1 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 rth-order system of partial differential equations (pdes) of n independent and m dependent variables, viz.,

$$\begin{aligned} E^{\beta }(x,u,u_{(1)},\ldots ,u_{(r)})=0, \qquad \beta =1,\ldots , m. \end{aligned}$$
(1.1)

A conservation law of (1.1) is the equation \( D_{i}T^{i}=0\,\) on the solutions of (1.1). Here, \( D_i={{\partial }\over {\partial x^i}}+u^\alpha _i\;{{\partial }\over {\partial u^\alpha }}+u^\alpha _{ij}\;{{\partial } \over {\partial u_j^\alpha }}+\cdots \, \) is the total differentiation operator and the tuple \(T=(T^1,\ldots ,T^n)\) is called a conserved vector (flow) of (1.1).

A vector field

$$\begin{aligned} X=\xi ^i\;{{\partial }\over {\partial x^i}}+\eta ^\alpha \;{{\partial }\over {\partial u^\alpha }}+\zeta _i^\alpha \;{{\partial }\over {\partial u_i^\alpha }}+\zeta _{i_1i_2}^\alpha \; {{\partial }\over {\partial u_{i_1i_2}^\alpha }}+\cdots , \end{aligned}$$
(1.2)

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 \({\delta }L/{\delta u^\alpha }=0\), \(\alpha =1,\ldots ,m\), where \(\delta /\delta u^\alpha \) is the Euler–Lagrange operator given by

$$\begin{aligned} {{\delta }\over {\delta u^\alpha }}={{\partial }\over {\partial u^\alpha }}+\sum _{s\ge 1}(-1)^sD_{i_1}\cdots D_{i_s}\;{{\partial }\over {\partial u^\alpha _{i_1\cdots i_s}}},\quad \alpha =1,\ldots ,m. \end{aligned}$$
(1.3)

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

$$\begin{aligned} \mathcal{L}=\int _{\Omega } L(x,u,u_{(1)},\ldots ,u_{(r)}) \textrm{d}x \end{aligned}$$
(1.4)

defined over \(\Omega \). If we include point dependent gauge terms \(f_1,\ldots ,f_n\), the Noether symmetries or gauge symmetries, X, are given by

$$\begin{aligned} XL+LD_i\xi ^i=D_if_i. \end{aligned}$$
(1.5)

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

$$\begin{aligned} {{\delta }\over {\delta u}}(QE)=0, \end{aligned}$$

then QE is a total divergence, i.e.,

$$\begin{aligned} QE=D_t T^t+D_x T^x, \end{aligned}$$

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,

$$\begin{aligned} Af =\sum ^{n}_{i=1}a_{i}(x_1,x_2... x_n){{\partial f}\over {\partial x_i}}=0, \end{aligned}$$
(1.6)

or its equivalent associated Lagrange’s system,

$$\begin{aligned} {{{d}x_{1}}\over {a_1}} = {{{d}x_{2}}\over {a_{2}}}=... ={{{d}x_{n}}\over {a_{n}}}. \end{aligned}$$
(1.7)

There are several ways to construct the multiplier. For eg., if we know all solutions less one, it is obtainable by

$$\begin{aligned} {{\partial (f,\omega _1,\omega _2... \omega _{n-1})}\over {\partial (x_1,x_2... x_n)}} = MAf, \end{aligned}$$
(1.8)

where

$$\begin{aligned} {{\partial (f,\omega _1,\omega _2,...,\omega _{n-1})}\over {\partial (x_1,x_2,...,x_n)}} = \det \begin{pmatrix} {{\partial f}\over {\partial x_1}} &{} \cdots &{} {{\partial f}\over {\partial x_n}} \\ {{\partial \omega _1}\over {\partial x_1}} &{} \cdots &{} {{\partial \omega _1}\over {\partial x_n}} \\ \vdots &{} \ddots &{} \vdots \\ {{\partial \omega _{n-1}}\over {\partial x_1}} &{} \cdots &{} {{\partial \omega _{n-1}}\over {\partial x_n}} \end{pmatrix} =0 \end{aligned}$$
(1.9)

and \(\omega _1,\omega _2,...,\omega _{n-1}\) 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

