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\def\rightheadline{EJDE--1999/49\hfil
Liouvillian first integrals of second order \hfil\folio}
\def\leftheadline{\folio\hfil Colin Christopher
\hfil EJDE--1999/49}
\voffset=2\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999) No.~49, pp. 1--7.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 34A05, 34C99.
\hfil\break
{\eighti Key words and phrases:} first integral, integrating factor,
Liouville, Darboux.
\hfil\break
\copyright 1999 Southwest Texas State University and
University of North Texas.\hfil\break
Submitted September 7, 1999. Published December 7, 1999. } }
\bigskip\bigskip
\centerline{LIOUVILLIAN FIRST INTEGRALS OF SECOND ORDER}
\centerline{POLYNOMIAL DIFFERENTIAL EQUATIONS}
\medskip
\centerline{Colin Christopher}
\bigskip\bigskip
{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
We consider polynomial differential systems in the plane
with Liouvillian first integrals. It is shown that all such systems
have Darbouxian integrating factors, and that the search for such
integrals can be reduced to a search for the invariant algebraic
curves of the system and their `degenerate' counterparts.
\bigskip}
\font\cpsc=cmcsc10
\def\balpha{{\bar\alpha}}
\def\bgamma{{\bar\gamma}}
\bigbreak
\centerline{\bf 1. Introduction} \medskip\nobreak
The purpose of this paper is to provide a more satisfactory conclusion
to the work of Singer on the existence of Liouvillian first integrals
of second order polynomial differential equations.
In his paper ([S] Theorem 1 and its corollary),
the following result is obtained:
\proclaim Theorem 1. If the second order polynomial differential equation
$$
{dx\over dt}=P(x,y),\qquad {dy\over dt}=Q(x,y),\eqno(1.1)
$$
has a local Liouvillian first integral, then there is a Liouvillian first
integral of the form
$$
\int_{(x_0,y_0)}^{(x,y)} RQ\,dx - RP\,dy ,
$$
where
$$
R=\exp\left\{\int_{(x_0,y_0)}^{(x,y)} U\,dx + V\,dy\right\} ,
$$
with $U$ and $V$ rational functions in $x$ and $y$ such that
$$
{\partial U\over\partial y}={\partial V\over\partial x}.
$$
Our aim here is to prove the following theorem, which reduces the
classification to a single quadrature.
\proclaim Theorem 2. If the system $(1.1)$ has an integrating factor
of the form
$$
\exp\left\{\int U\,dx +V\,dy\right\} , \qquad U_y=V_x , \eqno(1.2)
$$
where $U$ and $V$ are rational function of $x$ and $y$, then there exists a
integrating factor of the system $(1.1)$ of the form
$$
exp(D/E)\prod C_i{}^{l_i},
$$
where $D$, $E$ and the $C_i$ are polynomials in $x$ and $y$.
The significance of this theorem is that the Darbouxian integrating factor
above defines a collection of invariant algebraic curves of the system,
$C_i(x,y)=0$, satisfying the equation
$$
{d\over dt}C_i(x,y)=C_i(x,y)L_i(x,y),\eqno(1.3)
$$
for some polynomial $L_i(x,y)$ of smaller degree than the system.
The other term in the product, $\exp(D/E)$, can be considered as a
degenerate counterpart to these curves which satisfies a similar equation.
In this way the search for Liouvillian integrating factors can be reduced
to the search for invariant algebraic curves and `degenerate
algebraic curves' or exponential factors of the system satisfying
the equation (1.3).
As an example, consider the Lokta-Volterra equations.
$$\dot x = x (a + bx + cy),\qquad \dot y = y (d + ex + fy).$$
If $af(e-b)=bd(c-f)$ then we can find an integrating factor
of the form $x^\alpha y^\beta$.
Another example is given by the Kukles' system
$$ \dot x = y,\qquad \dot y = -x+x^2-2y^2
-{\textstyle {1\over 3}}x^3/3-k^{-1}x^2y+{\textstyle{1\over 3}}k^{-1}y^3,$$
with $k = \pm2^{1/2}$, examined in [Ch\&L]. This has an integrating factor
$$
(x(y+kx)+3k(1-x))^{-3}\exp(x(1-{\textstyle{1\over 2}}x)).
$$
It may be enquired whether the other integration is also unnecessary.
That is, whether there is an elementary first integral whenever there
is a Darbouxian integrating factor. However, this is not always the case.
