### Book

## Banach Center Publications

Painlevé equations, holomorphic vector fields and normal forms, summability of WKB solutions, Gevrey order and summability of formal solutions for ordinary and partial differ- ential equations,

• Stokes phenomena of formal solutions of non-linear PDEs, and the small divisors phenomenon,

• summability of solutions of difference equations,

• applications to integrable systems and mathematical physics.

Applying methods of plane Power Geometry we are looking for the asymptotic expansions of solutions to the fifth Painleve ́ equation in the neighbourhood of its singular and nonsingular points.

We consider a model quantum Hamiltonian of a charge in a resonance electromagnetic trap. Using the operator averaging method, we obtain an effective quantum operator that asymptotically describes the anharmonic part of the Hamiltonian. We show that the operator becomes a second-order difference operator in a specially chosen quantum action-angle representation. Using the discrete WKB method for this difference equation, we obtain the semiclassical WKB asymptotics of the spectrum and stationary states of the charge.

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has two purely imaginary eigenvalues, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to pseudonormal form. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

By means of Power Geometry we obtained all asymptotic expansions of solutions to the equation P5 of the following five types: power, power-logarithmic, complicated, exotic and half-exotic for all values of 4 complex parameters of the equation. They form 16 and 30 families in the neighbourhood of singular points z = infty and z = 0 correspondingly. There exist 10 families in the neighbourhood of nonsingular point. Over 20 families are new.

In this work, the methods of power geometry are used to find asymptotic expansions of solutions to the fifth Painlevй equation as *x * 0 for all values of its four complex parameters. We obtain 30 families of expansions, of which 22 are obtained from published expansions of solutions to the sixth Painlevй equation. Among the other eight families, one was previously known and two can be obtained from the expansions of solutions to the third Painlevй equation. Three families of half-exotic expansions and two families of complicated expansions are new.

In this work we construct and discuss special solutions of a homogeneous problem for the Laplace equation in a domain with the cone-shaped boundaries. The problem at hand is interpreted as that describing oscillatory linear wave movement of a uid under gravity in such a domain. These so- lutions are found in terms of the Mellin transform and by means of the reduction to some new functional-difference equations solved in an explicit form (in quadratures). The behavior of the so- lutions at far distances is studied by use of the saddle point technique. The corresponding eigenoscil- lations of a uid are then interpreted as eigenfunctions of the continuous spectrum.

In this paper, in a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has one zero eigenvalue, while the other eigenvalues lie outside the imaginary axis. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.