A Lagrangian Description of the Higher-Order Painlevé Equations

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by  A. Ghose Choudhury

A Lagrangian Description of the Higher-Order
Painlev´e Equations
A. Ghose Choudhury, Partha Guha,
2 Nikolai A. Kudryashov
1
0
2 Department of Physics, Surendranath College, 24/2 Mahatma

n
Gandhi Road, Calcutta-700009, India;
a
J
S.N. Bose National Centre for Basic Sciences, JD Block, Sector

8
1 III, Salt Lake, Kolkata - 700098, India;
Department of Applied Mathematics, National Research
]
I Nuclear University MEPHI, 31 Kashirskoe Shosse, 115409
S
. Moscow, Russian Federation
n
i
l
n
[
Abstract

1
v We derive the Lagrangians of the higher-order Painlev´e equations
3 using Jacobi’s last multiplier technique. Some of these higher-order
0
differential equations display certain remarkable properties like pass-
7
3 ing the Painlev´e test and satisfy theconditions stated by Jur´asˇ, (Acta
1. Appl. Math. 66 (2001) 25–39), thus allowing for a Lagrangian de-
0 scription.
2
1
:
v 1 Introduction
i
X
r The study of higher-order Painlev´e equations is interesting from the mathe-
a
matical point of view because of the possibility of existence of new transcen-
dental functions beyond the six Painlev´e transcendents. In addition such
higher-order Painlev´e often have interesting physical and mathematical ap-
plications. For example it is known that special solutions of equations for
the Korteweg de Vries hierarchy which are used for describing water waves
can be expressed via the higher-order Painlev´e equations.
The first Painlev´e hierarchy was first introduced in [1]. Thereafter many
resultswereobtainedintheanalysisofthehigherPainlev´eequations. Scaling
similarity solutions of three integrable PDEs namely the Sawada-Kotera,
1
fifth order KdV and Kaup-Kupershmidt equations were considered in [2]
where it was shown that these fourth-order ordinary differential equations
(ODEs) may be written as non-autonomous Hamiltonian equations for time
dependent generalizations of integrable cases of the H´enon-Heiles system.
In [3] it was proved that higher-order members for the first and second
Painlev´e hierarchies do not have polynomial first integrals and that their
solutions can determine new transcendental functions. Lax pairs for some
equations of these hierarchies are presented in [4] and the Cauchy problem
forequationsofthesehierarchiescanbesolvedbyananalogywiththeCauchy
problem of the well known Painlev´e equations. The Painlev´e tests for higher-
order Painlev´e equations were demonstrated in [5–7].
In [8] two new hierarchies of nonlinear ODEs were introduced which were
called the K and K hierarchies and which may be considered as new higher
1 2
Painlev´e hierarchies. The equations of these hierarchies have all the proper-
ties that are unique to the famous Painlev´e equations.
Shimomura in[9] presented aninteresting expression forthe first Painlev´e
hierarchy which allows us to consider new properties of equations. Poles and
α - points of the meromorphic solutions of the first Painlev´e hierarchy was
studied by Shimomura in[10], where a lower estimate for the number of poles
of meromorphic solution is also given.
In [11] instanton-type solutions and some leading expressions for the sec-
ond member of the first hierarchy were constructed using multiple-scale anal-
ysis. Recently Mo in [12] has applied a twistor description of the similarity
reductions to the case of the KdV hierarchy to obtain the twistor spaces of
the Painlev´e I and Painlev´e II hierarchy. Dai and Zhang [13] have extended
the results by Boutroux [14,15] for the first Painlev´e equation to the case of
the first Painlev´e hierarchy. The authors have shown that there are solutions
characterized by divergent asymptotic expansions near infinity in specified
sectors of the complex plane for higher-order analogue of the first Painlev´e
equation.
Some important results connected with higher-order Painlev´e equations
were also obtained in the papers [16,17]. In [16] Claeys and Vanlessen proved
the universality of the correlation kernel in a double scaling limit near sin-
gular edge points in random matrix models that were built out of functions
associated with a special solution of the second member for the first Painlev´e
hierarchy. In [17] the authors established the existence of real solution of
the fourth-order analogue of the Painlev´e equation and obtained the solv-
ability of an associated Riemann - Hilbert problem through the approach of
a vanishing lemma and found additionally the asymptotics of solutions.
The Hamiltonian structure of the second Painlev´e hierarchy was consid-
ered in [18]. Here the authors introduced new canonical coordinates and
2
obtained the Hamiltonian for evolutions. They also gave an explicit formu-
lae for these Hamiltonians and demonstrated that these Hamiltonians are
polynomials in the canonical coordinates.
