For a second-order linear differential equation with two irregular singular points of rank three, multiple Laplace-type contour integral solutions are considered. An explicit formula in terms of the Stokes multipliers is derived for the characteristic exponent of the multiplicative solutions. The Stokes multipliers are represented by converging series with terms for which limit formulas as well as more detailed asymptotic expansions are available. Here certain new, recursively known coefficients enter, which are closely related to but different from the coefficients of the formal solutions at one of the irregular singular points of the differential equation. The coefficients of the formal solutions then appear as finite sums over subsets of the new coefficients. As a by-product, the leading exponential terms of the asymptotic behaviour of the late coefficients of the formal solutions are given, and this is a concrete example of the structural results obtained by Immink in a more general setting. The formulas displayed in this paper are not of merely theoretical interest, but they also are complete in the sense that they could be (and have been) implemented for computing accurate numerical values of the characteristic exponent, although the computational load is not small and increases with the rank of the singular point under consideration.
We will study a monodromy preserving deformation with an irregular singular point and determine the $τ$ function of the monodromy preserving deformation by the elliptic $θ$ function moving the argument $z$ and the period $Ω$.
N. Katz introduced the notion of the middle convolution on local systems. This can be seen as a generalization of the Euler transform of Fuchsian differential equations. In this paper, we consider the generalization of the Euler transform, the twisted Euler transform, and apply this to differential equations with irregular singular points. In particular, for differential equations with an irregular singular point of irregular rank 2 at $x=\infty$, we describe explicitly changes of local datum caused by twisted Euler transforms. Also we attach these differential equations to Kac-Moody Lie algeb…
A system of nonlinear differential equations $x^{1+γ}\frac{dY}{dx}= F_0(x)+A(x)Y+F(x,Y)$ is considered. We study more precisely the meaning of asymptotic expansion of transformations and solutions than preceding pioneering works, by using the theory of Borel summable functions in asymptotic analysis, and apply results to Painlevé equations.
We consider a second order reducible equation having non-resonant irregular singularities at $x=0$ and $x=\infty$. Both of them are of Poincaré rank 1. We introduce a small complex parameter $\varepsilon$ that splits together $x=0$ and $x=\infty$ into four different Fuchsian singularities $x_L=-\sqrt{\varepsilon}, x_R=\sqrt{\varepsilon}$ and $x_{LL}=-1/\sqrt{\varepsilon}, x_{RR}=1/\sqrt{\varepsilon}$, respectively. The perturbed equation is a second order reducible Fuchsian equation with 4 different singularities, i.e. a Heun type equation. Then we prove that when the perturbed equation has ex…
We study an irregular singularity of Poincaré rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter $\varepsilon\in (\mathbb{R}_+, 0)\,(\varepsilon < 1)$ which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.
In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the modu…
We introduce middle convolution for systems of linear differential equations with irregular singular points, and we presend a tentative definition of the index of rigidity for them. Under some assumption, we show a list of terminal patterns of irreducible systems of linear differential equations by iterated application of middle convolution when the index is positive or zero.
We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the exponent of ramified type is generic, we give an algebraic symplectic form on the moduli space.
We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conform…
We consider fractional differential equations of order $α\in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order of growth of $α$-entire solutions is given. An analog of the Frobenius method for systems with regular singularity is developed. For a model example of an equation with a kind of an irregular singularity, a series for a formal solution is shown to be convergent for $t>0$ (if $α$ is an irrational number poorly approximated by rational ones) but divergent in the di…
We give a complete classification of analytic equivalence of germs of parametric families of systems of complex linear differential equations unfolding a generic resonant singularity of Poincare rank 1 in dimension $n = 2$ whose leading matrix is a Jordan bloc. The moduli space of analytic equivalence classes is described in terms of a tuple of formal invariants and a single analytic invariant obtained from the trace of monodromy, and analytic normal forms are given. We also explain the underlying phenomena of confluence of two simple singularities and of a turning point, the associated Stokes…
We consider a Cauchy problem for some family of q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables t and z for given formal power series initial conditions. Under suitable conditions and by the application of certain q-Borel and Laplace transforms (introduced by J.-P. Ramis and C. Zhang), we are able to deal with the small divisors phenomenon caused by the Fuchsian singularity, and to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the…
In this, paper, we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank 1 at the origin over a fixed neighborhood $D_r$. The unfolding parameter $ε$ is taken in a sector S pointed at the origin of opening larger than $2 π$ in the complex plane, thus covering a whole neighborhood of the origin. For each parameter value in S, we cover $D_r$ with two sectors and, over each sector, we construct a well chosen basis of solutions of the unfolded linear differential systems. This basis is used to find…
We provide a unique normal form for rank two irregular connections on the Riemann sphere.In fact, we provide a birational model where we introduce apparent singular points and where the bundlehas a fixed Birkhoff-Grothendieck decomposition. The essential poles and the apparent poles provide twoparabolic structures. The first one only depend on the formal type of the singular points. The latter one determine the connection (accessory parameters). As a consequence, an open set of the corresponding moduli space of connections is canonically identified with an open set of some Hilbert scheme of po…
A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized $L^p$ spaces with variable exponent.
In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sec…
Mathieu ordinary differential equation is of Fuchsian types with the two regular and one irregular singularities. In contrast, Heun equation of Fuchsian types has the four regular singularities. Heun equation has the four kind of confluent forms: (1) Confluent Heun (two regular and one irregular singularities), (2) Doubly confluent Heun (two irregular singularities), (3) Biconfluent Heun (one regular and one irregular singularities), (4) Triconfluent Heun equations (one irregular singularity). Mathieu equation in algebraic forms is also derived from the Confluent Heun equation by changing all …
We consider a 3-dimensional Pfaffian system, whose z-component is a differential system with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlevé equation PVI. The coefficients of the system will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [44, 61]. We then express in terms of the Stokes matrices of the 3x3 irregular system the monodromy invariants p_{jk}=Tr(M_jM_k) of the 2-dimensional isomonodromic Fuchsian system with four singularit…
We study the problem of decomposition (non-commutative factorization) of linear ordinary differential operators near an irregular singular point. The solution (given in terms of the Newton diagram and the respective characteristic numbers) is known for quite some time, though the proofs are rather involved. We suggest a process of reduction of the non-commutative problem to its commutative analog, the problem of factorization of pseudopolynomials, which is known since Newton invented his method of rotating ruler. It turns out that there is an "automatic translation" which allows to obtain the …
