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 distribution sense.
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 sufficient conditions, on data including the monodromy representation, the Stokes matrices and the Poincare ranks at prescribed singularities, to solve the generalized Riemann-Hilbert problem with irregular singularities. We recover in particular the irreducibility condition on the monodromy given by Bolibrukh and Kostov in the classical case. We apply the above criteria to solve the inverse problem in differential Galois theory with a better control of the singularities.
We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,...,u_n), which are eigenvalues of the leading matrix at the irregular singuilarity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u_1,...,u_n. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singulariti…
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…
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 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 extend the result of Kitaev and Korotkin to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $τ$-function whose deformation parameters are the positions of regular singularities and the parameter $t$ of an irregular singularity. Furthermore, the $τ$-function is expressed by the hyperelliptic $Θ$ function moving the argument $\z$ and the period $\B,$ where $t$ and the positions of regular singularities move $z$ and $\B,$ respectively.
This note explains a construction of a Poisson manifold whose symplectic foliation describes a deformation of a moduli space of meromorphic connections with unramified irregular singularities. In particular, this deformation of the moduli space corresponds to the unfolding of irregular singularities of the meromorphic connections. This is an announcement of some results in the forthcoming paper.
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.
We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions.
A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices …
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…
Using the Fourier-Laplace transform, we describe the isomonodromy equations for meromorphic connections on the Riemann sphere with unramified irregular singularities as those for connections with a (possibly ramified) irregular singularity and a regular singularity. This generalizes some results of Harnad and Woodhouse.
Borel summable divergent series usually appear when studying solutions of analytic ODE near a multiple singular point. Their sum, uniquely defined in certain sectors of the complex plane, is obtained via the Borel--Laplace transformation. This article shows how to generalize the Borel--Laplace transformation in order to investigate bounded solutions of parameter dependent non-linear differential systems with two simple (regular) singular points unfolding a double (irregular) singularity. We construct parametric solutions on domains attached to both singularities, that converge locally uniforml…
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 determine the position and the type of spontaneous singularities of solutions of generic analytic nonlinear differential systems in the complex plane, arising along antistokes directions towards irregular singular points of the system. Placing the singularity of the system at infinity we look at equations of the form $\mathbf{y}'=\mathbf{f}(x^{-1},\mathbf{y})$ with $\mathbf{f}$ analytic in a neighborhood of $(0,\mathbf{0})$, with genericity assumptions; $x=\infty$ is then a rank one singular point. We analyze the singularities of those solutions $\mathbf{y}(x)$ which tend to zero for $x\to …
This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painleve equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of…
We study the tau-function and theta-divisor of an isomonodromic family of linear differential (2x2)-systems with non-resonant irregular singularities. In some particular case the estimates for pole orders of the coefficient matrices of the family are applied.
We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincaré rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $Δ$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in Δ$. We analyse the case when the leading matrix remains diagonalisable at $Δ$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as …
