Contact Geometry and Non-linear Differential Equations

Abstract

Introduction Part I. Symmetries and Integrals: 1. Distributions 2. Ordinary differential equations 3. Model differential equations and Lie superposition principle Part II. Symplectic Algebra: 4. Linear algebra of symplectic vector spaces 5. Exterior algebra on symplectic vector spaces 6. A Symplectic classification of exterior 2-forms in dimension 4 7. Symplectic classification of exterior 2-forms 8. Classification of exterior 3-forms on a 6-dimensional symplectic space Part III. Monge-Ampere Equations: 9. Symplectic manifolds 10. Contact manifolds 11. Monge-Ampere equations 12. Symmetries and contact transformations of Monge-Ampere equations 13. Conservation laws 14. Monge-Ampere equations on 2-dimensional manifolds and geometric structures 15. Systems of first order partial differential equations on 2-dimensional manifolds Part IV. Applications: 16. Non-linear acoustics 17. Non-linear thermal conductivity 18. Meteorology applications Part V. Classification of Monge-Ampere Equations: 19. Classification of symplectic MAEs on 2-dimensional manifolds 20. Classification of symplectic MAEs on 2-dimensional manifolds 21. Contact classification of MAEs on 2-dimensional manifolds 22. Symplectic classification of MAEs on 3-dimensional manifolds.