| In many problems in physics and engineering one needs to find solutions for (nonlinear) partial differential equations (PDEs). This project is situated at the border between PDEs and geometry. We develop geometric methods for finding solutions of PDEs and study special PDEs that govern geometric structures. The main subjects of the project are: (1) Poisson structures originate in the Hamiltonian formulation of classical mechanics and are solutions of a certain natural PDE on manifolds. Poisson structures are of fundamental importance for many areas of mathematics and physics. (2) Backlund transformations are a powerful tool to obtain explicit solutions for nonlinear PDEs. (3) Lie pseudogroups and Lie groupoids represent modern approaches to the basic concept of symmetry for geometric structures and PDEs. We plan to study these subjects by a unified approach, using jet spaces and Lie theory, in the following directions. (1) A recent work of S. Igonin on a geometric description of Backlund transformations in terms of Lie algebras provides significant progress in the long-standing problem of understanding the intrinsic meaning of PDEs integrability. Using Igonin's results, jet spaces, and symmetries, we plan to develop an efficient method to obtain Backlund transformations for PDEs (and, therefore, to obtain a large number of explicit solutions for nonlinear PDEs). (2) Using Lie groupoids and jet spaces, we introduce and plan to elaborate a generalization of the classical notion of Lie pseudogroups. This generalization is suitable for studying geometric structures with singularities, such as Poisson structures. We intend to study also symmetry-invariant Poisson structures. |