  1. (i)

    If one chooses a different set of \(n-1\) independent solutions \(\eta _1, \eta _2,...,\eta _{n-1}\) for the equation \(Af=0\), then the corresponding Last Multiplier N is related to M as

    $$\begin{aligned} N=M{{\partial (\eta _1, \eta _2... \eta _{n-1})}\over {\partial (\omega _1,\omega _2...\omega _{n-1})}}. \end{aligned}$$
    (1.10)
  2. (ii)

    Given a nonsingular transformation of variables,

    $$\begin{aligned} \tau : (x_1,x_2... x_n) \rightarrow (x_{1}',x_{2}'... x_{n}'), \end{aligned}$$
    (1.11)

    the Multiplier \(M'\) of \(A'f=0\) is given by

    $$\begin{aligned} M'=M{{\partial (x _1', x _2'... x_n')}\over {\partial (x_1,x_2...x_n)}}, \end{aligned}$$
    (1.12)

    where M is obtained from the \(n-1\) solutions of \(Af=0\) corresponding to those chosen for \(A'f=0\) through the inverse transformation \(\tau ^{-1}\).

  3. (iii)

    One can prove that every Multiplier M is a solution of the linear partial differential equation,

    $$\begin{aligned} {{\partial (Ma_1)}\over {\partial x_1}}+{{\partial (Ma_2)}\over {\partial x_2}}+ \cdots +{{\partial (Ma_n)}\over {\partial x_n}}=0, \end{aligned}$$
    (1.13)

    or of its equivalent,

    $$\begin{aligned} \sum ^{n}_{i=1} a_i{{\partial (\log M)}\over {\partial x_i}}+ \sum ^{n}_{i=1}{{\partial a_i}\over {\partial x_i}}=0; \end{aligned}$$
    (1.14)

    vice versa every solution M of this equation is a Jacobi Last Multiplier.

  4. (iv)

    If one knows two Jacobi Last Multipliers, \(M_1\) and \(M_2\), of (1.6) then their ratio is a solution \(\omega \) 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

    $$\begin{aligned} \Gamma _i= \sum ^{n}_{j=1} \xi _{ij}(x_1,x_2... x_n) \partial _{x_{j}}, \qquad i=1...n-1, \end{aligned}$$
    (1.15)

    then Jacobi’s Last Multiplier is given by \(M=\Delta ^{-1}\), provided that \(\Delta \ne 0\) where

    $$\begin{aligned} \Delta = \det \begin{pmatrix} a_1 &{} \cdots &{} a_n \\ \xi _{1,1} &{} \cdots &{} \xi _{1,n} \\ \vdots &{} \ddots &{} \vdots \\ \xi _{n-1,1} &{} \cdots &{} \xi _{n-1,n} \end{pmatrix} \end{aligned}$$
    (1.16)

    For second-order odes,

    $$\begin{aligned} u''=F(x,u,u') \end{aligned}$$
    (1.17)

    the Jacobi Last Multiplier is as follows.

From property (iii) of the previous section we deduce that we can write

$$\begin{aligned} {{d}\over {{d} t}}(\log M)+ {{\partial F}\over {\partial u'}}=0 \Longrightarrow M=\exp \left[ - \int {{\partial F}\over {\partial u'}}{d}x \right] \end{aligned}$$
(1.18)

that is to be satisfied by the Multiplier. The Euler–Lagrange equation

$$\begin{aligned} -{{\textrm{d}}\over {\textrm{d}t }} {{\partial L}\over {\partial u'}}+{{\partial L}\over {\partial u}}=0 \end{aligned}$$
(1.19)

is differentiated with respect to \(u'\) and then it can be shown that

$$\begin{aligned} M={{\partial ^2 L}\over {\partial u'^2}} \end{aligned}$$
(1.20)

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

$$\begin{aligned} L= \int \left( \int M {d}u' \right) {d}u'+f_1(x,u)u'+f_2(x,u) \end{aligned}$$
(1.21)