Generically, the Lokta-Volterra system has a first integral of the form
$$x^{\alpha+1}y^{\alpha+1}L(x,y),$$
with $L$ a degree polynomial of degree one, but, in the second
example the first integral is
$$
\eqalign{
&(y^2(x+1)+2kyx(x-2)+6(3x5))(x(y+kx)+3k(1-x))^{-2}\exp(x(1-{\textstyle{1\over 2}}x))\cr
&+\int_0^x \exp(u(1-{\textstyle{1\over 2}}u))\,du ,\cr}
$$
which is not elementary. However, it is known that for `generic' classes
of Darbouxian integrating factors there also exists a Darbouxian first
integral [Ch1].
In the general \.Zo{\l}\c{a}dek has highlighted two classes
of systems with Darbouxian integrating factors whose first integrals are
not Darbouxian: those with so-called {\it Darboux-Schwatz-Christoffel} and
{\it Darboux-Hyperelliptic} first integrals. Each of these can be
distinguished from the Darbouxian case by their holonomy groups.
The Darboux-Hyperelliptic case is also elementary (an example of which
can be found in the counter-example of Prelle and Singer [P\&S] p216),
whilst the Darboux-Schwatz-Christoffel is non-elementary.
It is not known whether these comprise all the possible first integrals.
In conclusion, the search for integrating factors by the
use of algebraic invariant curves, which has been in use since the time
of Darboux captures all first integrals (1.1) which can be expressed in
closed form with quadratures.
It is interesting to consider what place other families of functions
play in the study of integrable systems. \.Zo{\l}\c{a}dek
[Z], for example, has found new classes of integrals which can be
defined using hypergeometric functions.
Similar results are known in the local analytic case.
The following proof has the advantage that it can be adapted easily to the case of
algebraic differential equations. I would like to thank the referee for helpful
comments on the first draft of the paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigbreak
\centerline{\bf 2. Proof of the Theorem } \medskip\nobreak
The proof of the theorem follows directly from evaluating the integral~(1.2).
In fact, the system (1.1) plays no role here at all.
Let $K$ be an algebraic extension of ${\bf C}(y)$ which is a splitting field
for the numerators and denominators of $U$
and $V$ considered as polynomials in $x$ over ${\bf C}(y)$. We can thus
rewrite $U$ and $V$ in their partial fraction expansions
$$
U=\sum_{i=1,\ldots,r\atop j=1,\ldots,n_i}
{\alpha_{i,j}\over(x-\beta_i)^j} + \sum_{i=0}^N\gamma_ix^i, \qquad
V=\sum_{i=1,\ldots,\bar r\atop j=1,\ldots,\bar n_i}
{\balpha_{i,j}\over(x-\beta_i)^j} + \sum_{i=0}^{\bar N}\bgamma_ix^i,
$$
where the $\alpha_{i,j}$, $\balpha_{i,j}$, $\beta_i$
and $\gamma_i$ are elements of $K$. By taking
$\alpha_{i,j}$, $\balpha_{i,j}$, and $\gamma_i$ to be zero outside their
defined values, we can neglect the explicit mention of the summation limits
without confusion.
We now apply the condition $U_y=V_x$ to the above expressions. Gathering
terms and using the uniqueness of the partial fraction expansion,
we see that
$$
\gamma_i'=\bgamma_{i+1}(i+1),\qquad
\alpha_{i,j+1}'+j\beta_i'\alpha_{i,j}+j\balpha_{i,j} = 0.\eqno(2.2)
$$
In particular we have $\alpha_{i,1}'=0$.
We now write down a function and show that it is indeed the integral of
the equation (2.1). Let $\phi$ be given by
$$
\phi = \sum \alpha_{i,1}\log(x-\beta_i) +
\sum {\alpha_{i,j}\over (x-\beta_i)^{j-1}}{-1\over j-1}
+ \sum {\gamma_ix^{i+1}\over i+1} + \int \bgamma_0\,dy.
$$
It is easy to verify that $\phi_x=U$ and $\phi_y=V$ using (2.2).
We now take the trace of $\phi$ divided by the number of terms in
the summation of the trace. Call this function $\bar\phi$.
It is clear that this function also satisfies $\phi_x=U$ and $\phi_y=V$.
Futhermore we have
$$
\bar\phi=\sum l_i\log(R_i(x,y)) + R(x,y) + \int S(y)\,dy,
$$
where $R_i$, $R$ and $S$ are rational functions. We can now evalute the
integral via the partial fraction expansion of $S$ as a sum
$$
\int S(y)\,dy=\sum \alpha_i\log(S_i(y))+S_0(y),
$$
Where the $S_i$ are polynomials in $y$. Taking exponentials,
the integrating factor obtained is of the form stated in the theorem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigbreak
\centerline{\bf 3. Applications} \medskip\nobreak
We want now to apply the existence of a Darbouxian first integral to
the system (1.1) to demonstrate how the theorem applies to the search
for Liouvillian first integrals.
