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Register a new userA study of a Fuchsian system of rank 8 in 3 variables and the ordinary differential equations as its restrictions
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A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equation, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions to these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series $_4F_3$ at 1. Keywords: Fuchsian differential equation, hypergeometric differential equation, middle convolution, Dotsenko-Fateev equation, recurrence formula, series solution
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We comprehensively study admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. For each class from the constructed chain of nested gauged classes of such systems, we single out its singular subclass, which appears to consist of systems being similar to the elementary (free particle) system whereas the regular subclass is the complement of the singular one. This allows us to exhaustively describe the equivalence groupoids of the above classes as well as of their singular and regular subclasses. Applying various algebraic techniques, we establish principal properties of Lie symmetries of the systems under consideration and outline ways for completely classifying these symmetries. In particular, we compute the sharp lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by systems from each of the above classes and subclasses. We also show how equivalence transformations and Lie symmetries can be used for reduction of order of such systems and their integration. As an illustrative example of using the theory developed, we solve the complete group classification problems for all these classes in the case of two dependent variables.
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
We study singularly perturbed linear systems of rank two of ordinary differential equations of the form $varepsilon xpartial_x psi (x, varepsilon) + A (x, varepsilon) psi (x, varepsilon) = 0$, with a regular singularity at $x = 0$, and with a fixed asymptotic regularity in the perturbation parameter $varepsilon$ of Gevrey type in a fixed sector. We show that such systems can be put into an upper-triangular form by means of holomorphic gauge transformations which are also Gevrey in the perturbation parameter $varepsilon$ in the same sector. We use this result to construct a family in $varepsilon$ of Levelt filtrations which specialise to the usual Levelt filtration for every fixed nonzero value of $varepsilon$; this family of filtrations recovers in the $varepsilon to 0$ limit the eigen-decomposition for the $varepsilon$-leading-order of the matrix $A (x, varepsilon)$, and also recovers in the $x to 0$ limit the eigen-decomposition of the residue matrix $A (0, varepsilon)$.
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the results.
In an earlier paper (A. N. Kochubei, {it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirovs fractional differentiation operator $D^alpha$, $alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^alpha$ that the change of an unknown function $u=I^alpha v$ reduces the Cauchy problem for a linear equation with $D^alpha$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we study nonlinear equations of this kind, find conditions of their local and global solvability.
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