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Register a new userIntroduction to the Painleve property, test and analysis
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This short survey presents the essential features of what is called Painleve analysis, i.e. the set of methods based on the singularities of differential equations in order to perform their explicit integration. Full details can be found in textit{The Painleve handbook} or in various lecture notes posted on arXiv.
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The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painleve property are explored for ODEs of order $n =2, dots ,5$. Among the 6 ODEs identifying the Painleve transcendents only $P_{III}$, $P_V$ and $P_{VI}$ have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painleve transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painleve test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.
We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani-Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by exact reduction of this PDE has the Painleve property. The Painleve property motivated motivated much research on obtaining exact solutions on nonlinear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.
We consider the generalized Painleve--Ince equation, begin{equation*} ddot{x}+alpha xdot{x}+beta x^{3}=0 end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $beta =alpha ^{2}/9~$the given differential equation is maximally symmetric and well-known that it pass the Painlev{e} test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlev{e}--Ince equation fails at the Painlev{e} test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable $x=1/y.$ We conclude that the Painlev{e}--Ince equation is integrable is terms of Lie symmetries and of the Painlev{e} test.
The classical Painleve equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymptotic results that may not be uniformly valid. To overcome these issues, we recently carried out asymptotic analysis in Okamotos geometric space of initial values for the first and second Painleve equations. In this paper, we review this method and indicate how it may be extended to the discrete Painleve equations.
We consider systems of ordinary differential equations (ODEs) of the form ${cal B}{mathbf K}=0$, where $cal B$ is a Hamiltonian operator of a completely integrable partial differential equation (PDE) hierarchy, and ${mathbf K}=(K,L)^T$. Such systems, whilst of quite low order and linear in the components of $mathbf K$, may represent higher-order nonlinear systems if we make a choice of $mathbf K$ in terms of the coefficient functions of $cal B$. Indeed, our original motivation for the study of such systems was their appearance in the study of Painleve hierarchies, where the question of the reduction of order is of great importance. However, here we do not consider such particular cases; instead we study such systems for arbitrary $mathbf K$, where they may represent both integrable and nonintegrable systems of ordinary differential equations. We consider the application of the Prelle-Singer (PS) method --- a method used to find first integrals --- to such systems in order to reduce their order. We consider the cases of coupled second order ODEs and coupled third order ODEs, as well as the special case of a scalar third order ODE; for the case of coupled third order ODEs, the development of the PS method presented here is new. We apply the PS method to examples of such systems, based on dispersive water wave, Ito and Korteweg-de Vries Hamiltonian structures, and show that first integrals can be obtained. It is important to remember that the equations in question may represent sequences of systems of increasing order. We thus see that the PS method is a further technique which we expect to be useful in our future work.
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