1. Extended symmetries of the 3D kinetic electron plasma waves
V. B. Taranov
Institute for Nuclear Research, Kyiv, Ukraine

2. Manifold of solutions
Lie point, conditional and extended symmetries:

3. A

4. B

5. C

6. D

7. E

8. 3

9. MAPLE 12 PROGRAM

10. 4

11. 5

12. 6

13. 7

14. 8

15. Mexican hat

16. 1. Hydrodynamic plasma theory.
CONCLUSIONS
1. Hydrodynamic plasma theory.
Continuous symmetries of the partial differential equations can be obtained by usual Lie group algorithm. Standard programs of Maple 12 package allow us to find the symmetries for nonlinear equations and systems of equations.
Conditional symmetries are possible when the manifold of solutions is reduced by certain additional conditions. From the physical point of view, these symmetry extensions are due to the simpler structure of nonlinear waves in the vicinity of their critical points.
2. Kinetic plasma theory.
Continuous symmetries of the integro-differential equations can be obtained by an indirect method (from the symmetries of the infinite system of PDE’s for the moments of distribution functions) and by the direct methods. Maple 12 standard programs can help us even in this case.
Conditional symmetries are possible in kinetic theory case, too.
Moreover, extended symmetries appear if some global conditions are weakened.
“Wavelet” type extended symmetries are possible for the systems containing finite order moments of distribution functions.

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