Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya

Not all isomonodromic deformations are described by Painleve equations, but each Painleve equation defines an isomonodromic deformation. Obtaining isomonodromy from Painleve equations is ideologically transparent and easy to verify. Obtaining Painleve equations from isomonodromy is difficult and not always possible. In the present paper, all Painleve equations are derived uniformly, without any restrictions on the parameters. To this aim, a special subclass of isomonodromic deformations (Schlesinger ones) is specialized, and only this subclass is considered. The Fuchsian case (Painleve-VI equation) is considered in details, the remaining Painleve equations are obtained from Painleve-VI by the confluence procedure. Isomonodromy is verified by calculation.

We show that a completely non-degenerate $B$-group is uniquely determined by its factor: two such groups with conformally equivalent factors are Möbius conjugate. A similar property is inherent to the quasi-Fuchsian groups but not to degenerate $B$-groups. We also study the factor of a $B$-group as a triple: the main factor, the marked characteristic complex, and a homotopy class of maps of the first to the second one.

We describe deformations of the classical principal chiral model and the $(1+1)$-dimensional Gaudin model related to the Lie group $\mathrm{GL}_N$. The deformations are generated by $R$-matrices satisfying the associative Yang–Baxter equation. Using the coefficients of the expansion for these $R$-matrices we derive equations of motion based on a certain ansatz for $U$–$V$ pair satisfying the Zakharov–Shabat equation. Another deformation comes from the twist function, which we identify with the cocentral charge in the affine Higgs bundle underlying the Hitchin approach to $2d$ integrable models.

We investigate a singularly perturbed $q$-difference differential Cauchy problem with polynomial coefficients in complex time $t$ and space $z$ and with quadratic non-linearity. We construct local holomorphic solutions on sectors in the complex plane with respect to the perturbation parameter $\varepsilon$ with values in some Banach space of formal power series in $z$ with analytic coefficients on shrinking domains in $t$. Two aspects of these solutions are addressed. One feature concerns asymptotic expansions in $\varepsilon$ for which a Gevrey type structure is unveiled. The other fact deals with confluence properties as $q>1$ tends to $1$. In particular, the built up Banach valued solutions are shown to merge in norm to a fully bounded holomorphic map in all the variables $t$, $z$ and $\varepsilon$ that solves a non-linear partial differential Cauchy problem.

We consider the problem of solvability of linear differential equations over a differential field $K$. We introduce a class of special differential field extensions, which widely generalizes the classical class of extensions of differential fields by integrals and by exponentials of integrals and which has similar properties. We announce the following result: if a linear differential equation over $K$ cannot be solved by generalized quadratures, then no special extension can help solve it. In the paper, we prove a weaker version of this result in which we consider only pure transcendental extensions of $K$. Our paper is self-contained and understandable for beginners. It demonstrates the power of Liouville's original approach to problems of solvability of equations in finite terms.