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LIE GROUP THEORY: APPLICATION TO DIFFERENTIAL EQUATIONS

1. Introduction. Lie groups of transformations

Groups. Lie groups. One-parameter local transformation group. The Lie equations.
Infinitesimal group operator. Group invariants. Invariant equation.

2. Symmetries and prolongation theory

Prolongation theory. Equations in the prolonged space. Prolongation of the infinitesimal operator. Prolongation of point transformations. Tangent and contact transformations.
Invariant differential equations. Infinitesimal invariance criteria. Determining equations and symmetry groups. Generalized symmetries.
Lie algebras and multiparameter symmetries.
Conservation laws and integrals of motion. Determining equations for the conservation laws. Relations between symmetries and integrals.

3. Group analysis of ordinary differential equations

Symmetries and integration methods. Determining equations. The method of canonical variables. The method of differential invariants.
Multiparameter symmetries and integration of differential equations. Two-parameter Lie symmetry groups and order reduction. Solvable Lie groups and algebras and order reduction using multiparameter symmetries.
Conservation laws for ODEs. Integrating factors for high-order equations. Determining equations for integrating factors. The bootstrap method. Combination of symmetries and first integrals.

4. Partial differential evolution equations

Symmetries of evolution equation. Determining equation for symmetries. Multiplication of solutions using the symmetry group.
invariant solutions and linearization, invariant solutions of solution equations. Linearization of system of the hydrodynamical type.
Recursion operators and higher symmetries. Recursion operator and the generation of exact solutions.

Literature:
1. P. Hydon. Symmetry Methods for Differential Equations. Cambridge University Press, 2000.
2. P. J. Olver, Applications of Lie Groups to Differential Equations, (Second Ed.), Springer, New York,1986.
3. L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York,1982.
4. N. H. Ibragimov. Elementary Lie Group Analysis and Ordinary Differential Equations (Mathematical Methods in Practice S.