Conservation Laws of a Family of Reaction-Diffusion-Convection Equations
Ibragimov introduced the concept of nonlinear self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi-self-adjoint equations. Gandarias defined the concept of weak self-adjoint. In this paper, we found a class of nonlinea
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Abstract Ibragimov introduced the concept of nonlinear self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi-self-adjoint equations. Gandarias defined the concept of weak self-adjoint. In this paper, we found a class of nonlinear self-adjoint nonlinear reaction-diffusion-convection equations which are neither self-adjoint nor quasi-self-adjoint nor weak self-adjoint. From a general theorem on conservation laws proved by Ibragimov we obtain conservation laws for these equations.
1 Introduction Many interesting chemical, biological and physical phenomena such as pattern formation, morphogenesis, animal coats and skin pigmentation, nerve impulse propagation in nerve fibres, wall propagation in liquid crystals, nucleation kinetics and neutron action in the reactor, are strongly related with the study of nonlinear reaction-diffusion-convection equations. We consider the class of nonlinear reaction-diffusion-convection equations ut D ŒA.u/ux x C B.u/ux C C.u/;
(1)
and K.u/ ¤ 0, B.u/ ¤ 0 where u D u.x; t/ is an unknown function, A D dK.u/ du and C.u/ is an arbitrary differentiable function. Equation (1) is a generalization of
M.S. Bruzón () M.L. Gandarias Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain e-mail: [email protected] R. de la Rosa Universidad de Cádiz, Cádiz, Spain e-mail: [email protected] R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__21, © Springer International Publishing Switzerland 2014
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many nonlinear evolution equations of second order. Lie symmetries of equation (1), where A.u/, B.u/, C.u/ are arbitrary smooth functions, were completely obtained by Cherniha and Serov [4–6]. Dorodnitsyn and Svirshchevskii [7] proved that Eq. (1) admits nontrivial conservation laws when B D 0, A D k and C D k1 u C k2 , and when A D dK.u/ d u and C D k1 K.u/ C k2 u C k3 , where K.u/ is an arbitrary function, k, k1 and k2 are arbitrary constants. Cherniha and King generalized these results to higher dimensions by inspection and they obtained a new conservation law when A D A.u/ and C D k1 K.u/ C k2 u C k3 in which K.u/ is the Kirchhoff variable K.u/ [3]. The classical theory of Lie point symmetries for differential equations describes the groups of infinitesimal transformations in a space of dependent and independent variables that leave the manifold associated with the equation unchanged [9,14,15]. The methods of point transformations are a powerful tool for find exact solutions for nonlinear partial differential equations (PDE’s) and for construct conservation laws [1]. The idea of a conservation law, or more particularly of a conserved quantity, has its origin in mechanics and physics. Since a large number of physical theories, including some of the “laws of nature”, are usually expressed as systems of nonlinear differential equations, it follows that conservation laws are useful in both general theor
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