Movable Singularities Research Articles

Evidence of a Natural Boundary and Nonintegrability of the Mixmaster Universe Model

The formal asymptotic analysis of Latifi et al. [4] suggests that the Mixmaster Universe model possesses movable transcendental singularities and thus is nonintegrable in the sense that it does not satisfy the Painleve property (i.e., singularities with nonalgebraic branching). In this paper, we present numerical evidence of the nonintegrability of the Mixmaster model by studying the singularity patterns in the complext-plane, wheret is the “physical” time, as well as in the complex τ-plane, where τ is the associated “logarithmic” time. More specifically, we show that in the τ-plane there appears to exist a “natural boundary” of remarkably intricate structure. This boundary lies at the ends of a sequence of smaller and smaller “chimneys” and consists of the type of singularities studied in [4], on which pole-like singularities accumulate densely. We also show numerically that in the complext-plane there appear to exist complicated, dense singularity patterns and infinitely-sheeted solutions with sensitive dependence on initial conditions.

The radius of convergence and the well-posedness of the Painlevé expansions of the Korteweg - de Vries equation

In this paper we obtain explicit lower bounds for the radius of convergence of the Painlevé expansions of the Korteweg - de Vries equation around a movable singularity manifold in terms of the sup norms of the arbitrary functions involved. We use this estimate to prove the well-posedness of the singular Cauchy problem on in the form of continuous dependence of the meromorphic solution on the arbitrary data.

The Moving Singularities of the Perturbation Expansion of the Classical Kepler Problem

The convergence properties of the perturbation expansion for the periodic solution of the classical Kepler problem are studied as a function of the perturbation parameter a, corresponding to “linear” amplitude. We find the convergence to be directly affected by an infinite number of movable singularities in the complex a-plane. The singularities occur at locations in the complex plane where the differential equation under consideration is singular. These singularities explain the nonuniform convergence of the perturbation series that Melvin [SIAM J. Appi. Math., 33 (1977), pp. 161–194] noted. They occur in complex conjugate pairs and move in the complex plane as a function of the independent variable, causing divergence of the perturbation series solution. Our analysis near one of these singularities indicates that distinct branches of solutions occur there. These solutions undergo transition at the singularity to develop into new solution branches with distinctively different properties. Moving singularities similar to those discussed here were also shown to affect the convergence of the perturbation expansion for the limit cycle of van der Pol's equation [SIAM J. Appi. Math., 44 (1984), pp. 881–895; SIAM J. Appi. Math., 50 (1990), pp. 1764–1779]. The recurrence of this phenomenon in the present simpler problem suggests that it is very likely to affect the convergence of perturbation expansions for periodic solutions of other problems.

The Coalescence Limit of the Second Painlevé Equation

In this paper, we study a well‐known asymptotic limit in which the second Painlevé equation (PII) becomes the first Painlevé equation (PI). The limit preserves the Painlevé property (i.e., that all movable singularities of all solutions are poles). Indeed it has been commonly accepted that the movable simple poles of opposite residue of the generic solution of PII must coalesce in the limit to become movable double poles of the solutions of PI, even though the limit naively carried out on the Laurent expansion of any solution of PII makes no sense. Here we show rigorously that a coalescence of poles occurs. Moreover we show that locally all analytic solutions of PI arise as limits of solutions of PII.

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth‐order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n − 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.

The Bianchi IX (mixmaster) cosmological model is not integrable

The perturbation of an exact solution exhibits a movable transcendental essential singularity, thus proving the nonintegrability. Then, all possible exact particular solutions which may be written in closed form are isolated with the perturbative PainlevtÉ test; this proves the inexistence of any vacuum solution other than the three known ones.

Painleve analysis of new soliton equations by Hu

We perform the singularity analysis of four new generalized bilinear equations by Hu (1993), of which two possess two-soliton solutions and the other two possess even three-soliton solutions. All four equations fail to pass the Painleve test for integrability, exhibiting non-dominant movable logarithmic singularities at highest resonances of generic branches.

The two-singular-manifold method: I. Modified Korteweg-de Vries and sine-Gordon equations

We address the problem of finding, by Painleve analysis only, the Backlund transformation of partial-differential equations (PDEs) having two families of movable singularities with opposite principal parts, such as the modified Korteweg-de Vries (MKdV), sine-Gordon or nonlinear Schrodinger equations. This first paper gives an almost algorithmic method which extends the singular-manifold method of Weiss(1986) that is unable to handle these equations. First, with only one singular manifold at a time, we obtain the Darboux transformation. Second, we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective Riccati system with undetermined coefficients; the Darboux transformation then generates a very small number of determining equations, admitting a unique solution. Equivalent to the Lax pair of the Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur scheme by the canonical linearization of the Riccati system. The method is here applied to the MKdV and sine-Gordon equations.

