by V V Zharinov (Steklov Mathematical Institute)

This book focuses on the properties of nonlinear systems of PDE with geometrical origin and the natural description in the language of infinite-dimensional differential geometry. The treatment is very informal and the theory is illustrated by various examples from mathematical physics. All necessary information about the infinite-dimensional geometry is given in the text.

• Introduction: Internal Geometry of PDE:
  • Differential Manifolds
  • Lie-Backlund Mappings
  • Lie-Backlund Fields and Infinitesimal Symmetries
  • Cartan Forms, Currents and Conservation Laws
  • C-Spectral Sequence. Further Properties of Conservation Laws
  • Trivial Equations. The Formal Variational Calculus
  • Evolution Equations
• External Geometry of PDE:
  • Differential Submanifolds
  • Normal Projection. External Fields and Forms
  • Trivial Ambient Differential Manifolds
  • The Characteristic Mapping
  • The Green's Formula
  • Low-Dimensional Conservation Laws
  • Backlund Correspondence
• Further Studies:
  • Lagrangian Formalism
  • Hamiltonian Equations
  • Example: The Nambu's String
• Appendix