The aim of this paper is to obtain the Lagrangians for the four higher
Painlev´e hierarchies using thesameapproach. Inrecent years much attention
has been paid to the Lagrangian framework of higher-order differential equa-
tions. Although a Lagrangian always exists for any second-order ordinary
differential equationitsconnectionwithJacobi’slastmultiplier(JLM)[19,20]
is perhaps not very widely known. The credit for resurrecting the JLM, in
recent years, must go to Leach and Nucci, who have shown how it may be
used to determine the first integrals and also Lagrangians of a wide variety
of nonlinear differential equations [21]. While it appears that the connection
of the Jacobi last multiplier to the existence of Lagrangian functions were
the subjects of investigation by a few authors in the early 1900’s, the precise
natureofthisinterrelation wasbrought outby Rao, inthe1940’s[22]. There-
after it does not appear to have attracted the attention of most researchers
working in the field of differential equations.
According to the classical theory of Darboux [23] every scalar second-
order ordinary differential equation is multiplier variational. The problem
of finding a Lagrangian for a given ODE is generally referred to as the in-
verse variational problem of classical mechanics. The necessary and sufficient
conditions for an equation y′′ = F(x,y,y′) to be derivable from the Euler-
Lagrange equation ∂L − d ∂L = 0, was enunciated by Helmholtz [24,25]
∂y dx ∂y′
(cid:16) (cid:17)
in the form of certain identities.
The variational multiplier problem for higher-order scalar ordinary dif-
ferential equations has been studied by Fels [26] and Jura´sˇ [27]. The inverse
problem for a fourth-order ODE was solved by Fels who investigated scalar
fourth-order ordinary differential equations of the form
d4u du d2u d3u
= f(x,u, , , ).
dx4 dx dx2 dx3
Fels approach for solving the fourth-order inverse problem was essentially
based on a modified version of Douglas’s [28] classical solution to the multi-
plier problem as refined by Anderson and Thompson in [29], who used the
variational bicomplex theory [30] to derive the multiplier and showed that
theexistence ofamultiplier was inadirect correspondence withtheexistence
of special cohomology classes arising in the variational bicomplex associated
with a differential equation. Fels conditions ensure the existence and unique-
ness of the Lagrangian in the case of a fourth-order equation and it has been
shown by Nucci and Arthurs [31] and more recently by us [32] that when
3
these conditions are satisfied, a Lagrangian can be derived from the Jacobi
last multiplier.
In fact Fels approached the problem using Cartan’s equivalence method,
and arrived at two differential invariants whose vanishing completely char-
acterizes the existence of a variational multiplier. Unlike the second-order
case, the multiplier is unique up to a constant multiple. The programme was
further developed by Jura´sˇ[27]who studied theinverse problem forsixthand
eighth-order equations. In fact Jura´sˇ obtained a similar solution by using,
however, a more direct approach in the spirit of the variational bicomplex;
the differential invariants becoming increasingly complicated for higher-order
systems. By analyzing the structure equations of the horizontal differential
he uncovered a two-form Π with the property dΠ ≡ 0 mod Π, if and only
if the equation
d2nu du d2n−1u
= f x,u, ,··· , ,
dx2n dx dx2n−1
(cid:0) (cid:1)
is multiplier variational. He proved that a Lagrangian, if it exists, is unique
uptothemultiplicationbyaconstant andfoundfunctionsI ,I ,...,I , whose
1 2 n
vanishing provides a necessary condition for the above equation to be varia-
tional. These functions are not relative contact invariants, but their simul-
taneous vanishing is a contact invariant condition.
In[32]theauthorsmadeuse oftheJacobi LastMultiplier (JLM) toderive
Lagrangians for a set of fourth-order ODEs which pass the Painlev´e test, i.e.,
their solutions are free of movable critical points. Recently the conjugate
Hamiltonian equations for such fourth-order equations passing the Painlev´e
test have also been derived in [33].
2 Four Painlev´e hierarchies
Now the first and the second Painlev´e hierarchy are well known and can be
written as the following
N
t L [u] = x, (1)
m m
mX=1
M
d
+u t L [u −u2]−xu−β = 0, (2)
m m x N
(cid:18)dx (cid:19)
mX=1
where N and M are integers, t , (m = 1,...,N) is the sequence of operators
m
L [u] that satisfies the Lenard recursion relation
m
1
d L [u] = d3 +4ud +2u L [u], L [u] = . (3)
x m+1 x x x m 0 2
(cid:0) (cid:1)
4
Taking the operator (3) into account we obtain
L [u] = u, (4)
1
L [u] = u +3u2, (5)
2 xx
L [u] = u +10uu +5u2 +10u3, (6)
3 xxxx xx x
L [u] = u +14uu +28u u +21u2 +
4 xxxxxx xxxx x xxx xx
(7)
70u2u +70uu2 +35u4.
xx x
Using the values of operators L , L , L , L and so on we can obtain the
1 2 3 4
equations of the first and the second Painlev´e hierarchies.
