Approximation to the Dissipative Klein-Gordon Equation

Transcription

1 International Journal of Mathematical Analysis Vol. 9, 215, no. 22, HIKARI Ltd, Approximation to the Dissipative Klein-Gordon Equation Edilber Almanza-Vasquez Campus Piedra de Bolivar, Avenue of Consulado Ana-Magnolia Marin-Ramirez Campus San Pablo, Avenue of Consulado Ruben-Dario Ortiz-Ortiz Campus San Pablo, Avenue of Consulado Copyright c 215 Edilber Almanza-Vasquez, Ana-Magnolia Marin-Ramirez and Ruben- Dario Ortiz-Ortiz. This is an open access article 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 In this paper we show that there exists a solution of the cubic nonlinear Klein-Gordon equation for a small parameter. We construct a traveling wave equation and we show that the corresponding system does not have periodic orbits for some real constants. Mathematics Subject Classification: 34A34 Keywords: Klein-Gordon equation, Dynamical systems

2 16 Edilber Almanza-Vasquez et al. 1 Introduction It was made some applications to the nonlinear Klein-Gordon equation [1]. It was found an antibound state for the Klein-Gordon equation [2]. In [3] was used a method with Green functions for constructing asymptotics of eigenvalues for the linear Klein-Gordon equation. In [4] was studied the Klein-Gordon equation. In [5] was given an explicit formula for the eigenvalue below the essential spectrum of discrete Klein-Gordon operator. We show there exists a solution for a small parameter and construct a traveling wave equation and a system without periodic orbits. 2 Preliminary Notes The Klein-Gordon equation u tt u + αu t + βu + γu 3 = (1) where α, β, γ are real constants. 3 Main Results These are the main results of the paper. Theorem 3.1. The solution of (1) is (5) for a small parameter γ. Proof. Taking Fourier transform we have û tt (p, t) + (p 2 + β)û(p, t) + αû t + γû2 û = (2) with solution û(p, t) =e 1 2 t( e 1 2 t( γf(p, ζ)e 1 2 ζ( +αζ dζ+ α γf(p, ζ)e 1 2 ζ( 2 4β 4p 2 α)+αζ dζ+ + k 1 (p)e 1 2 t( + k 2 (p)e 1 2 t( (3) where f = û2 û. Rewriting this equation we have û(p, t) = γt û(p, t) + k 1 (p)e 1 2 t( + k 2 (p)e 1 2 t( (4)

3 Approximation to the dissipative Klein-Gordon equation 161 where T û(p, t) =e 1 2 t( e 1 2 t( f(p, ζ)e 1 2 ζ( +αζ dζ+ α f(p, ζ)e 1 2 ζ( 2 4β 4p 2 α)+αζ dζ Then by Neumann series û(p, t) = γ n T n (k 1 (p)e 1 2 t( α 2 4β 4p 2 α) + k 2 (p)e 1 2 t( α 2 4β 4p 2 α) ) (5) n= 4 Traveling wave solution If the Klein-Gordon equation has a traveling wave solution of the form u(x, y, t) = u(ξ), ξ = kx + ly λt (6) where k, l, λ are real constants. Substituting (6) into (1) we obtain u ξξ αλ λ 2 k 2 l 2 u ξ + 5 Dynamical system β λ 2 k 2 l 2 u + γ λ 2 k 2 l 2 u3 = (7) Taking u ξ = y and u = x we have the following system xξ = y y ξ = σy µx νx 3 (8) αλ where σ = <, µ = λ 2 k 2 l 2 Hamiltonnian system β λ 2 k 2 l 2 >, ν = γ λ 2 k 2 l 2 >. The xξ = H y y ξ = H x (9) has this solution H = y2 x2 + σyx µ ν x4 = K for some constant K For the next results we are going to use the Poincaré-Bendixson theorem and the following quasi-differential equation [ ( h h f1 f 1 + f 2 = h C(x 1, x 2 ) + f )] 2. (1)

4 162 Edilber Almanza-Vasquez et al. Theorem 5.1. The dynamical system (8) can be generalized to (11) and both do not have periodic orbits for y + σ > and x R. Proof. Taking K = and x ξ = y, and supposing that h =, C(x 1, x 2 ) = σ + y >. Using (1) we have y h = h [C(x 1, x 2 ) σ] and h = h. So h = e x. If f 2 = σ then f 2 = σy + C 2 (x). From equation (1), we get the ordinary differential equation f 1 = σ + y ( f 1 + σ). Then its solution is f 1 = y + c 1 (y)e x. We obtain the generalized dynamical system x1ξ = y + c 1 (y)e x (11) x 2ξ = σy + C 2 (x). Taking ν = 1 into (8) and using the following Poincaré transformation dt z = dτ, x 2 1 = 1 z, x 2 = u, (z ) (12) z we obtain uτ = u 2 z 2 µz σuz 2 z τ = uz 3. (13) Theorem 5.2. The system (13) can be generalized to (14) and both do not have periodic orbits for x 2 >. Proof. Taking x 1 = u, x 2 = z. Suppose f 1 = 2x 1 x σx 2 2, then f 1 = x 2 1x σx 2 2x 1 + C 1 (x 2 ). From (1), and taking C = σx 2 2 < and h = 1 with x 5 2 x 2 >. Then h x 2 = 5 and we have an ordinary differential equation x 6 2 Then its solution is 2x 1 x 3 2 x 2 f 2 = 5f 2. f 2 = C 2 (x 1 )x 5 2 x 1 x 3 2 Also, it holds (f 1 h) + (f 2 h) = σ >. x 3 2 We have the following generalized dynamical system x1τ = x 2 1x σx 2 2x 1 + C 1 (x 2 ) x 2τ = C 2 (x 1 )x 5 2 x 1 x 3 2. (14) Acknowledgements. The authors express their deep gratitude to Universidad de Cartagena for partial financial support.

5 Approximation to the dissipative Klein-Gordon equation 163 References [1] Z. Feng, G. Chen and S. B. Hsu, A qualitative study of the damped Duffing equation and applications, Discrete and Continuous Dynamical System B, 6 (5) (26), [2] A. M. Marin, R. D. Ortiz and R. Zabaleta, Antibound state for Klein- Gordon equation, International Journal Of Mathematical Analysis, 8 (59) (214), [3] A. M. Marin, R. D. Ortiz and J. A. Rodriguez, Asymptotics of the Klein-Gordon equation, Far East Journal Of Applied Mathematics, 7 (2) (212), [4] A. M. Marin, R. D. Ortiz and J. A. Rodriguez, Asymptotics for Klein- Gordon equation, Proyecciones (Antofagasta) - Revista De Matemática, 32 (3) (213), [5] A. M. Marin, R. D. Ortiz and J. A. Rodriguez, Asymptotics of eigenfunctions for discrete Klein-Gordon equation, International Journal Of Mathematical Sciences And Engineering Applications, 7 (3) (213), Received: February 21, 215; Published: March 27, 215