In the case where the integral is elementary, It follows from the
work of Prelle and Singer [P\&S] that there is an integrating factor
in the form
$$
\prod C_i^{l_i}, \eqno(3.1)
$$
for some polynomials $C_i$ which satisfy the equation
$$
{d\over dt}C_i=C_{ix}P+C_{iy}Q=C_iL_i, \eqno(3.2)
$$
for some polynomial $L_i$, which we call the {\it cofactor} of $C_i$.
If the degrees of $P$ and $Q$ are at most $m$, then the degree of the
cofactor will be at most $m-1$. This is equivalent to the fact that
the curves $C=0$ are invariant algebraic curves of the complex system [Ch2].
The search for elementary solutions to the system is therefore equivalent
to the search for invariant algebraic curves of the system. Furthermore,
for a fixed system, there is either a finite number of irreducible algebraic
curves or a rational first integral. Therefore for any system there exists
a bound on the degree of its irreducible algebraic curves.
However, there is not an effective way to compute this as yet.
The problem of integration in elementary terms is thus reduced to the following
question (Prelle and Singer [P\&S] p227):
\medskip
{\it Give an effective proceedure for computing, given a system of the form
(1.1), a bound for the degree of any irreducible polynomials $C(x,y)$ which
satisfy (3.2).}
\medskip
Substantial progress has been made recently in this direction by
Campillo and Carnicer [C\&C].
In the Liouvillian case, Theorem 1 is not powerful enough to give
such a formulation. It was the lack of this information which motivated
the work here. Now that we have a Darbouxian integrating factor, we are
able to finish the characterisation problem. To do this we need only
introduce the concept of a `degenerate algebraic curve', or exponential
factor, to cover the term $\exp(D/E)$ which does not appear in (3.1).
It is worth noting that the difference between the integrating factor (3.1)
and the Darbouxian integrating factor we obtained in the previous section
is more than just the term $\exp(D/E)$, because in the
elementary case the $l_i$ can also be chosen to be rational.
For example, in the Lokta-Volterra system considered in the introduction
we can also find an integrating factor of the form $(x y L)^{-1}$.
In the Liouvillian case this is no longer true. For example, the system
$$
\dot x = -x(1-x)s,\qquad \dot y =
(1+rx+sy)((\alpha+1)(1-x)-(\beta+1)x) + rx(1-x) + 1,
$$
has an integrating factor $x^\alpha(1-x)^\beta$ and first integral
$$
x^{\alpha+1}(1-x)^{\beta+1}(1+rx+sy) + \int_0^x t^\alpha(1-t)^\beta\,dt.
$$
The Kukles' system considered in the introduction would give an example of a
system which requires the exponential factor in its integrating factor.
From Theorem 2, we have a first integral
$$
R=\exp(D/E)\prod C_i^{l_i},
$$
which therefore satisfies the differential equation
$$
R_xP+R_y+R(P_x+Q_y)=0.
$$
Without loss of generality, we can assume that $D$ and $E$ are coprime
and that the $C_i$ are distinct and irreducible.
After some rearrangement, we then have
$$
\displaylines{\quad
\Bigl((D_xP+D_yQ)E-(E_xP+E_yQ)D\Bigr)\prod C_i \hfill\cr
\hfill{}+E^2\Bigl( \sum l_i(C_{ix}P+C_{iy}Q)\prod_{j\neq i}C_i
+ (P_x+Q_y)\prod C_i \Bigr) = 0.\quad(3.3) \cr}
$$
First suppose that $E$ has an irreducible factor $F$ with multiplicity $n$.
If we take $E=F^nG$, then it is clear from the equation above that
$$
F^{2n}|\bigl((D_xP+D_yQ)E-(E_xP+E_yQ)D\bigr)\prod C_i .
$$
since all the $C_i$ are distinct, this implies that
$$
F^{2n-1}|(D_xP+D_yQ)E-(E_xP+E_yQ)D
$$
and since $2n-1\geq n$ for $n\geq 1$, we have
$$
F^n|E_xP+E_yQ=n(F_xP+F_yQ)F^{n-1}+(G_xP+G_yQ)F^n,
$$
and so
$$
F|F_xP+F_yQ.
$$
Thus $F=0$ is an invariant algebraic curve. Suppose now that $C_i$ is
not one of the irreducible factors of $E$, then equation (3.3) implies that
$$
C_i|C_{ix}P+C_{iy}Q,
$$
And so all the $C_i$ define invariant algebraic curves of the system.
