Conservation laws for Kawahara equations Igor L. Freire Júlio C. S. Sampaio Centro de Matemática, Computação e Cognição, CMCC, UFABC 09210-170, Santo André, SP E-mail: igor.freire@ufabc.edu.br, julio.sampaio@ufabc.edu.br Abstract: In this talk we present some results regarding conservation laws for evolution equations. Namely we revisit some recent results that we have obtained and we establish some new conservation laws for some evolution equations of fifth-order. Keywords: Conservation laws, Ibragimov s theorem, nonlinearly self-adjointness 1 Introduction To begin with, we would like to observe that the present text follows closely our recent communication [5], where we have applied some new developments to a class of fifth-order evolution equations. Since Ibragimov s seminal paper concerning conservation laws [8] intense research has been carried out in order to find classes of equations admiting special properties the so-called selfadjoint equations [7, 8], quasi-self-adjoint equations [9] and, more recently, weak self-adjoint equations [6] and nonlinear self-adjoint equations [12]. To these special equations, the Ibragimov s theorem on conservation laws [8] provides an elegant way to establish local conservation laws for the equations under consideration. For instance, in [10] Ibragimov, Torrisi and Tracinà found a class of quasi-self-adjoint semilinear 1 + 1) wave equations. In [11] the same authors determined that the system derived from a 2 + 1) generalized Burgers equation is quasi-self-adjoint. In both cited papers, conservation laws for the equations considered were established using the Ibragimov s theorem [8]. In [1] Bruzón, Gandarias and Ibragimov determined the self-adjoint classes of a generalized thin film equations and some conservation laws were established. In addition, in [4] one of us I.L. Freire) determined the self-adjoint classes of a third-order quaselinear evolution equations. In [13] Ibragimov, Khamitova and Valenti determined the quasi-self-adjoint classes of a generalization of the Camassa-Holm equation. Further examples can be found in [2, 3] and references therein. Recently Maria Gandarias [6] and Nail Ibragimov [12] have developed new concepts generalizing quasi-self-adjoint and self-adjoint equations. Namely, Gandarias proposed the concept of weak self-adjoint equation and Ibragimov generalized himself introducint the definition of nonlinearly self-adjoint equations. Inspired by these developments we have determined [5] the nonlinear self-adjoint class of the generalized fifth-order equation u t + αuu x + βu 2 u x + γu xxx + µu xxxxx = 0 1) and, by using the Ibragimov s theorem, some local conservation laws for 1) are established. We recall that setting α = γ = 1 and β = µ = 0, equation 1) becomes the well known Korteweg-de Vries KdV) equation bolsista de Mestrado da UFABC u t = u xxx + uu x. 2) 287
In [14] it was studied particular cases of equation 1) from the point of view of Lie point symmetry theory. Namely, there the authors found the Lie point symmetry generators of equations up to notation) General Kawahara equation Simplified Kawahara equation and Simplified modified Kawahara equation u t + αuu x + γu xxx + µu xxxxx = 0; 3) u t + αuu x + µu xxxxx = 0 4) u t + βu 2 u x + µu xxxxx = 0. 5) The remaining of the text we revisit some main results obtained in [5]. In the next section we determine the class of nonlinear self-adjoint equations of the type 1). Next, in the section 3 we establish conservation laws for some particular cases of 1). 2 The class of nonlinear self-adjoint equations of the type 1) Let δ δu = u + j=1 1) j D i1 D ij u i1 i j, 6) be the Euler-Lagrange operator, x = x 1,, x n ) be n independent variables, u = ux) be a dependent variable, F x, u,, u s) ) = 0 7) be a given differential equation and L = vf, where v = vx). Definition 1. Equation 7) is said to be nonlinearly self-adjoint if the equation obtained from the adjoint equation F x, u, v,, u s), v s) ) := δ δu L = 0 8) by the substitution v = φx, u) with a certain function φx, u) 0 is identical with the original equation 7), that is, F x, u, v, u 1), v 1),, u s), v s) ) v=φx,u) = 0. 9) Whenever 9) holds for a certain differential function φ such that φ u 0 and φ x i 0, equation 8) is called weak self-adjoint. In other words: equation 8) is said to be nonlinear self-adjoint if there exists a function φ = φx, u) such that F v=φ = λx, u, )F, 10) for some differential function λ = λx, u, ). Theorem 1. Equation 1), with γ, µ) 0 and α, β) 0, 0), is nonlinearly self-adjoint. Proof: See [5]. 288
