The moving frame method and invariant subspace under parametric group actions

Subject Areas : Differential geometry

Y. Alipour Fakhri ¹ , Y. Azadi ²

1 - Department of Mathematics, Payame Noor University, Tehran, Iran
2 - Department of Mathematics, Payame Noor University, Tehran, Iran

Received: 2021-07-22 Accepted : 2021-09-28 Published : 2021-09-01

Keywords: moving frame, Differential equation, First integral, group action, invariant subspace,

Abstract :

‎For a given subspace as a solution space of a linear ODE‎, ‎we define a special linear parametric group action and prolong it to the jet bundle‎. ‎We determine these group parameters by moving frame method and prove that these group parameters are the first integrals of the given ODE‎. ‎These first integrals are used to construct the general form of operators which preserve given subspace invariant‎.

References:

[1] M. Fels, P. J. Olve, Moving coframes. I,II. A practical algorithm, Acta. Appl. Math. 51 (1998), 161-213.

[2] V. A. Galaktionov, S. A. Posashkov, S. R. Svirshchevskii, Generalized separation of variables for differential equations with polynomial nonlinearities, Differ. Equat. 31 (1995), 233-240.

[3] V. A. Galaktionov, S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Taylor & Francis Group, 2007.

[4] N. Kamran, R. Milson, P. J. Olver, Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems, Adv. Math. 156 (2000), 286-319.

[5] P. J. Olver, Applications of Lie Groups to Differential equations, Springer-Verlag, New York, 1993.

[6] S. R. Svirshchevskii, Invariant linear spaces and exact solutions of nonlinear evolution equations, J. Nonl. Math. Phys. 3 (1996), 164-169.