This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. We proof that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shrödinger equations. Also, as an auxiliary tool, we improve a previously known criterium for second order linear differential equations to admit a polynomial solution.
We present a semi-decision procedure to tackle first order differential equations, with Liouvillian functions in the solution (LFOODEs). As in the case of the Prelle-Singer procedure, this method is based on the knowledge of the integrating factor structure.
In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic's algorithm and asymptotic iteration method we find a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouiville integrable differential equations. For each fixed degree, we prove that the spectral set is a countable union of non accumulating algebraic varieties. This algebraic description of the spectral set allow us to bound the number of eigenvalues for algebraically quasi-solvable …
Let $k$ be a differential field of characteristic zero and $E$ be a liouvillian extension of $k$. For any differential subfield $K$ intermediate to $E$ and $k$, we prove that there is an element in the set $K-k$ satisfying a linear homogeneous differential equation over $k$. We apply our results to study liouvillian solutions of first order non linear differential equations and provide generalisations and new proofs for several results of M. Singer and M. Rosenlicht on this topic.
For a field k$with an automorphism σand a derivation δ, we introduce the notion of liouvillian solutions of linear difference-differential systems {σ(Y) = AY, δ(Y) = BY} over k and characterize the existence of liouvillian solutions in terms of the Galois group of the systems. We will give an algorithm to decide whether such a system has liouvillian solutions when k = C(x,t), σ(x) = x+1, δ= d/dt$ and the size of the system is a prime.
The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular ones and propose some criteria of solvability for systems whose (formal) exponents are sufficiently small.
Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In \cite{JCAM}, we have introduced the basis for the present implementation. The particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE). We present a set of software routines in Maple 10 for solving rational first order ODEs. The package present commands permitting research incursions of some algebraic properties of the system that is being studied.
We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponential of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. That is, solutions are obtainable by suitable differential field extensions construction. In particular, we have established a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation wh…
A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen-van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar…
I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of resonance i.e. the appearance of a free constant in solutions. Secondly, for any singular differential equation in a differential field of the form $Q(y)\dot y=P(y)$, $P,Q$ polynomials. Then the greatest Picard-Vessiot extension exists. It is shown that the arising set of solutions is obtained from algebraic equations labelled by constants. Sufficient conditions …
An ordering for Laurent polynomials in the algebraic torus $(\mathbb C^*)^D$, inspired by the Cantero-Moral-Velázquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus $\mathbb T^D$. The Gauss-Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. Christoffel type perturbations of the measure given by the multiplicatio…
Differential Galois theory has played important roles in the theory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order nonlinear differential equation. We will define for the differential equation the differential Galois group, will study the structure of the group, and will prove the equivalent between the existence of the Liouvillian first integral and the solvability of the corresponding differential Galois group.
We extend Kovacic's algorithm to compute the differential Galois group of some second order parameterized linear differential equation. In the case where no Liouvillian solutions could be found, we give a necessary and sufficient condition for the integrability of the system. We give various examples of computation.
A classical result due to Bochner classifies the orthogonal polynomials on the real line which are common eigenfunctions of a second order linear differential operator. We settle a natural version of the Bochner problem on the unit circle which answers a similar question concerning orthogonal Laurent polynomials and can be formulated as a bispectral problem involving CMV matrices. We solve this CMV bispectral problem in great generality proving that, except the Lebesgue measure, no other one on the unit circle yields a sequence of orthogonal Laurent polynomials which are eigenfunctions of a li…
We show that the algebraic curve $a_0(x)(y-r(x))+p_2(x)a'(x)=0,$ where $r(x)$ and $p_2(x)$ are polynomial of degree 1 and 2 respectively and $a_0(x)$ is a polynomial solution of the convenient Fucsh's equation, is an invariant curve of the quadratic planar differential system. We study the particular case when $a_0(x)$ is an orthogonal polynomials. We prove that that in this case the quadratic differential system is Liouvillian integrable.
We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second order as the convergence of formal solutions and the existence of Liouvillian solutions. A characterization of regular singularities is given in terms of the space of solutions. An analytic classification of such linear homogeneous ODEs is obtained. This is done by associating to such an ODE a Riccati differential equation and therefore a global holonomy gr…
It is shown, how to generate infinite sequences of differential equations of the second order based on some standard equations, using Euler-Imshenetsky-Darboux (EID) transformation. For all this, factorizations of differential operators and operational identities are used.Some generalizations of integrable cases of Schrödinger's equations are finded. The example of integrable equation with liouvillian coefficients, that, apparently, can not be solved by Singer and Kovacic's algorithm (and its modifications), was built. The alogorithm and the program for solving conctructed classes of equations…
We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations.
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated…
We present $\text{Fuchsia}$ $-$ an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $\partial_x\,\mathbf{f}(x,ε) = \mathbb{A}(x,ε)\,\mathbf{f}(x,ε)$ finds a basis transformation $\mathbb{T}(x,ε)$, i.e., $\mathbf{f}(x,ε) = \mathbb{T}(x,ε)\,\mathbf{g}(x,ε)$, such that the system turns into the epsilon form: $\partial_x\, \mathbf{g}(x,ε) = ε\,\mathbb{S}(x)\,\mathbf{g}(x,ε)$, where $\mathbb{S}(x)$ is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimens…
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem which have interesting consequences…
This paper gives a classification of first order polynomial differential operators of form $\mathscr{X} = X_1(x_1,x_2)δ_1 + X_2(x_1,x_2)δ_2$, $(δ_i = \partial/\partial x_i)$. The classification is given through the order of an operator that is defined in this paper. Let $X=\mathscr{X}y$ to be the differential polynomial associated with $\mathscr{X}$, the order of $\mathscr{X}$, $\mathrm{ord}(\mathscr{X})$, is defined as the order of a differential ideal $Λ$ of differential polynomials that is a nontrivial expansion of the ideal $\{X\}$ and with the lowest order. In this paper, we prove that th…
The Painlevé classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification still remains an open problem especially for the higher-order equations. One of the main classical methods of Painlevé analysis is based on considering the resonance numbers corresponding to the possible indices of arbitrary coefficients in the Laurent expansion of the general solution in a neighborhood of a movable singularity. Complex and non-integer values of…
We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iteratated solutions to the differential equations associated with hypergeometric functions to prove the following result (Theorem 1): The epsilon-expansion of a generalized hypergeometric function with integer values of parameters is expressible in terms of generalized polylogarithms with coefficients that are ratios of polynomials. The method used in this proof provides an effici…
Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length…
