Contact Geometry and Nonlinear Differential Equations Encyclopedia of Mathematics and its Applications Series

Methods from contact and symplectic geometry can be used to solve highly non-trivial nonlinear partial and ordinary differential equations without resorting to approximate numerical methods or algebraic computing software. This book explains how it's done. It combines the clarity and accessibility of an advanced textbook with the completeness of an encyclopedia. The basic ideas that Lie and Cartan developed at the end of the nineteenth century to transform solving a differential equation into a problem in geometry or algebra are here reworked in a novel and modern way. Differential equations are considered as a part of contact and symplectic geometry, so that all the machinery of Hodge-deRham calculus can be applied. In this way a wide class of equations can be tackled, including quasi-linear equations and Monge-Ampere equations (which play an important role in modern theoretical physics and meteorology).

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.

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.