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

$$\begin{aligned} f_1= & {} {{\partial g}\over {\partial u}} \end{aligned}$$
(1.22)
$$\begin{aligned} f_2= & {} {{\partial g}\over {\partial x}}+f_3(x,u). \end{aligned}$$
(1.23)

Hence, we get the Lagrangian

$$\begin{aligned} L= \int \left( \int M {d}u' \right) {d}u'+{{\partial g}\over {\partial u}}u'+{{\partial g}\over {\partial x}}f_3, \end{aligned}$$
(1.24)

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

$$\begin{aligned} u''+A(u)u'^{2}+B(x,u)u'+C(x,u)=0 \end{aligned}$$
(1.25)

and the equation defining the Last Multiplier (1.18) is

$$\begin{aligned} {{d}\over {{d}x}}(\log M)-2A(u)u'-B(x,u)=0 \Longrightarrow M= \exp \left[ \int (2A(u)u'+B(x,u)){d}x \right] . \end{aligned}$$
(1.26)

2 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.

2.1 Painleve classes and first integrals

For \(u=u(x)\), the ‘conserved vector’ is reduced to a scalar differential function \(T(x,u,u',\ldots )\) being a ‘first integral’. An attempt is made to keep the labelling of the Painlevé classes as that adopted in [10].

For the ode

$$\begin{aligned} u'' -\frac{1}{2} {{ u'^2}\over {u }}-4u^2-\alpha u+ {{ 1}\over {2u }} =0, \end{aligned}$$

we have a lagrangian \(L=\frac{1}{2}{{ u'^2}\over {u }}\,+\alpha \,u_{{}}+2\,{u_{{}}}^{2}+\frac{1}{2} \,{u_{{}}}^{-1} \) with a variational symmetry \({{\partial }\over {\partial x}}\) and corresponding first integral, via Noether’s theorem,

$$\begin{aligned} T=\frac{1}{2}{{ u'^2}\over {u }}-\alpha u-2u^2-{{1}\over {2u }}. \end{aligned}$$

The ode generates a single multiplier \(Q=-{{ u'}\over {u }}\).

II. \(u''=6u^2\) with Lagrangian \(L=\frac{1}{2}u'^2+2u^3\)

The Lagrangian L leads to a single variational symmetry \({{\partial }\over {\partial x}}\) with first integral

$$\begin{aligned} T=\frac{1}{2}u'^2-2u^3 \end{aligned}$$

and the multiplier is \(Q=-u'\).

III. \(u''=6u^2+\frac{1}{2}\) and \(L=\frac{1}{2}u'^2+2u^3+\frac{1}{2} u\) has first integral

$$\begin{aligned} T=\frac{1}{2}u'^2-2u^3-\frac{1}{2} u \end{aligned}$$

from \({{\partial }\over {\partial x}}\) with multiplier \(Q=-u'\).

IV. \(u''=6u^2+x\), with Lagrangian \(L=xu+\frac{1}{2}u'^2+2u^3\) generates no first integrals.

VI. \(u''=-3uu'-u^3+q(x)(u'+u^2)\) has Lagrangian \(L={{1 }\over {2u^2+2u' }}e^{2\int q(x) \textrm{d}x }\) which generates no first integrals.

VIII. \(u''=2u^3\), \(L=\frac{1}{2}u'^2+\frac{1}{2}u^4\) has first integral

$$\begin{aligned} T=\frac{1}{2}u'^2-\frac{1}{2}u^4 \end{aligned}$$

with variational symmetry \({{\partial }\over {\partial x}}\) and multiplier \(Q=-u'\).