A method of proving the convergence of the Painleve expansions of partial differential equations

The Painleve property of a partial differential equation, that is, that all solutions are single-valued around all non-characteristic movable singularity manifolds, is widely used as an indicator of its complete integrability (meaning exact solvability). The usual technique employed to investigate the Painleve property seeks formal expansions around movable singularity manifolds. In this paper, we announce a method of proving their convergence. Results are stated for Burgers' and the Korteweg-de Vries equations. For the latter, we outline the proof.

The Bianchi IX (mixmaster) cosmological model is not integrable

The perturbation of an exact solution exhibits a movable transcendental essential singularity, thus proving the nonintegrability. Then, all possible exact particular solutions which may be written in closed form are isolated with the perturbative PainlevtÉ test; this proves the inexistence of any vacuum solution other than the three known ones.

The antiplane problem of a crack with edges touching planes where the constants of elasticity change

The antiplane problem of a crack whose edges touch planes where the constants of elasticity change is considered. The special feature of this problem is that it reduces to a singular integral equation whose kernel contains two fixed singularities as well as the traditional movable singularity of the Cauchy kernel. An efficient approximate method of solving equations of this type is presented, based on the construction of a closed solution of a special form of this equation. A method of solving integral equations in the case of a single fixed singularity was presented in [1].

Integrability and singularity structure of predator-prey system

The ‘‘extended’’ ARS (Ablowitz, Ramani, and Segur) algorithm is introduced to characterize a dynamical system as Painlevé or otherwise; to that end, it is required that the formal series—the Laurent series, logarithmic, algebraic psi series about a movable singularity—are shown to converge in the deleted neighborhood of the singularity. The determinations thus obtained are compared with those following from the α method of Painlevé. An attempt is made to relate the structure of solutions about a movable singularity with that of first integrals (when they exist). All these ideas are illustrated by a comprehensive analysis of the general two-dimensional predator-prey system.

Painlevé test for the integrable dispersive long wave equations in two space dimensions

The integrable dispersive long wave equations in two space dimensions, which possess a weak Lax pair and a Kac-Moody-Virasoro type Lie algebra, have a single-valued expansion and still fail the Painlevé test in the ARS algorithm and/or in the WTC approach because multiple-valued expansions about some movable characteristic singularity manifolds are allowed for this model.

Partial differential equations with solutions having movable first-order singularities

This paper deals with the examination of nonlinear partial differential equations with solutions which have movable singularities of first order. Multi-phase and rational solutions of the family of nonintegrable equations are demonstrated. The equations can be integrable in other cases depending on the correlations of the coefficients of these equations.

WHAT CAN COMPLEX TIME TELL US ABOUT REAL TIME DYNAMICS?

In this tutorial we will review the successes and limitations of singularity analysis of dynamical systems, the progress made to date and the prospects for future research. By singularity analysis we mean the identification of the nature of all movable (i.e., initial condition dependent) singularities of a dynamical system in the complex time plane, and the study of the (generally complex) solutions of the system, in the vicinity of these singularities. The emphasis will be on the main ideas, methods and techniques, and the presentation will focus on examples and applications. However, a description of the most important theoretical developments and an outline of the main rigorous results will also be given. Our central theme throughout this review will be the connection between complex time singularities and the properties of the solutions of a dynamical system in real time.

Movable singularities in Sigma and Skyrme models

The singularity structure of the solutions of the equations of motion for the Sigma and Skyrme model Lagrangians with the hedgehog ansatz is investigated. In both cases the solutions present superposed a polelike term and a logarithmic branch point. The set of solitonic configurations of the Skyrme model is characterized by the sequence of locations of their singularities on the negative real axis of the dimensionless variable z = (eFπr)2, with an accumulation point at z=0. The first few terms of the Laurent series representing the Skyrme soliton profile are shown to reproduce well the exact values in an interval about the origin. Padé approximants to the residual power series obtained after subtraction of the dominant pole term are modified in order to satisfy the constraints imposed by the asymptotic power series expansion, and approximate representations are built for the Skyrme soliton configuration, which are shown to combine simplicity with accuracy.