It now remains to characterise the polynomial $D$. This is best done by
considering the expression $\exp(D/E)$ as a whole. It is clear from
equation (3.3), using equation (3.2) for the $C_i$, that we have:
$$
\exp(D/E)_xP+\exp(D/E)_yQ=\exp(D/E)M, \eqno(3.4)
$$
where $M$ is a polynomial of degree at most $m-1$. That the form of this
equation is the same as (3.2) is no coincidence, and in fact the term
$\exp(D/E)$ can be considered as a limit of coalescing curves---in the same
way as the coalescence of two exponential solutions in a linear ODE give rise
to the `degenerate' solution $xe^{\lambda x}$ (see [Ch2] for a more detailed
justification of this). Rewriting equation (3.4), we obtain
$$
D_xP+D_yQ-DL = EM, \eqno(3.5)
$$
Where $L$ is the cofactor of $E$.
Suppose we have more than $m(m+1)/2$ linearly independent rational functions
$D_i/E_i$ such that
$$
{d\over dt}\exp(D_i/E_i)=\exp(D_i/E_i)M_i.
$$
with $M_i$ of degree at most $m-1$, then we can choose constants $l_i$
such that
$$
{d\over dt}\prod(\exp(D_i/E_i))^{l_i} = 0;
$$
whence
$$
\sum l_i(D_i/E_i)
$$
is a rational first integral of the system.
Thus, if (1.1) does not have a rational first integral, there exists a
bound on the number of independent $D_i/E_i$ which
satisfy (3.5) and hence on the degree of the numerator and denominator of
any rational function $D/E$ which satisfies (3.4).
Hence the search for Liouvillian first integrals is reduced to the search
for invariant algebraic curves, and exponential factors of the form
$\exp(D/E)$. Of course to make the whole thing algorithmic, we need to be
able to answer the following question in the spirit of Prelle and Singer:
\medskip
{\it (i) Give an effective proceedure for computing, given a system of the form
(1.1), a bound for the degree of any irreducible polynomials $C(x,y)$ which
satisfy (3.2).}
\smallskip
{\it (ii) Give an effective proceedure for computing, given a particular system
$(1.1)$ without rational first integral, a bound on the degree of any
polynomials $D$ and $E$ which satisfy equation (3.5), given that $E$
also satifies $(3.2)$ with cofactor $L$.}
\medskip
The question of the existence of a rational first integral required in (ii) can
be effectively decided from (i) and so the whole process of seeking Liouvillian
first integrals would be algorithmic.
\bigbreak
\centerline{\bf References} \medskip\nobreak
\item{[C\&C]} {{\cpsc Campillo A. \& Carnicer M.M.},
Proximity inequalities and bounds for the degree of invariant
curves by foliations of ${\bf P}_{\bf C}^2$,
{\it Trans.\ Amer.\ Math.\ Soc.} {\bf 349}(6), (1997), 2211--2228.}
\item{[Ch1]} {{\cpsc Christopher C.J.},
Polynomial systems with invariant algebraic cur\-ves,
Pre\-print, University of Plymouth.}
\item{[Ch2]} {{\cpsc Christopher C.J.},
Invariant algebraic curves and conditions for a centre,
{it Proc.\ Royal Soc.\ Edinburgh} {\bf 124A}, (1994), 1209--1229.}
\item{[Ch\&L]} {{\cpsc Christopher C.J. \& Lloyd N.G.},
On the paper of Jin and Wang concerning the conditions for a centre
in certain cubic systems,
{\it Bull.\ London.\ Math.\ Soc.} {\bf 22}, (1990), 5--12.}
\item{[P\&S]} {{\cpsc Prelle M.J. \& Singer M.F.},
Elementary first integrals of differential equations,
{\it Trans.\ Amer.\ Math.\ Soc.} {\bf 279}(1), (1983), 215--229.}
\item{[S]} {{\cpsc Singer M.F.},
Liouvillian first integrals of Differential Equations,\break
{\it Trans.\ Amer.\ Math.\ Soc.} {\bf 333 No.2}, (1992), 673--688.}
\item{[Z]} {{\. Zo\l\c adek H.},
The problem of center for resonant singular points of
polynomial vector fields,
{\it J. Differential Equations} {\bf 137}, (1997), 94--118.}
\bigskip
\noindent
{\cpsc Colin Christopher} \hfill\break
School of Mathematics and Statistics \hfill\break
University of Plymouth, PL4 8AA, UK \hfill\break
e-mail: C.Christopher@plymouth.ac.uk
\bye