Conservation Laws of a Family of Reaction-Diffusion-Convection Equations

  1. Bruzón, M. S. 1
  2. Gandarias, M. L. 1
  3. de la Rosa, R. 2
  1. 1 Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain
  2. 2 Universidad de Cádiz, Cádiz, Spain
Libro:
Nonlinear Systems and Complexity

Editorial: Springer Link

ISSN: 2195-9994,  2196-0003

ISBN: 9783319020563,  9783319020570

Año de publicación: 2013

Páginas: 403-417

Tipo: Capítulo de Libro

DOI: 10.1007/978-3-319-02057-0_21 GOOGLE SCHOLAR lock_openAcceso abierto editor

Resumen

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.

Referencias bibliográficas

  • M.S. Bruzón, M.L. Gandarias, Symmetry reductions and traveling wave solutions for the Krichever–Novikov equation. Math. Meth. Appl. Sci. (2012). doi:10.1002/mma.1578
  • M.S. Bruzón, M.L. Gandarias, Conservation laws for a family of Benjamin-Bona-Mahony-Burgers equations, in Proceedings 4th IEEE International Conference on Nonlinear Science and Complexity, Budapest, 2012
  • R. Cherniha, J.R. King, Lie symmetries and conservation laws of nonlinear multidimensional reaction-diffusion systems with variable diffusivities. IMA J. Appl. Math. 71, 391–408 (2006)
  • R. Cherniha, M. Serov, Lie and Non-Lie symmetries of nonlinear diffusion equations with convection term. Symmetry Nonlinear Math. Phys. 2, 444–449 (1997)
  • R. Cherniha, M. Serov, Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection term. Eur. J. Appl. Math. 9, 527–542 (1998)
  • R. Cherniha, M. Serov, Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection term, II. Eur. J. Appl. Math. 17, 597–605 (2006)
  • V.A. Dorodnitsyn, S.R. Svirshchevskii, in Conservation Laws: Lie Group Analysis of Differential Equations, vol. 1, ed. by N.H. Ibragimov (CRC Press, Boca Raton, 1994), pp. 137–139
  • M.L. Gandarias, Weak self-adjoint differential equations. J. Phys. A Math. Theor. 44, 262001–262007 (2011)
  • N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics (Reidel, Dordrecht, 1985)
  • N.H. Ibragimov, The answer to the question put to me by L.V. Ovsyannikov 33 years ago. Arch. ALGA 3, 53–80 (2006)
  • N.H. Ibragimov, A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)
  • N.H. Ibragimov, Quasi self-adjoint differential equations. Arch. ALGA 4, 55–60 (2007)
  • N.H. Ibragimov, Nonlinear self-adjointness and conservation laws. J. Phys. A Math. Theor. 44, 432002–432010 (2011)
  • P. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993)
  • L.V. Ovsyannikov, Group Analysis of Differential Equations (Academic, New York, 1982)
  • A.M. Vinogradov, Local symmetries and conservation laws. Acta Appl. Math. 2, 21–78 (1984)