6,022 research outputs found
Movable algebraic singularities of second-order ordinary differential equations
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 curves are of
the algebraic type just described. This work generalizes previous results of S.
Shimomura. The only other possible kind of movable singularity that might occur
is an accumulation point of algebraic singularities that can be reached by
analytic continuation along infinitely long paths ending at a finite point in
the complex plane. This behaviour cannot occur for constant coefficient
equations in the class considered. However, an example of R. A. Smith shows
that such singularities do occur in solutions of a simple autonomous
second-order differential equation outside the class we consider here
Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0.Agencia Estatal de InvestigaciónMinisterio de Economía y CompetitividadAustrian Science Fun
Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0.Ministerio de Ciencia, Innovación y UniversidadesMinisterio de Economía, Industria y CompetitividadAustrian Science Fun
Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlev\'{e} Series
The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to
identify new integrable nonlinear dynamical systems. For a system which passes
the Painlev\'{e} test, the calculation of the integrals relies on a variety of
methods which are independent from Painlev\'{e} analysis. The present paper
proposes an explicit algorithm to build first integrals of a dynamical system,
expressed as `quasi-polynomial' functions, from the information provided solely
by the Painlev\'{e} - Laurent series solutions of a system of ODEs.
Restrictions on the number and form of quasi-monomial terms appearing in a
quasi-polynomial integral are obtained by an application of a theorem by
Yoshida (1983). The integrals are obtained by a proper balancing of the
coefficients in a quasi-polynomial function selected as initial ansatz for the
integral, so that all dependence on powers of the time is
eliminated. Both right and left Painlev\'{e} series are useful in the method.
Alternatively, the method can be used to show the non-existence of a
quasi-polynomial first integral. Examples from specific dynamical systems are
given.Comment: 16 pages, 0 figure
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Quantum Curve and the First Painlev\'e Equation
We show that the topological recursion for the (semi-classical) spectral
curve of the first Painlev\'e equation gives a WKB solution for the
isomonodromy problem for . In other words, the isomonodromy system
is a quantum curve in the sense of [Dumitrescu O., Mulase M., Lett. Math. Phys.
104 (2014), 635-671, arXiv:1310.6022] and [Dumitrescu O., Mulase M.,
arXiv:1411.1023]
Solving first order autonomous algebraic ordinary differential equations by places
Given a first order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial values, classified in terms of the number of distinct formal power series solutions extending them. In addition, if a particular initial value is given, we present a second algorithm that computes all the formal power series solutions, up to a suitable degree, corresponding to it. Furthermore, when the ground field is the field of the complex numbers, we prove that the computed formal power series solutions are all convergent in suitable neighborhoods.Agencia Estatal de InvestigaciónAustrian Science Fund (FWF
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