Lie point symmetries, partial Noether operators and first integrals of the Painlevé-Gambier equations

We utilize the Lie-Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé-Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé-Gambier equations. It is also shown that those Painlevé-Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé-Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.     

Noether, partial Noether operators and first integrals for a linear system

We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.     

Seeking (and finding) Lagrangians

It is well known that for any second-order ordinary differential equation (ODE), a Lagrangian always exists, and the key to its construction is the Jacobi last multiplier. Is it possible to find Lagrangians for first-order systems of ODEs or for higher-order ODEs? We show that the Jacobi last multiplier can also play a major role in this case.     

On Lagrangians and Hamiltonians of some fourth-order nonlinear Kudryashov ODEs

We derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov, when they satisfy the conditions stated by Fels [Fels ME, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations. Trans Am Math Soc 1996;348:5007–29] using Jacobi’s last multiplier technique. In addition the Hamiltonians of these equations are derived via Jacobi–Ostrogradski’s theory. In particular, we compute the Lagrangians and Hamiltonians of fourth-order Kudryashov equations which pass the Painlevé test.     

Ordinary differential equations invariant under translation in the independent variable and rescaling: The Lagrangian formulation

The Lagrangian formulation of the class of general second-order ordinary differential equations invariant under translation in the independent variable and rescaling is presented. The differential equations arising from this analysis are analysed using the Painlevé test. The well-known differential equation, y^″+yy^′+ky³=0, is a unique member of this class when k=3 since it is linearisable by a point transformation. A wider subset is shown to be linearisable by means of a nonlocal transformation.     

Equivalence classes of second-order ordinary differential equations with only a three-dimensional Lie algebra of point symmetries and linearisation

There are seven equivalence classes of second-order ordinary differential equations possessing only three Lie point symmetries and hence not linearisible by means of a point transformation. We examine the representatives of these classes for linearisibility by means of other types of transformation. In particular we compare the potential for linearisibility and the possession of the Painlevé property. The complete symmetry group is realised in the standard algebra for each of the equivalence classes.     

Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007-5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.     

Symmetry and singularity properties of second-order ordinary differential equations of Lie's type III

We extend the work of Abraham-Shrauner [B. Abraham-Shrauner, Hidden symmetries and linearization of the modified Painlevé-Ince equation, J. Math. Phys. 34 (1993) 4809-4816] on the linearization of the modified Painlevé-Ince equation to a wider class of nonlinear second-order ordinary differential equations invariant under the symmetries of time translation and self-similarity. In the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations.     

The Hessian and Jacobi morphisms for higher order calculus of variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler-Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.     

On a Painlevé II Model in Steady Electrolysis: Application of a Bäcklund Transformation

A model equation of Painlevé II type was introduced by Bass in 1964 in connection with a boundary value problem which describes the electric field distribution in a region x > 0 occupied by an electrolyte. This is possibly the earliest explicit physical application of a Painlevé equation to be found in the literature. Here we return to this problem informed by the subsequent discovery of a Bäcklund transformation for Painlevé II. This enables us to construct exact representations for the electric field and ion distributions for boundary value problems wherein the ratio of fluxes of the positive and negative ions at the boundary adopts one of an infinite sequence of values.     

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.

     

Two-dimensional systems that arise from the Noether classification of Lagrangians on the line

Noether-like operators play an essential role in writing down the first integrals for Euler-Lagrange systems of ordinary differential equations (ODEs). The classification of such operators is carried out with the help of analytic continuation of Lagrangians on the line. We obtain the classification of 5, 6 and 9 Noether-like operators for two-dimensional Lagrangian systems that arise from the submaximal and maximal dimensional Noether point symmetry classification of Lagrangians on the line. Cases in which the Noether-like operators are also Noether point symmetries for the systems of two ODEs are mentioned. In particular, the 8-dimensional maximal Noether algebra is remarkably obtained for the simplest system of the free particle equations in two dimensions from the 5-dimensional complex Noether algebra of the standard Lagrangian of the scalar free particle equation. We present the effectiveness of Noether-like operators for the determination of first integrals of systems of two nonlinear differential equations which arise from scalar complex Euler-Lagrange ODEs that admit Noether symmetry.     

Choquet order for spectra of higher Lamé operators and orthogonal polynomials

We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine–Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated with the aforementioned polynomials. As a consequence we obtain new inequalities for the moments and logarithmic potentials of the corresponding root-counting measures and their weak-^* limits in the semi-classical and various thermodynamic asymptotic regimes. We also prove analogous results for systems of orthogonal polynomials such as Jacobi polynomials.     

On generalized Sundman transformation method,first integrals,symmetries and solutions of equations of Painlevé–Gambier type

We employ the generalized Sundman transformation method to obtain certain new first integrals of autonomous second-order ordinary differential equations belonging to the Painlevé–Gambier classification scheme. This method not only yields systematically the known first integrals of a large number of the Painlevé–Gambier equations but also some time dependent ones, which greatly simplify the computation of their corresponding solution. In addition we also compute the Sundman symmetries of these equations.     

Value distribution of the fifth Painlevé transcendents in sectorial domains

This article is concerned with a value distribution of the fifth Painlevé transcendents in sectorial domains around a fixed singular point. We show that the cardinality of the 1-points of a fifth Painlevé transcendent in a sector has an asymptotic growth of finite order, thereby giving an improvement of the known estimates.     

CANONICAL FORMS FOR PARTICLE LAGRANGIANS OF THE LINEAR SECOND-ORDER EQUATION

Abstract

A particle Lagrangian of a linear scalar second-order ordinary differential equation can admit maximally one of 1,2,3 or 5 Noether point symmetries. Moreover, canonical forms of particle Lagrangians of the linear equation are presented according to the number (and algebra) of Noether point symmetries they admit.     

Painlevé Hierarchies and the Painlevé Test

In a series of recent papers, we derived several new hierarchies of higher-order analogues of the six Painlevé equations. Here we consider one particular example of such a hierarchy, namely, a recently derived fourth Painlevé hierarchy. We use this hierarchy to illustrate how knowing the Hamiltonian structures and Miura maps can allow finding first integrals of the ordinary differential equations derived. We also consider the implications of the second member of this hierarchy for the Painlevé test. In particular, we find that the Ablowitz–Ramani–Segur algorithm cannot be applied to this equation. This represents a significant failing in what is now a standard test of singularity structure. We present a solution of this problem.