Contact Geometry and Nonlinear Differential Equations

Alexei Kushner is a Professor and Dean of the Department of Mathematics and Computer Science, and a Senior Researcher at the Russian Academy of Sciences. Valentin Lychagin is a Professor at the Institute of Mathematics and Statistics, Tromsø University, and a Senior Researcher at the Institute for Theoretical and Experimental Physics in Moscow. Vladimir Rubtsov is a Professor at the Département de Mathématiques, Angers University, and a Senior Researcher at the Institute for Theoretical and Experimental Physics in Moscow.

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-Ampère Equations: 9. Symplectic manifolds; 10. Contact manifolds; 11. Monge-Ampère equations; 12. Symmetries and contact transformations of Monge-Ampère equations; 13. Conservation laws; 14. Monge-Ampère 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-Ampère 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.