Multiseries Lie groups and asymptotic modules for characterizing and solving integrable models

Abstract

A multiseries integrable model (MSIM) is defined as a family of compatible flows on an infinite-dimensional Lie group of N-tuples of formal series around N given poles on the Riemann sphere. Broad classes of solutions to a MSIM are characterized through modules over rings of rational functions, called asymptotic modules. Possible ways for constructing asymptotic modules are Riemann–Hilbert and ∂̄ problems. When MSIM’s are written in terms of the ‘‘group coordinates,’’ some of them can be ‘‘contracted’’ into standard integrable models involving a small number of scalar functions only. Simple contractible MSIM’s corresponding to one pole, yield the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. Two-pole contractible MSIM’s are exhibited, which lead to a hierarchy of solvable systems of nonlinear differential equations consisting of (2+1)-dimensional evolution equations and of quite strong differential constraints.