The sixth-order ordinary differential equations of the first and the second
Painlev´e hierarchies have the form
t u +14uu +28u u +21u2 +70u2u +
4 xxxxxx xxxx x xxx xx xx
+7(cid:0)0uu2 +35u4 + t u +10uu +5u2 +10u3 + (A)
x 3 xxxx xx x
+(cid:1)t u (cid:0)+3u2 +t u = x, (cid:1)
2 xx 1
(cid:0) (cid:1)
t u −14u2u −56uu u −28u2u −42uu2 +
3 xxxxxx xxxx x xxx x xx xx
(cid:0)+70u4u +140u3u2 −20u7 +t u −10u2u − (B)
xx x 2 xxxx xx
−10uu2 +6u5 +t u −(cid:1) 2u3(cid:0)−xu−β = 0,
x 1 xx 3
(cid:1) (cid:0) (cid:1)
We see that equations of the first and the second hierarchy have even
integer orders 2N −2 and 2M respectively.
Equations (A) and (B) are important and interesting because setting the
constants t = t = 0 one recovers the Painlev´e equations. When t = t = 0
3 2 1 3
these yield equations which we have studied recently. In the case t = t = 0
1 2
they reduce to sixth-order equations which are the third members of the
first and second Painlev´e hierarchies. The general case of these equations
correspond to the first and second Painlev´e hierarchies.
5
There are two other hierarchies of nonlinear ordinary differential equa-
tions that have the properties similar to Painlev´e equations. These hierar-
chies were introduced in [8] and were referred to in [34] as the K and K
1 2
hierarchies. These hierarchies can be presented as the following
N
t H [u] = x, (8)
m m
mX=1
M
d 1
+u t H u − u2 −xu−β = 0, (9)
m m x M
(cid:18)dx (cid:19) (cid:20) 2 (cid:21)
mX=1
where N and M are integers, t are parameters of the equation and the
m
operator H may be calculated by means of the formulae
m
H = J[v]Ω[v]H , (10)
n+2 n
under the conditions
H [v] = 1, H [v] = v +4v2, (11)
0 1 xx
where the operators Ω[v] and J[v] are determined by the relations
d
Ω = D3 +2vD+v , D = , (12)
x
dx
J = D3 +3(vD +Dv)+2(D2vD−1 +D−1vD2)+
(13)
+8(v2D−1 +D−1v2), D−1 = dx.
Z
Taking conditions (11) and operators (12), (13) into account we have the
operators H and H as the following
2 3
32
H [v] = v +12vv +6v2 + v3, (14)
2 xxxx xx x 3
6
H [v] = v +20vv +60v v +134v v +
3 xxxxxxxx xxxxxx x xxxxx xx xxxx
+136v2v +84v2 +544vv v +408vv2 +396v2v +
xxxx xxx x xxx xx x xx (15)
1120 256
+ v3v +560v2v2 + v5.
3 xx x 3
Note that hierarchies (8) and (9) can also be presented using another
operator G [u]. In terms of this operator these hierarchies take in the form
k
N
t G [u] = x. (16)
k k
Xk=1
M
1 d
u− t G [−2u −2u2]−xu−β = 0. (17)
k k x M
(cid:18) 2 dx(cid:19)
Xk=1
Hierarchy (16) can be transformed to (8) but hierarchy (17) coincides
with hierarchy (9). The recursion relation G is determined by the nonlinear
k
operator
G = J [v]Ω[v]G , (18)
k+2 1 k
under the conditions
1
G [v] = 1, G [v] = v + v2. (19)
0 1 xx
4
The operator J [v] takes the form
1
1 1
J = D3 + (D2vD−1 +D−1vD2)+ (v2D−1 +D−1v2). (20)
1
2 8
Hierarchies K and K though similar to the first and the second Painlev´e
1 2
hierarchies have a fundamental difference in the sense that we cannot trans-
form equations of hierarchies (16) and (17) to hierarchies (1) and (2). More-
over the hierarchy K has even integer order except 6k (k = 1,2,...) and
1
hierarchy K also has even integer order except 6k (k = 1,2,...).
