Multiseries Lie groups and asymptotic modules for characterizing and solving integrable models. (English) Zbl 0673.22009
Summary: 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 \({\bar \partial}\) 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.
MSC:
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 37C10 | Dynamics induced by flows and semiflows |
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
| 37N99 | Applications of dynamical systems |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
Keywords:
multiseries integrable model; compatible flows; infinite-dimensional Lie group; asymptotic modules; Riemann-Hilbert and \({\bar \partial }\) problems; nonlinear differential equations; evolution equationsReferences:
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