Nonlinear Partial Differential Equations and Related Problems of Pade Approximations

摘要

The focus of the authors' work is the understanding and possibly explicit mathematical solution of the physically realistic classical and quantum models of field theories in dimensions one to four. They aim at the description of hidden symmetries and analytic structures of the solutions of various physical systems in dimensions one-four described by systems of nonlinear p.d.e. in cases when a system is suspected of being completely integrable, or at the investigation of obstructions to complete integrability. We continued and expanded our earlier studies of the structure of infinite-dimensional Lie algebras generated by Baecklund transformations and the geometric interpretation of the structure of S-matrices, applied to the extension of the Bethe Ansatz determining the solutions of completely integrable quantum systems to three- and four-dimensional systems. Very little is currently known in these cases, and there arise new multidimensional generalizations of factorization equations of S-matrices that have to be solved. A totally new area for mathematical studies has been opened by recent progress in the physics of superstrings, where we hope to apply our methods of multi-particle and multi-string interactions, and where the use of methods from algebraic topology and theta functions (some of which were introduced by us in lower-dimensional cases earlier) seems very promising.