solutions to forced Sharma-Tasso-Olver equation by using the Cole-Hopf transformation. We think that some of the solutions given in this paper are new in the open literature. References [1] A.S. Sharma, H., Connection between wave envelope and explicit solution of a nonlinear dispersive equation. Report IPP 6/158. 1977. [2] P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977 1212 1215. [3] J. Cole 1951. On a quasilinear parabolic equation occuring in aerodynamics. Quarterly Journal of Applied Mathematics, 9:225 236. [4] E. Hopf. The partial differential equation u t + uu x = µu xx. Communications in Pure and Applied Mathematics, 1950, 3:201 230. [5] L. Zeng-Ju, S. Y. Loua, Symmetries and exact solutions of the Sharma Tass Olver equation, Nonlinear Analysis 63 (2005 e1167 e1177. [6] A. H. Salas, Symbolic Computation of Solutions for a forced Burgers equation, Applied Mathematics and Computation 216 (2010 18 26. [7] A. H. Salas, Exact solutions for the general fifth KdV equation by the exp function method, Applied Mathematics and Computation 205 (2008. [8] A. H. Salas, New solutions to Korteweg De Vries (kdv equation by the Riccati equation expansion method, International Journal of Applied Mathematics (IJAM 22(2009, pp. 1169-1177. [9] A. Salas, Some solutions for a type of generalized Sawada Kotera equation, Applied Mathematics and Computation 196 (2008 812 817. [10] A. H. Salas, Symbolic Computation of exact solutions to KdV equation, Canadian Applied Mathematics Quarterly, vol. 16, No. 4(2008. [11] A. H. Salas, Exact solutions of coupled sine Gordon equations, Nonlinear Analysis: Real World Applications 11 (2010 3930 3935. [12] A. H. Salas, Computing solutions to a forced KdV equation, Nonlinear Analysis: Real World Applications 12 (2011 1314 1320.
Solutions to STO 2295 [13] A. H. Salas, Exact solutions to mkdv equation with variable coefficients, Applied Mathematics and Computation 216 (2010 2792 2798. [14] A. H. Salas, Computing exact solutions to a generalized Lax seventh-order forced KdV equation (KdV7, Applied Mathematics and Computation 216 (2010 2333 2338. [15] A. H. Salas, About the general KdV6 and its exact solutions, Applied Mathematical Sciences,Vol. 5, 2011, no. 13, 631 638. [16] A. H. Salas, Construction of N soliton solutions to (2+1-dimensional Konopelchenko Dubrovsky (KD equations, Applied Mathematics and Computation 217 (2011 7391 7399. [17] A. H. Salas, L. J. Martinez and O. Fernandez, Reaction Diffusion Equations: A Chemical Application, Scientia et Technica, Universidad Tecnológica de Pereira, Risaralda Colombia XVII, No 46, 2010. Received: February, 2011