On the first integrals of the Painlevé classes of equations

The role of symmetries and first integrals are well known mechanisms for the reduction of ordinary differential equations (odes) and, used in conjunction, lead to double reductions of the odes. In this article, we attempt to construct the first integrals of a large class of the well known second-order Painlevé equations. In some cases, variational and/or gauge symmetries have additional applications following a known Lagrangian in which case the first integral is obtained by Noether’s theorem. Sometimes, it is more convenient to adopt the ‘multiplier’ approach to find the first integrals. In a number of cases, we can conclude that the class is linearizable.


Introduction
It is well known that there is strong interplay between invariance and conservation laws of differential equations (des) and that this relationship can be physical, for e.g., in variational systems, energy conservation is associated with time invariance and linear momentum conservation is linked to space invariance.Often, though, the relationship may be purely mathematical and this has its merits in the analysis and integrability of the system (see [1,2,5,12], for e.g.).In the first two, the authors use the underlying invariance with specific algorithms to deal with diffusion systems such as the KdV, Fizhugh-Nagumo and Fisher equations.These equations are non variational so that Noether's Theorem cannot be appealed to for first integrals, among other things.Nevertheless, a particular reduction of such systems may be variational.
In this paper, we will consider a well known class of equations known as the Painlevé type equations ( [7,10]) which are second-order odes.In [10], Lagrangians are constructed for the odes using the 'Jacobi Last Multiplier' (see also [11]).In principle, one would utilise the Lagrangians to determine the variational symmetries and Noether's Theorem to construct the first integrals.We will adopt the Lagrangian approach or the 'multiplier method' ([3,6,9] to attain first integrals.In some cases, we use both approaches as the former, although more convenient, only delivers the zero-gauge symmetries.For eg., for second-order odes that are linearizable, there exists five independent first integrals (corresponding to five Noether symmetries of which at least two have non-zero gauge dependence) -the multiplier method would lead to all five.
We summarise some concepts below.In an r th-order system of partial differential equations (pdes) of n independent and m dependent variables, viz., A conservation law of (1.1) is the equation D i T i = 0 on the solutions of (1.1).Here, is the total differentiation operator and the tuple T = (T 1 , . . ., T n ) is called a conserved vector (flow) of (1.1).
A vector field is a Lie point symmetry generator, if it leaves invariant (1.1)-X represents a one parameter Lie group of transformations.
The Euler-Lagrange equations, if they exist, associated with (1.1) are the system δL/δu α = 0, α = 1, . . ., m, where δ/δu α is the Euler-Lagrange operator given by L is referred to as a Lagrangian and a Noether symmetry operator X of L arises from a study of the invariance properties of the associated functional defined over .If we include point dependent gauge terms f 1 , . . ., f n , the Noether symmetries or gauge symmetries, X , are given by For the gauge terms equal to zero, we refer to X as 'variational symmetries'.Corresponding to each X , a conserved flow is obtained via Noether's Theorem.Moreover, for the scalar case, if there exists a nontrivial differential function Q, called a 'multiplier', such that δ δu for some (conserved) vector (T t , T x ).Thus, a knowledge of each multiplier Q leads to a conserved vector determined by, inter alia, a Homotopy operator.See details and references in [3,6,9].' It is well known that for a variational ode, a single Noether symmetry leads to double reduction via the Noether first integral' which in the case of second-order odes, lead to a solution.This is so because the first integral is also invariant under the Noether symmetry.We will demonstrate this for one of the cases below as an illustration.We now present a method devised by Jacobi to derive Lagrangians of any second-order differential equation which involves finding the 'Last Multiplier'.This method provides a procedure [8] to determine the solutions of the pde, or its equivalent associated Lagrange's system, There are several ways to construct the multiplier.For eg., if we know all solutions less one, it is obtainable by where and ω 1 , ω 2 , ..., ω n−1 are the solutions of (1.8).Thus, M is a function of the variables (x 1 , ..., x n ) and depends on the n − 1 solutions chosen.The main properties of the Jacobi Last Multiplier are (i) If one chooses a different set of n − 1 independent solutions η 1 , η 2 , ..., η n−1 for the equation A f = 0, then the corresponding Last Multiplier N is related to M as . (1.10) (ii) Given a nonsingular transformation of variables, the Multiplier M of A f = 0 is given by where M is obtained from the n − 1 solutions of A f = 0 corresponding to those chosen for A f = 0 through the inverse transformation τ −1 .(iii) One can prove that every Multiplier M is a solution of the linear partial differential equation, or of its equivalent, vice versa every solution M of this equation is a Jacobi Last Multiplier.(iv) If one knows two Jacobi Last Multipliers, M 1 and M 2 , of (1.6) then their ratio is a solution ω of (1.6) or a first integral of (1.7).Since the existence of a solution is a result of the existence of a symmetry, an alternative formulation in terms of symmetries is provided in [4].If we know n − 1 symmetries of (1.6)/(1.7),suppose then Jacobi's Last Multiplier is given by M = −1 , provided that = 0 where For second-order odes, the Jacobi Last Multiplier is as follows.
From property (iii) of the previous section we deduce that we can write that is to be satisfied by the Multiplier.The Euler-Lagrange equation is differentiated with respect to u and then it can be shown that where L = L(x, u, u ) is the Lagrangian sought.This means that, if we know one Last Multiplier, then we can obtain L by two successive quadratures where f 1 and f 2 are arbitrary functions.Since every pair of Lagrangians which differ only by a total derivative with respect to time of a differentiable function give rise to the same equations, we can define equivalence classes among Lagrangians up to total derivatives of an arbitrary function (gauge function).Then Hence, we get the Lagrangian where g is the arbitrary gauge function and using the Euler equation (1.19), it turns out that f 3 is not arbitrary.Using the above properties of the Last Multiplier, the Lagrangians of the second-order ordinary differential equations of Painlev'e type were performed [10].Each of them can be written as a second-degree polynomial in u with coefficients analytical in x and u and the equation defining the Last Multiplier (1.18) is

Lagrangians, multipliers and first integrals
In [10,11], the authors established a connection between the Jacobi Last Multiplier and the Lagrangians of second-order ordinary differential equations and, in the latter, the Lagrangians of some fifty second-order ordinary differential equations of Painlevé type are constructed.We consider a large class of these and show how we can obtain first integrals using either the Lagrangian formulation or the multiplier approach.Since the former sometimes require a knowledge of gauge terms, the latter may produce a larger class of independent first integrals.
Since there are two additional nonzero gauge symmetries, there exists two more nontrivial first integrals.The ode is linearizable.
which generate the zero gauge (variational) symmetries 4 ], u) ∂ ∂u which corresponds to a complicated first integral and the Noether symmetries ) .
Here too, there are two additional nonzero gauge symmetries with nontrivial first integrals.The ode is linearizable.XLIX.u =

Conclusion
In this paper, we constructed the first integrals of a large class of the well known second-order Painlevé equations.In some cases, variational and/or gauge symmetries have additional applications following a known Lagrangian in which case the first integral is obtained by Noether's theorem.In some convenient cases, we also adopted the 'multiplier' approach to find the first integrals.