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Integrable systems and isomonodromy deformations. (English) Zbl 0795.35077
The authors consider three classes of deformation problems associated with integrable systems. The first two are related to the scaling invariance of the $$n\times n$$ AKNS hierarchies and Gel’fand-Dikij hierarchies. The corresponding isomonodromy equations are generally of order greater than 2, so they are not the classical Painlevé transcendents. However they do have the Painlevé property, in fact the stronger property that any solution has a single-valued meromorphic extension to the entire plane.
The third class of monodromy problems considered in the paper is connected with the equation of M. R. Douglas [Phys. Lett. B 238, 176-180 (1990)] in two-dimensional quantum gravity. These problems are obtained by replacing $$\dot L$$ by $$\hbar I$$ in the Lax equation for the Gel’fand-Dikij hierarchies. The monodromy data is constructed in each case.
Reviewer: P.Holod (Kiev)

##### MSC:
 35Q15 Riemann-Hilbert problems in context of PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 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.) 35Q55 NLS equations (nonlinear Schrödinger equations)
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