Integrable Evolution Equations on the N-Dimensional Sphere

Abstract:

 The problem of a classification of integrable evolution equations on the\break N-dimensional sphere is considered. We modify the main notions of the symmetry approach such as the formal symmetry and the canonical series of conserved densities to the case of such equations. Using these theoretical results, we solve several special classification problems. The main result is a complete classification of integrable isotropic evolution equations of third order on the sphere. An important class of anisotropic equations is also considered.