# Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras

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## Abstract

We give a complete point-symmetry classification of all third-order evolution equations of the form *u* _{ t }=*F*(*t*,*x*,*u*,*u* _{ x },*u* _{ xx })*u* _{ xxx }+*G*(*t*,*x*,*u*,*u* _{ x },*u* _{ xx }) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.

## Keywords

Symmetries Lie algebras Equivalence group Semi-simple Lie algebras Solvable Lie algebras## Notes

### Acknowledgements

V. Lahno was partially supported by the Swedish Research Council (grant number 624-2004-1073). The research of F. Güngör was supported by the Research Council of Turkey (TÜBİTAK).

Faruk Güngör and Peter Basarab-Horwath wish to acknowledge the great debt they owe to Viktor Lahno, for his friendship, mathematical insights and patient collaboration over the years. Viktor died on June 5th 2011, while we were preparing this article for submission. We shall miss him greatly.

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