2
The fourth order equation corresponding to the hierarchy K takes the
1
form
7
32
t u +12uu +6u2 + +t u +4u2 = 0. (C)
2 (cid:18) xxxx xx x 3 (cid:19) 1 xx
(cid:0) (cid:1)
At t = 0 equation (C) is the first Painlev´e equation but at t 6= 0 the
2
forth order equation differs from the the fourth order equation of the first
Painlev´e equation and we hope that this one may give a new transcendal
function.
On the other hand the sixth-order equation from hierarchy K can be
2
written as
t u +7u u −7u2u +14u u −28uu u −
2 xxxxxx x xxxx xxxx xx xxx x xxx
(cid:0) 28
−28u2u −21uu2 − uu3 −14u2u u +14u4u +
x xx xx 3 x x xx xx
(D)
4
+28u3u2 − u7 +t u +5u u −5u2u −
x 3 (cid:19) 1 xxxx x xx xx
(cid:0)
−5uu2 +u5 −xu−β = 0.
x 2
(cid:1)
Equation(D)isasixth-ordernonlinearordinarydifferential equationwith
properties similar to the Painlev´e equations but cannot be transformed to
the equation of the second Painlev´e hierarchy. This equation does not have
a first integral in the polynomial form and it is possible that it determines a
new transcendental function.
InthefollowingsectionwefindtheLagrangiansforthenonlinearordinary
differential equations (A), (B) and (D).
3 Inverse problem for sixth-order equations
and their Lagrangians
Considerasixth-orderequationinthenormalform,u = f(x,u,u ,u ,u ,u ,u ).
6 1 2 3 4 5
Here we introduce the abridged notation u = dku/dxk. The following theo-
k
remduetoJura´sˇgivesthenecessaryandsufficientconditionsforasixth-order
equation to admit a variational multiplier [27].
Theorem. A sixth-order ordinary differential equation admits a varia-
tional multiplier and non-degenerate third-order Lagrangian if and only if
following two conditions are satisfied
2 ∂f 10 ∂f ∂f ∂f 20 ∂f ∂f
0 = − D4 + D3 +D3 + D D2
3 x ∂u 9 ∂u x ∂u x ∂u 9 x ∂u x ∂u
(cid:0) 5(cid:1) 5 (cid:0) 5(cid:1) (cid:0) 4(cid:1) (cid:0) 5(cid:1) (cid:0) 5(cid:1)
8
20 ∂f ∂f 1 ∂f ∂f ∂f ∂f ∂f
− 2D2 − D2 − D2 −D2
27 ∂u x ∂u 3∂u x ∂u ∂u x ∂u x ∂u
(cid:0) 5(cid:1) (cid:0) 5(cid:1) 4 (cid:0) 5(cid:1) 5 (cid:0) 4(cid:1) (cid:0) 3(cid:1)
10 ∂f ∂f ∂f ∂f 20 ∂f ∂f
2 3
− D −D D + D
x x x x
9 ∂u ∂u ∂u ∂u 81 ∂u ∂u
5(cid:0) (cid:0) 5(cid:1)(cid:1) (cid:0) 5(cid:1) (cid:0) 4(cid:1) (cid:0) 5(cid:1) (cid:0) 5(cid:1)
1 ∂f ∂f 1 ∂f ∂f ∂f 1 ∂f ∂f 2 ∂f ∂f
2
+ D + D + D + D
x x x x
3 ∂u ∂u 3∂u ∂u ∂u 3∂u ∂u 3∂u ∂u
(cid:0) 5(cid:1) (cid:0) 4(cid:1) 5 4 (cid:0) 5(cid:1) 3 (cid:0) 5(cid:1) 5 (cid:0) 3(cid:1)
∂f 2 ∂f 1 ∂f ∂f 1 ∂f ∂f 1 ∂f ∂f ∂f
5 3 2
+D − − − − − ,
x
∂u 243 ∂u 27 ∂u ∂u 9 ∂u ∂u 3∂u ∂u ∂u
(cid:0) 2(cid:1) (cid:0) 5(cid:1) (cid:0) 5(cid:1) 4 (cid:0) 5(cid:1) 3 5 2 1
and
5 ∂f 5 ∂f ∂f ∂f
0 = D2 − D −2D +
3 x ∂u 3∂u x ∂u x ∂u
(cid:0) 5(cid:1) 5 (cid:0) 5(cid:1) (cid:0) 4(cid:1)
5 ∂f 2 ∂f ∂f ∂f
3
+ + .
