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

Document Type : Research Paper

Authors

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.

Abstract

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. 

Keywords

References

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Computational Methods for Differential Equations
Volume 11, Issue 2
April 2023
Pages 399-411

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History

  • Receive Date: 03 March 2022
  • Revise Date: 27 June 2022
  • Accept Date: 27 July 2022

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