Applying moving frames to finding conservation laws of the nonlinear Klein-Gordon equation

Document Type : Research Paper


1 Department of Mathematics, Islamic Azad University, Naghadeh Branch, Naghadeh, Iran.

2 Department of Pure Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran.

3 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.


In this paper, we use a geometric approach based on the concepts of variational principle and moving frames to obtain the conservation laws related to the one-dimensional nonlinear Klein-Gordon equation. Noether’s First Theorem guarantees conservation laws, provided that the Lagrangian is invariant under a Lie group action. So, for calculating conservation laws of the Klein-Gordon equation, we first present a Lagrangian whose Euler-Lagrange equation is the Klein-Gordon equation, and then according to Gon¸calves and Mansfield’s method, we obtain the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under a hyperbolic group action. 


  • [1] A. Bihlo, J. Jackaman, and F. Valiquette, Invariant variational schemes for ordinary differential equations, Studies in Applied Mathematics, 148 (2022), 991–1020.
  • [2] M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Mathematics and Computers in Simulation, 79 (2008), 700–715.
  • [3] M. Fels and P. J. Olver, Moving coframes: II. Regularization and theoretical foundations, Acta Applicandae Mathematica, 55 (1999), 127–208.
  • [4] T. M. N. Gon¸calves and E. L. Mansfield, Moving frames and conservation laws for Euclidean invariant La- grangians, Studies in Applied Mathematics, 130 (2013), 134–166.
  • [5] T. M. N. Gon¸calves and E. L. Mansfield, Moving Frames and Noether’s Conservation Laws - the General Case, Vol. 4, Forum of Mathematics, Sigma, Cambridge University Press, 2016.
  • [6] T. M. N. Gon¸calves and E. L. Mansfield, On moving frames and Noether’s conservation laws, Studies in Applied Mathematics, 128 (2012), 1–29.
  • [7] D. Kaya and S. M. El-Sayed, A numerical solution of the Klein-Gordon equation and convergence of the decom- position method, Applied mathematics and computation, 156 (2004), 341–353.
  • [8] P. Kim, Invariantization of numerical schemes using moving frames, BIT Numerical Mathematics, 47 (2007), 525–546.
  • [9] I. A. Kogan and P. J. Olver, Invariant Euler–Lagrange equations and the invariant variational bicomplex, Acta Applicandae Mathematica, 76 (2003), 137–193.
  • [10] E. L. Mansfield, A practical guide to the invariant calculus, Vol. 26, Cambridge University Press, New York, 2010.
  • [11] E. Noether, Invariant variation problems, Transport theory and statistical physics, 1 (1971), 186–207.
  • [12] P. J. Olver, Applications of Lie groups to differential equations, Vol. 107, Springer Science and Business Media, 2000.
  • [13] E. Ozbenli and P. Vedula, Construction of invariant compact finite-difference schemes, Physical Review E, 101 (2020), 023303.
  • [14] E. Ozbenli and P. Vedula, High order accurate finite difference schemes based on symmetry preservation, Journal of Computational Physics, 349 (2017), 376–398.
  • [15] E. Ozbenli and P. Vedula, Numerical solution of modified differential equations based on symmetry preservation, Physical Review E, 96 (2017), 063304.
  • [16] F. Shakeri and M. Dehghan, Numerical solution of the Klein-Gordon equation via He’s variational iteration method, Nonlinear Dynamics, 51 (2008), 89–97.
  • [17] Sirendaoreji, Auxiliary equation method and new solutions of Klein-Gordon equations, Chaos Solitons and Fractals, 31 (2007), 943–950.
  • [18] A. M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 889–901.
  • [19] A. M. Wazwaz, The modified decomposition method for analytic treatment of differential equations, Applied mathematics and computation, 173 (2006), 165–176.