Equivalent Lagrangians: generalization, transformation maps, and applications. (English) Zbl 1322.35001

Wilson, N.; Kara, A. H.

J. Appl. Math. 2012, Article ID 860482, 19 p. (2012).

Summary: Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as the Kummer equation and the combined gravity-inertial-Rossby wave equation and certain classes of partial differential equations related to multidimensional wave equations.

MSC:

35A30

Geometric theory, characteristics, transformations in context of PDEs

References:

[1]	P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1993. · Zbl 0785.58003
[2]	N. H. Ibragimov, A. H. Kara, and F. M. Mahomed, “Lie-Bäcklund and Noether symmetries with applications,” Nonlinear Dynamics, vol. 15, no. 2, pp. 115-136, 1998. · Zbl 0912.35011
[3]	A. H. Kara, “Equivalent lagrangians and the inverse variational problem with applications,” Quaestiones Mathematicae, vol. 27, no. 2, pp. 207-216, 2004. · Zbl 1089.34514
[4]	A. H. Kara and F. M. Mahomed, “Equivalent lagrangians and the solution of some classes of nonlinear equations q\"+p(t)q\?+r(t)q=\mu q\?2q-1+f(t)qn,” International Journal of Non-Linear Mechanics, vol. 27, no. 6, pp. 919-927, 1992. · Zbl 0760.34011
[5]	F. M. Mahomed, A. H. Kara, and P. G. L. Leach, “Lie and Noether counting theorems for one-dimensional systems,” Journal of Mathematical Analysis and Applications, vol. 178, no. 1, pp. 116-129, 1993. · Zbl 0783.34002
[6]	A. D. McDonald, “Properties of the confluent hypergeometric function,” Tech. Rep. 84, Research Laboratory of Electronics, M.I.T., Cambridge, Mass, USA, 1984.
[7]	J. McKenzie, “Instability of combined gravity-inertial-rossby waves in atmospheres and oceans,” Annales Geophysicae, vol. 29, no. 6, pp. 997-1003, 2011.
[8]	N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1, Symmetries and Exact Solutions and Conservation Laws, Boca Raton, Fla, USA, 1994. · Zbl 0864.35001
[9]	S. Jamal and A. H. Kara, “New higher-order conservation laws of some classes of wave and Gordon-type equations,” Nonlinear Dynamics, vol. 67, no. 1, pp. 97-102, 2012. · Zbl 1242.35162