Symmetry reductions and exact solutions for the Vakhnenko equation

Transcription

1 XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, 1-5 septiembre 009 (pp. 1 6) Symmetry reductions and exact solutions for the Vakhnenko equation M.L. Gandarias 1, M.S. Bruzón 1, Dpto. Matematicas, Universidad de Cádiz, Rio San Pedro, Puerto Real, Cádiz. s: marialuz.gandarias@uca.es, matematicas.casem@uca.es. Palabras clave: symmetries Resumen The integrable Vakhnenko equation is analysed from the point of view of symmetry groups theory. Using classical Lie symmetries, we consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. It is shown that the reduced ordinary differential equation with the special parameters can be solved and many novel solutions will be derived in terms of Jacobi elliptic functions, where some known solutions will be recovered when the modulus arrives its limiting value. The corresponding solutions of the Vakhnenko equation can be expressed in terms of Jacobi elliptic functions, and involve an arbitrary smooth function. 1. Introduction To describe gravity waves propagating down a channel under the influence of Coriolis force Ostrovsky [1] derived the approximate nonlinear equation (u t + (u ) x βu xxx ) x = γu, (1) with β R and γ > 0, where β and γ are dispersion coefficients. Vakhnenko and Parkes [5] proved that the reduced Ostrovsky equation can be transformed to the new integrable equation uu xxt u x u xt + u u t = γu () which is known as Vakhnenko equation. In a recent paper [4] the tanh and sine cosine methods were used to construct exact periodic and soliton solutions of nonlinear evolution equations such as () with γ = 0. 1

2 M.L. Gandarias, M.S. Bruzón In this work, we study equation () with γ = 0 from the point of view of the theory of symmetry reductions in partial differential equations. Then, we use the transformations groups to reduce the equations to ordinary differential equations. In [3] Kudryashov pointed out that many of the so called new travelling wave solutions could be derived from the solutions of a simple second order nonlinear ordinary differential equation. By using this equation we obtain for the Vakhnenko equation () many exact solutions that can be expressed in terms of the Jacobi elliptic functions and consequently in terms of trigonometric and hyperbolic functions. Hence, we get for () plenty of periodic waves, solitary waves, with an arbitrary function.. Classical symmetries In this section we perform Lie symmetry analysis for the Vakhnenko equation (). Let us consider a one-parameter Lie group of infinitesimal transformations in (x, t, u) given by x = x + εξ(x, t, u) + O(ε ), t = t + ετ(x, t, u) + O(ε ), u = u + εφ(x, t, u) + O(ε ), (3) where ε is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of the equation (). This yields to the overdetermined, linear system of eleven equations for the infinitesimals ξ(x, t, u), τ(x, t, u) and φ(x, t, u). The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form v = ξ x + τ t + φ u. (4) Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition Φ ξ u x + τ u φ = 0. (5) t By solving this system we get that ξ = k 1 x + k, τ = τ(t), (6) φ = k 1 u. We obtain that the admitted group of symmetries has generators: v 1 = x, v = α(t) t, v 3 = x x u u. Our aim in this paper is to use the theory of symmetry reductions to find solutions for (). In order to obtain these solutions, we consider the following reductions arising from the following vector fields.

3 Symmetry reductions for the Vakhnenko equation Reduction 1 By using the generator k 1 v 3 + k v 1 + v we obtain the similarity variables and similarity solution and the PDE E 1 where β(t) = z = log(k 1x + k ) k 1 β(t), u = h(z) (k 1 x + k ), (7) hh h h + h h + k 1 ((h ) 3hh ) + k 1hh = 0, (8) dt α(t). Reduction By using the generator v 1 + v we obtain the similarity variables and similarity solution z = x β(t), u = h(z), (9) and the ODE E hh h h + h h = 0. (10) 3. Travelling wave solutions The Korteweg-de-Vries equation u t + 6uu x + u xxx = 0 (11) is the most known nonlinear partial differential equation. To find travelling wave solutions we take u(x, t) = h(z), z = x λt, (1) and we obtain h zz + 3h λh + c 1 = 0. (13) It was pointed out in [3] that equation (13) is very important and the author exhibited many examples of nonlinear partial differential equations describing physical phenomena where equation (13) arises. Multiplying equation (13) by h z and integrating once with respect to z we get the nonlinear equation We are considering the equation (h z ) + h 3 λh + c 1 h + c = 0. (14) h zz + bh 3 + ch + dh + e = 0, (15) and multiplying equation (15) by h z and integrating once with respect to z we get the nonlinear equation (h z ) + b h4 + c 3 h3 + dh + eh = 0. (16) We observe the following: Equation (10) admits any solution of equation (15) for b = 0, e = 0 and c = 1. Hence, for equation (10) we get the following solutions: 3

4 M.L. Gandarias, M.S. Bruzón Case 1 with h = a sn (k z, p) + a = b = 6k p, ( ) a 0 = k p + 14p p + 1. b sn (k z, p) + a 0 (17) For special values of p we get 6k h = cosh (kz) sinh (kz), p = 1, h = 4k 6k sin (kz), p = 0. (18) Case with h = b sn (k z, p) + a 0 (19) Case 3 with b = 6k, a 0 = k p 6 k 4 4 k + d, d = 4 k ( 5 k p 18 k 4 p + 4 k p + 3 d p + 9 k 6 3 d k 4 k ). a = 6k p, For special values of p we get h = a sn (k z, p) + a 0 (0) a 0 = k p p k (p + 1). h = 6k sech (kz), p = 1, h = 4k (1) 6k sin (kz), p = 0. From solutions of (10) and using (9) we obtain the following solutions for () in terms of the Jacobi elliptic function sn: 4

5 Symmetry reductions for the Vakhnenko equation Figura 1: Solution u = 6sech (x β(t)) with β(t) = Airy Ai(t). u = a sn (k(x β(t)), p) + u= sn sn b + a0, (k (x β(t)), p) b + a0, (k (x β(t)), p) () u = a sn (k (x β(t)), p) + a0, and the corresponding solutions in terms of the degenerate trigonometric and hyperbolic functions u = 6k sech (k(x β(t))cosech (k(x β(t)), u = 4k 6k cosec (k(x β(t)), (3) u = 6k sech (k(x β(t)). Some particular cases of these solutions, with β(t) = λt, were derived in [4] by using the tanh method and the sine cosine method. 4. Conclusions By using classical Lie symmetries, we have considered travelling-wave reductions with a variable velocity depending on the form of an arbitrary function for the integrable Vakhnenko equation. We have shown that the reduced ordinary differential equation with 5

6 M.L. Gandarias, M.S. Bruzón the special parameters can be solved in terms of Jacobi elliptic functions. The corresponding solutions of the Vakhnenko equation have been expressed in terms of Jacobi elliptic functions, and involve an arbitrary smooth function. 5. Acknowledgments The support of DGICYT project MTM , Junta de Andalucía group FQM- 01 and project P06-FQM are gratefully acknowledged. Referencias [1] L.A. Ostrovsky Okeanologiya, (1978), [] O.A. Gilman, et al. Stud Appl Math, (1995), 95, [3] N. Kudryashov Commun Nonlinear Sci Numer Simulat 14 (009) [4] E.Yusufoglu, A. Bekir Chaos Solitons and Fractals (008) 3, [5] V.O. Vakhnenko, E.J. Parkes. Nonlinearity (1998), 11,