27 ∂u 3∂u ∂u ∂u
(cid:0) 5(cid:1) 5 4 3
Prove. Suppose the sixth-order equation
u = f(x,u,u ,u ,u ,u )
6 1 2 3 4
is independent of u . Then it admits a variational multiplier and a non-
5
degenerate third-order Lagrangian if and only if the following two conditions
are satisfied:
0 = D3 ∂f −D2 ∂f +D ∂f − ∂f and 0 = −2D ∂f + ∂f .
x ∂u4 x ∂u3 x ∂u2 ∂u1 x ∂u4 ∂u3
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
3.1 The Jacobi Last Multiplier and construction of La-
grangians for sixth-order equations
In this section we describe the connection of the Jacobi Last Multiplier with
the Lagrangian function for sixth-order ODEs.
Proposition. Let u = f(x,u,u ,u ,u ,u ,u ) bea sixth-order ordinary
6 1 2 3 4 5
differential equation which admits a Lagrangian L. Then the function M :=
3
∂2L ,is aJacobi last multiplier, i.e., it satisfies theequation dM+ ∂f M =
(cid:16)∂u23(cid:17) dx ∂u5
0, where u = u .
5 xxxxx
Proof : Considering the higher-order Euler operator, E, the Euler-
Lagrange equation of motion for the ODE u = f(x,u,u ,...,u ) is given
6 1 5
by
∂L ∂L ∂L ∂L
E(L) = −D +D2 −D3 = 0, (21)
∂u x ∂u x ∂u x ∂u
(cid:0) 1(cid:1) (cid:0) 2(cid:1) (cid:0) 3(cid:1)
9
where L = L(x,u,u ,u ,u ) is a third-order Lagrangian. It is obvious
1 2 3
that the partial derivatives of L, namely L ,L ,....L are all functions of
u u1 u3
x,u,...,u . Upon expanding the Euler-Lagrange equation we find that
3
0 = E(L) = u L −[2u L +u L +f(x,u,u ,...,u )L +u D (L )+
5 u2u3 5 u3u3x 5 u2u3 1 5 u3u3 5 x u3u3
+2u u L +2u u L +2u u L +2u u L ]+terms independent of u .
4 5 u3u3u3 5 1 u3u3u 2 5 u3u3u1 3 5 u3u3u2 5
Here the subscripts denote partial derivatives with respect to the indicated
variables. Since the partial derivative of this equation with respect to u
5
must also be identically zero, we find that 3D (L )+ ∂f (L ) = 0.
x u3u3 ∂u5 u3u3
Let be M = L , then the above equation, E(L) = 0 is expressible as
(3) u3u3
be D logM3 + ∂f = 0, showing thereby that the Jacobi Last multiplier
x(cid:16) (3)(cid:17) ∂u5
is given by
M = M3 . (cid:3)
(3)
Remark: Note that for the fourth-order ODE, u = f(x,u,...,u ), ad-
4 3
mitting a second-order Lagrangian the analog of (3.1) is the following equa-
tion:
∂f
D logM2 + = 0,
x (2) ∂u
(cid:0) (cid:1) 3
so that the JLM is M = M2 where M = L . On the other hand for
(2) (2) u2u2
the second-order ODE, u = f(x,u,u ), it is the solution of
2 1
∂f
D logM + = 0,
x (1)
∂u
(cid:0) (cid:1) 1
with M = M = L .
(1) u1u1
Equation (??) provides us a tool for determining the Lagrangian of a
fourth-order equation once a solution of the defining equation for the JLM
, M, is obtained from (??). In fact in the event f is independent of u , so
5
that the condition (??) is trivially satisfied one obtains the solution M =
(3)
constant, which may be set equal to unity, without loss of generality. In such
a situation the Jura´sˇ conditions are also considerably simplified as evident
from the Corollary 1.
3.2 Determination of the Lagrangians
Wewishtodetermineanondegeneratethird-orderLagrangianL = L(x,u,u ,u ,u )
1 2 3
such that E(L) = 0, where ∂2L 6= 0, where E is the Euler-Lagrange op-
∂u32
erator E = ∂ − D ∂ + D2 ∂ − D3 ∂ , and D denotes the total
∂u x ∂u1 x ∂u2 x ∂u3 x
derivative operator D(cid:0) =(cid:1) ∂ +(cid:0)u ∂(cid:1) + u (cid:0)∂ +(cid:1) u ∂ . If there is a third-
x ∂x 1∂u 2∂u1 3∂u2
order Lagrangian satisfying the conditions stated in Theorem 3.1, one says
10