University of TabrizComputational Methods for Differential Equations2345-398211220230401Applying moving frames to finding conservation laws of the nonlinear Klein-Gordon equation3994111499910.22034/cmde.2022.50659.2101ENYousefMasoudiDepartment of Mathematics, Islamic Azad University, Naghadeh Branch, Naghadeh, Iran.0000-0001-8346-5695MehdiNadjafikhahDepartment of Pure Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran.0000-0002-1354-9786MegerdichToomanianDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.Journal Article20220303In 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. https://cmde.tabrizu.ac.ir/article_14999_a1db7d08cd72cfdabb168f0f0ca66806.pdf