IX. \(u''=2u^3+xu+y\) is not conserved.

XI. \(u''={{u'^2 }\over { u }}\) with Lagrangian \(L=\frac{1}{2} {{u'^2 }\over { u^2 }} \) leading to variational symmetries

$$\begin{aligned} u\partial _u, \qquad 2x\partial _x +u\ln u \partial _u, \qquad \partial _x \end{aligned}$$

leading to first integrals, respectively,

$$\begin{aligned} -{{u' }\over { u }}, \qquad -{{u'(u\ln u -2xu') }\over { u^2 }},\qquad \frac{1}{2} {{u'^2 }\over { u^2 }}. \end{aligned}$$

The multiplier approach yields the multipliers

$$\begin{aligned} Q_1=\frac{1}{2} {{x\ln u }\over { u }}-\frac{1}{2} u' {{x^2 }\over {u^2 }},\quad Q_2=\frac{1}{2} {{\ln u }\over { u }}- u' {{x }\over {u^2 }}, \quad Q_3= {{u' }\over { u^2 }},\quad Q_4= {{x }\over { u }}, \quad Q_5={{1}\over { u }} \end{aligned}$$

for which a couple of the first integrals are

$$\begin{aligned} T_3={\frac{u' \left( u_{{}}u''-u'^{2} \right) x}{{u_{{} }}^{3}}}, \qquad T_5={\frac{x \left( u_{{}}u''-{u'}^{2} \right) }{u'^{2}} }. \end{aligned}$$

Alternatively, if the ode is written as \({{uu'' -u'^3 }\over {u^3 }} =0 \), we obtain the multipliers to match the characteristics of the variational symmetries, viz.,

$$\begin{aligned} Q_1^c=\frac{1}{2} {{xu\ln u }}-\frac{1}{2} u' {{x^2 }},\quad Q_2^c=\frac{1}{2} {{u\ln u }}- xu',\quad Q_3^c=-u',\quad Q_4^c=xu,\quad Q_5^c=u \end{aligned}$$

of which the last two correspond to the Noether symmetries which are gauge dependent and the first integrals are

$$\begin{aligned} T_4^c=\frac{1}{2}\,{\frac{{x}^{2} \left( u_{{}}u''-u'^{2} \right) }{{u _{{}}}^{2}}}, \qquad T_5^c={\frac{x \left( u_{{}}u''-u'^{2} \right) }{{u_{{}}}^{2}} } \end{aligned}$$

Clearly, the ode \(u''={{u'^2 }\over { u }}\) is linearizable.

XII. \(u'' = {{u'^2 }\over { u }}+\alpha u^3 +\beta u^2 +\gamma +{{\delta }\over {u }} \) with Lagrangian \(L=\frac{1}{2}\,{\frac{u'^{2}}{{u_{{}}}^{2}}}+\frac{1}{2}\,\alpha \,{u_{{}}}^{2}+ \beta \,u_{{}}-\frac{1}{2}\,{\frac{\delta }{{u_{{}}}^{2}}}-{\frac{\gamma }{u_{{ }}}} \) admits a single symmetry \(\partial _x\) with first integral

$$\begin{aligned} T= {{u'^2 }\over { u^2 }}-{{\frac{1}{2}\,\alpha \,{u_{{}}}^{4}+\beta \,{u_{{}}}^{3}-\gamma \,u_{{}}+\frac{1}{2}\,u'^{2}-\frac{1}{2}\delta }\over {u^2 }} \end{aligned}$$

and \( u^2u'' = u^2[ {{u'^2 }\over { u }}+\alpha u^3 +\beta u^2 +\gamma +{{\delta }\over {u }} ] \) admits a single multiplier \(Q=-u'\) corresponding to the characteristic of the symmetry \(\partial _x\).

XIII. \(u'' = {{u'^2 }\over { u }}-{{u' }\over { u }}+\alpha {{u^2 }\over { x }}+\beta {{1 }\over { x }}+\gamma u^3+\delta {{1 }\over { u }} \) admits no symmetry and no first integrals.

XXXVII. \( u''= ({{1}\over { 2u }}+{{1 }\over { u-1 }} ){{u'^2 }\over { u }} \) has Lagrangian \(L= \frac{1}{2}\,{\frac{u'^{2}}{u_{{}} \left( u_{{}}-1 \right) ^{2}}} \) which generate the zero gauge (variational) symmetries

$$\begin{aligned} \left( u-1 \right) \sqrt{u}\partial _u,\qquad -x\partial _x+ \left( u-1 \right) \sqrt{u}\arctan \left( \sqrt{u} \right) \partial _u,\qquad \partial _x \end{aligned}$$