Many-Valuedness of First Integrals, Their Domains and Many-Valued Solutions

Many-valuedness of first integral is studied through the 2-dimension Lotka-Volterra model and it is proved that this many·valuedness brings nonexistence of the Puiseux expansions of the general solutions. In other words, a system with a certain many-valued first integral does not have the weak Painleve property. This impli~s that many-valuedness is necessary for their general solutions. In physical case, we consider the influence of the logarithmic branchings of first integrals on real motion and show that a first integral including logarithmic functions becomes single-valued almost every­ where in the real domain under a certain condition of singularity sets of the first integral. If a dynamical system has a sufficient number of global single-valued first inte­ grals, it is obvious that it has only regular motion. Even if a system has irregular motion including chaos, there is a sufficient number of global first integrals. However they are many~valued globally because they are yielded by the analytic prolongation of local single-valued first integrals which always exist. In other words, many­ valuedness of first integral is a necessary condition for the irregular motion. Then, in this many-valuedness, there must be a root of the difference between two phases of motion. Nevertheless there are few studies of the relations between this many­ valuedness and the behavior of the system as yet, although, there are many analytical or numerical studies 1 )-5) of some relations between many-valued solutions and the behavior of the system. Then we need the considerations of infinitely many­ valuedness of first integral in addition to that of solutions in order to grasp the irregular motion as global structure. In this paper, on two regions in complex phase space, we discuss many-valuedness of first integral. One region is a neighborhood of the dominant behavior to connect this many-valuedness to the singular point analysis of solution. On this region, we show that a many-valued first integral brings many-valued solutions. Another region is a neighborhood of the real part of complex phase space to see the influence of many-valuedness of first integral on physical real motion. On this region, we find that this many-valuedness vanishes under a certain condition of singularity sets of the first integrals. This paper is organized as follows. In § 2, we show that a first integral of the 2-dimension Lotka-Volterra model is infinitely many-valued essentially in the com­ plex phase space and show that this system has a single-valued general solution around its movable singularity if this first integral has no logarithmic branchings around the dominant behavior. In § 3 we generalize this result and present the theorem The Puiseux expansion of general solution is not possible if a many-valued

Analytic structure of the damped driven Morse oscillator

An investigation of the analytic structure of the solution of the damped and driven Morse oscillator is carried out after effecting an exponential transformation. From the psi series representation, the local behaviour in the neighbourhood of a movable singularity is mapped onto a simple solvable differential equation admitting hyperbolic function solution. The inverse map reveals the existence of a two-armed infinite sheeted Riemann structure of the singularities for the system.

Exact second-order Born approximation with correct boundary conditions for symmetric charge exchange.

Symmetric (homonuclear) charge transfer between completely stripped projectiles and hydrogenlike atoms is studied by means of the second-order Born approximation (CB2) with the correct boundary conditions. Along the integration path, the transition amplitude ${\mathit{T}}_{\mathit{i}\mathit{f}}^{\mathrm{CB}2}$ exhibits so-called movable singularities, such as branch points and poles. A powerful method is presented which demonstrates that these singularities are integrable, not only for the resulting cross sections, but also for every individual matrix element. The resulting algorithm is very efficient, since the exact differential cross sections of the CB2 method are readily obtained through only two-dimensional numerical quadratures. The present theory is applied to symmetric resonant charge exchange in ${\mathrm{H}}^{+}$+H(1s)\ensuremath{\rightarrow}H(1s)+${\mathrm{H}}^{+}$ collisions at several impact energies, and the results are found to be in satisfactory agreement with the experimental data of Martin et al. [Phys. Rev. A 23, 3357 (1981)] and Vogt et al. [Phys. Rev. Lett. 57, 2256 (1986)].

Painleve Property and Algebraic Integrability of Single Variable Ordinary Differential Equations with Dominants

If the general solution of an integrable nonlinear system has a movable singularity, we can often expand it in the Laurent series with enough arbitrary constants around the singular point in the complex domain. The property that a single valued general solution exists around a movable pole singular point is called property. There are many examples whose first integrals are found out when their parameters are chosen so that they have this property.l)-4) How does the property around a singular point relate to an integrability? In this regard, there are Ziglin's studY;) for a monodoromymatrix of Hamilton systems and Yoshida's study6) for Kowalevski's exponent (resonance) of similarity invariant systems. In this work we consider n-th order single variable ordinary differential equations (Unx-A(u, Ux, ', U(n-l)x)=O, AE e[u, ', U(n-l)x]) which have dominants, and we prove by an algebraic method that the equations satisfy Painleve property's necessary condition if they have enough algebraic first integrals.