Solutions for the Combined sinh-cosh-gordon Equation

Transcription

1 International Journal of Mathematical Analysis Vol. 9, 015, no. 4, HIKARI Ltd, Solutions for the Combined sinh-cosh-gordon Equation Ana-Magnolia Marin-Ramirez Verónica-Patricia Jaramillo-Camacho Ruben-Dario Ortiz-Ortiz Copyright c 015 Ana-Magnolia Marin-Ramirez, Verónica-Patricia Jaramillo-Camacho and Ruben-Dario Ortiz-Ortiz. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper is based on finding solutions for the combined sinh-cosh- Gordon equation using Hamiltonnian systems at a specific region on the plane in which these do not have periodic orbits. Mathematics Subject Classification: 34A34 Keywords: dynamical systems, Hamiltonnian systems, Poincaré Bendixson theorem, periodic orbits, combined sinh-cosh-gordon equation

2 1160 A. M. Marin, R. D. Ortiz and V. P. Jaramillo 1 Introduction The search of solutions to the combined sinh-cosh-gordon equation (1) has been of great interest. The projective Ricatti equation method has been used to show twelve new exact solutions [1] while it was claimed that these can not be considered new because these can be obtained as a particular case of the general solution using elementary methods []. The objective of this work is to seek traveling wave solutions through Hamiltonian systems and to find non periodic solutions at a specific region on the plane [3, 4, 5]. Combined sinh-cosh-gordon Equation The combined sinh-cosh-gordon equation is given by: u tt κu xx + α sinh(u) + β cosh(u) = 0 (1) where u is a real scalar function of two independents variables x and t, and α and β are two different constants from zero. Theorem.1. Traveling waves solutions for (1) are given by (6) and (7). Proof. We introduce the transformations: sinh u = V V 1, cosh u = V + V 1, V = e u. Hence, we obtain the equation: V (V tt κv xx ) + (κv x V t ) + (β + α)v 3 + (β α)v = 0. () Substituting V = υ(ξ) = υ(x + λt), we have that (α + β)υ 3 (α β)υ + (λ κ)υυ (λ κ)(υ ) = 0. (3) We will study the case, α β = 0. So, we obtain that Taking µ = λ κ, then we have αυ 3 + (λ κ)υυ (λ κ)(υ ) = 0. (4) αυ 3 + µυυ µ(υ ) = 0. (5)

3 Solution for the combined sinh-cosh-gordon equation 1161 Its solutions are given by: υ(ξ) = c 1µ + c 1 µ tanh ( 1 ( c 1 c c 1 ξ)) α υ(ξ) = c 1µ + c 1 µ tanh ( 1( c 1 ξ + c 1 c )). (7) α (6) 3 Solutions through Hamiltonnian System Theorem 3.1. The solutions for the equation (3) with α = β through Hamiltonnian system are given by (8). Proof. Considering the following variable changes: υ = y, υ = y, x = υ and µ = λ κ we obtain the system: { x = y, y = y x α µ x. Its Hamiltonnian system is given by { x = H y, y = H x then for some constant C. H = y + y ln x α x 3 µ 3 = C (8) 4 Solutions through Dynamical Systems Theorem 4.1. There are no periodic solutions for the equation (1) at a specified region on the plane. Proof. We consider the system { x = y, y = y x µ x with x 0, µ > 0, α < 0. To ensure the non existence of periodic orbits we will make use of the Poincaré- Bendixson theorem { x 1 = f(x 1, x ) x = g(x 1, x )

4 116 A. M. Marin, R. D. Ortiz and V. P. Jaramillo then it does not have periodic orbits when (hf 1 ) + (hf ) 0 x 1 x where h is a Dulac function, we will go to find this function, then h x1 f 1 + h x f > h( f 1 x 1 f x ) [ ( h h f1 f 1 + f = h C(x 1, x ) + f )]. x 1 x x 1 x Let x 1 = x, x = y then f 1 = y and f = µy αx 3 µx hence y h x 1 + f 1 x 1 = 0, ( µy αx 3 µx f = µy x µx = y x ) ( ) h y + h = hc(x 1, x ). x x If h x = 0, then y h ( ) y + h = hc(x 1, x ). x 1 x If h = h, then h = x ex. Substituting into the above equation we have: ( y 1 + ) = C. x Hence C > 0. Therefore, the region on the plane in which there are no periodic orbits is y > 0 and x > 0. Acknowledgements. The authors express their deep gratitude to Universidad de Cartagena for partial financial support. References [1] C.A. Gómez, A. H. Salas, New exact solutions for the combined sinh-cosh- Gordon equation, Lecturas Matemáticas, 7 (006), [] J. C. López-Carreño, R. Mendoza-Suarez, Un comentario sobre New exact solutions for the combined sinh-cosh-gordon equation, Lecturas Matemáticas, 3 (011), 3-7.

5 Solution for the combined sinh-cosh-gordon equation 1163 [3] A. M. Marín, R. D. Ortiz, J. A. Rodriguez-Ceballos, On the nonexistence of periodic orbits of some quadractic systems, Far East Journal of Mathematical Sciences, 89 (1) (014), [4] A. M. Marín-Ramirez, R. D. Ortiz-Ortiz, J. A. Rodriguez- Ceballos, Quadratic Systems without periodic orbits, International Journal of Mathematical Analysis, 8(4) (014), [5] L. Perko, Differential Equations and Dynamical Systems, Springer, Berlin, (006). Received: March 10, 015; Published: April 14, 015