with corresponding first integrals

$$\begin{aligned} -{\frac{u'}{\sqrt{u_{{}}} \left( u_{{}}-1 \right) }},\qquad -{{u'[u^{\frac{3}{2}}{\arctan }h( \sqrt{u} )- \sqrt{u}{\arctan }h( \sqrt{u} )+xu' ] }\over { u_{{}} \left( u_{{}}-1 \right) ^{2} }}+\frac{1}{2} {{xu'^2 }\over {u_{{}} \left( u_{{}}-1 \right) ^{2} }},\qquad \frac{1}{2}\,{\frac{{u'}^{2}}{u_{{}} \left( u_{{}}-1 \right) ^{2}}}. \end{aligned}$$

Since there are two additional nonzero gauge symmetries, there exists two more nontrivial first integrals. The ode is linearizable.

XLIII. \(u''=\frac{3}{4}(\frac{1}{u}+\frac{1}{u-1})u'^{2}\), has Lagrangian \(L=\frac{u'^{2}}{2(u(u-1))^\frac{3}{2}}\) which generate the zero gauge (variational) symmetries

\( \left[ x(u-\frac{1}{2}){hypergeom}([\frac{1}{2},1],[\frac{5}{4}],u)+\frac{4xu(u-1) {hypergeom}([\frac{3}{2},2],[\frac{9}{4}],u)}{5}\right] \frac{\partial }{\partial x}+u(u-1){hypergeom}([\frac{1}{2},1],[\frac{5}{4}],u)\frac{\partial }{\partial u}\) which corresponds to a complicated first integral and the Noether symmetries

$$\begin{aligned} (u(u-1) )^{\frac{3}{4}} \partial _u, \qquad \partial _x \end{aligned}$$

have correspnding first integrals

$$\begin{aligned} \frac{-u'(u(u-1))^{\frac{1}{4}}}{u(u-1)}, \qquad \frac{u'^{2}}{2u(u-\sqrt{u(u-1)})}. \end{aligned}$$

Here too, there are two additional nonzero gauge symmetries with nontrivial first integrals. The ode is linearizable.

XLIX. \( u''=\frac{1}{2}( {{1}\over { u }}+{{1 }\over { u-1 }} +{{1 }\over { u-\alpha }} ){u'^2 } +(u^2-u)(u-\alpha )(\beta +{{\gamma }\over {u^2 }}+{{\delta }\over {(u-1)^2 }} +{{\epsilon }\over {(u-\alpha )^2 }} ) \) has Lagrangian \(L=\frac{1}{2}\,{\frac{u'^{2}}{u_{{}} \left( u_{{}}-1 \right) \left( u_{ {}}-\alpha \right) }}+\beta \, \left( u_{{}}+{\frac{\alpha }{\alpha +1}} \right) -\gamma \, \left( {u_{{}}}^{-1}+ \left( \alpha +1 \right) ^{-1} \right) -{\frac{\delta \, \left( u_{{}}+\alpha \right) }{ \left( u_{{ }}-1 \right) \left( \alpha +1 \right) }}+{\frac{\epsilon \, \left( u_{ {}}+1 \right) }{ \left( -u_{{}}+\alpha \right) \left( \alpha +1 \right) }} \) with a single variational symmetry \(\partial _x\) generating a first integral

$$\begin{aligned} T=-\frac{1}{2}{\frac{{u'}^{2}}{u_{{}} \left( u_{{}}-1 \right) \left( -u _{{}}+\alpha \right) }}-{\frac{\beta \, \left( \alpha \,u_{{}}+\alpha +u _{{}} \right) }{\alpha +1}}+{\frac{\gamma \, \left( \alpha +1+u_{{}} \right) }{u_{{}} \left( \alpha +1 \right) }}+{\frac{\delta \, \left( u _{{}}+\alpha \right) }{ \left( u_{{}}-1 \right) \left( \alpha +1 \right) }}-{\frac{\epsilon \, \left( u_{{}}+1 \right) }{ \left( -u_{{ }}+\alpha \right) \left( \alpha +1 \right) }} \end{aligned}$$

3 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.