3 Conservation laws for equation 1) The Ibragimov s theorem on conservation law [8] provides a conservation law D t C 0 +D x C 1 = 0 for equation 1) and its adjoint equation F := v t αu + βu 2 )v x γv xxx µv xxxxx = 0, where C 0 = τl + W u t, L = vf, W = η τu t ξu x and [ C 1 = ξl + W + D 2 x + Dx 4 u x u xxx [ D x W ) [ +DxW 2 ) D x D x + Dx 3 u xxx + Dx 2 u xxx u xxxxx u xxxxx DxW 3 )D x + D 4 x, u xxx u xxxxx X = τx, t, u) t is any Lie point symmetry of 1). Let us now find conservation laws for 1). 3.1 Kawahara equation In [14] is found that u xxxxx + ξx, t, u) + ηx, t, u) x u X α = αt x + u is a Lie point symmetry generator of equations 3) and 4). From 11) and 12) we obtain C 0 = 1 αtu x )v, ] ] ] 11) 12) C 1 = αtvu t + αuu x + γu xxx ) + 1 αtu x )αvu + γv xx + µv xxxx ) 13) +αtγv x + µv xxx )u xx αtγv + µv xx )u xxx + αγtu xxxx v x as a conserved vector for 12). From Theorem 1 it is possible to find a local conservation law for 3). Substituting v = u in the components 13) and after reckoning, we obtain ) ) C 0 = u D x αt u2, C 1 = D t αt u2 + αu 2 + γu xx + µu xxxx. 2 2 By transfering the terms D x ) from C 0 to C 1, we obtain the conserved vector C γ,µ = u, α u2 2 + γu xx + µu xxxx ). 14) We observe that whenever α = 1, C 1,0 provides a conservation law for KdV equation 2), whereas C 0,µ provides a conservation law for 4). 289
3.2 Simplified Kawahara equation Concerning equation 4), let us establish a conservation law using the dilational symmetry From 11) we obtain C 0 = 4vu xvu x + 5αtvuu x + 5µtvu xxxxx, X = x x + 5t t 4u u. 15) C 1 = xvu t 4αvu 2 4µuv xxxx µxu x v xxxx 5αtvuu t 5µtu t v xxxx + 5µu x v xxx +µxu xx v xxx + 5µtu xt v xxx 6µv xx u xx µxv xx u xxx 5µtv xx u xxt + 7µv x u xxx +µxv x u xxxx + 5µtv x u xxxt 8µvu xxxx 5µtvu xxxxt. Substituting v = x αtu into the components above, we obtain C 0 = 2xu + αtu 2 + D x x 2 u + 3αxtu 2 5 ) 3 α2 t 2 u 3 +D x 5µxtu xxxx 5µtu xxx 5αµt 2 uu xxxx + 5αµt 2 u x u xxx 5 ) 2 αµt2 u 2 xx, C 1 = αxu 2 + 2 3 α2 tu 3 2µxu xxxx + 2µu xxx + 2αµtuu xxxx 2αµtu x u xxx + αµtu 2 xx + D t x 2 u 3αxtu 2 + 5 3 α2 t 2 u 3 ) D t 5µxtu xxxx + 5µtu xxx + 5αµt 2 uu xxxx 5αµt 2 u x u xxx + 5 2 αµt2 u 2 xx ) After transfering the term D x ) from C 0 to C 1, we find the conserved vector C = C 0, C 1 ), C 0 = 2xu + αtu 2, C 1 = αxu 2 + 2 3 α2 tu 3 2µxu xxxx + 2µu xxx + 2αµtuu xxxx 16) 2αµtu x u xxx + αµtu 2 xx, for the simplified Kahawara equation. 3.3 Simplified modified Kawahara equation Let us establish conservation laws for equation 5) using the Lie point symmetry generator X = x x + 5t t 2u u. 290
A conserved vector for 5) and its corresponding adjoint is C = C 0, C 1 ), where C 0 = 2vu xvu x + 5βtvu 2 u x + 5µtvu xxxxx, C 1 = 2βu 3 v + xvu t + µxvu xxxxx 2µuv xxxx µxu x v xxxx 5µtu t v xxxx +3µu x v xxx + µxu xx v xxx + 5µtu xt v xxx 4µu xx v xx µxu xxx v xx 17) 5µtu xxt v xx + 5µv x u xxx + µxu xxxx v x + 5µtv x u xxxt 6µvu xxxx µxvu xxxxx 5µtvu xxxxt 5βvu 2 u t. Substituting v = 1 into 17) it is obtained C 0 = u D x xu 5 3 βtu3 5µtu xxxx ) C 1 = µu xxxx 5 3 βu3 + D t xu 5µtuxxxx 5 3 βtu3). Transfering the term D x ) from C 0 to C 1 and changing the signal, we obtain the conserved vector C = u, 5 3 βu3 + µu xxxx ). 18) Let now allow v = u in 17). Thus, after reckoning, C 0 = 3 2 u2 + D x xu2 2 + 5 4 βtu4 + 5µtuu xxxx 5µtu x u xxx + 5 ) 2 µtu2 xx, C 1 = 3 4 βu4 3µuu xxxx + 3µu x u xxx 3 2 µu2 xx D t xu2 2 + 5 4 βtu4 + 5µtuu xxxx 5µtu x u xxx + 5 ) 2 µtu2 xx. Since D t C 0 + D x C 1 = D t 3 ) 2 u2 + D x 3 4 βu4 3µuu xxxx + 3µu x u xxx 3 ) 2 µu2 xx after reckoning and multiplication by 2/3 it follows that C = u 2, 1 ) 2 βu4 + 2µuu xxxx 2µu x u xxx + µu 2 xx is a conserved vector for 5). Acknowledgements Júlio Cesar Santos Sampaio would like to thank the Pró-reitoria de pós-graduação da UFABC PROPG-UFABC for financial support. References [1] M. S. Bruzón, M. L. Gandarias and N. H. Ibragimov, Self-adjoint sub-classes of generalized thin film equations, J. Math. Anal. Appl., 357 2009) 307 313. 291
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