The decomposition method and Maple procedure
for finding first integrals of nonlinear PDEs
of any order with any number of independent variables
Yu. N. Kosovtsov
Lviv Radio Engineering Research Institute, Ukraine
email: kosovtsov@escort.lviv.net
Abstract
In present paper we propose seemingly new method for finding solu-
tions of some types of nonlinear PDEs in closed form. The method is
based on decomposition of nonlinear operators on sequence of operators
of lower orders. It is shown that decomposition process can be done by
iterative procedure(s), each step of which is reduced to solution of some
auxiliary PDEs system(s) for one dependent variable. Moreover, we find
on this way the explicit expression of the first-order PDE(s) for first inte-
gral of decomposable initial PDE. Remarkably that this first-order PDE
is linear if initial PDE is linear in its highest derivatives.
The developed method is implemented in Maple procedure, which can
really solve many of different order PDEs with different number of in-
dependent variables. Examples of PDEs with calculated their general
solutions demonstrate a potential of the method for automatic solving of
nonlinear PDEs.
1 Introduction
Nonlinear partial differential equations (PDEs) play very important role in many
fields of mathematics, physics, chemistry, and biology, and numerous applica-
tions. If for nonlinear ordinary differential equations (ODEs) one can observe
incontestable progress in their automatic solving, the situation for nonlinear
PDEs seems as nearly hopeless one.
Despite the fact that various methods for solving nonlinear PDEs have been
developed in 19-20 centuries as the suitable groups of transformations, such as
point or contact transformations, differential substitutions, and Backlund trans-
formations etc., the most powerful method for explicit integration of second-
order nonlinear PDEs in two independent variables remains the method of Dar-
boux [1]-[4]. The original Darboux method (as already Darboux stated in [1])
is extendable in principle to equations of all orders in an arbitrary number of
independent variables, even to systems of equations; however, in [1]-[2] and sub-
sequent papers by many authors, the detailed calculations were performed only
http://arxiv.org/abs/0704.0072v1
for a single second-order equation with one dependent and two independent
variables.
The Darboux method was refined in recent years into more precise and effi-
cient (although not completely algorithmic) form [5]-[8] and references therein.
Nevertheless this approaches suffer from high complexity and necessitate to use
some tricks.
There were some partially successful attempts to extend modern variants of
the Darboux method based on Laplace cascade method on higher-order PDEs
and PDEs in the space of more than two independent variables [10]-[13] but
they suffer from high complexity too.
There is an original approach to the problem, based on the special type
of local change of variables which leads to the order reduction of initial PDE,
proposed in [14], which is suitable for high dimensions problems but of very
special class though.
In present paper we propose seemingly new method for finding solutions
of some types of nonlinear PDEs in closed form. The method is based on
decomposition of nonlinear operators on sequence of operators of lower orders. It
is shown that decomposition process can be done by iterative procedure(s), each
step of which is reduced to solution of some auxiliary PDEs system(s) for one
dependent variable. Moreover, we find on this way the explicit expression of the
first-order PDE(s) for first integral of decomposable initial PDE. Remarkably
that this first-order PDE is linear if initial PDE is linear in its highest derivatives.
The developed method is implemented in Maple procedure, which can really
solve many of different order PDEs with different number of independent vari-
ables. Examples of PDEs with calculated their general solutions demonstrate a
potential of the method for automatic solving of nonlinear PDEs.
2 Bases of the method
2.1 Decomposable PDEs
The simplest second-order non-linear PDE for w = w(t, x)
can be easily transformed to the following decomposed form
) = 0 , (2)
from which we can without difficulty obtain the general solution to PDE (1) in
two steps. First step gives us
d ln(G(x))
, (3)
where G(x) is an arbitrary function. And then, solving the equation (3) on the
second step, we obtain
w(t, x) = F (t)G(x) , (4)
where F (t) is one more arbitrary function.
The main observations on analyzing the grounds of solvability of the PDE
(1) by the above method are that
1. The PDE (1) is ”decomposable”, i.e., it can be represented as a composi-
tion of successive differential operators of type (5) (not necessarily linear). It is
clear that such type of decomposition can be done for some PDEs of any order
and with any number of independent variables in the following manner
D1(w) = u1 ,
D2(u1) = u2 ,
. . . . . . , (5)
Dn(un−1) = 0 ,
where ~x = (x1, . . . , xm), w = w(~x), ui = ui(~x) and
Di(u) = Vi(~x, u,
, . . . ,
Assuming that Vi are arbitrary functions, and eliminating ui by successive sub-
stitutions in system (5), we get a family of PDEs for w of nth order
Dn(Dn−1(. . . D1(w) . . . )) = 0 . (6)
which are ”decomposable” and in principle their solutions general or particular
can be obtained by integration of split system (5). The PDE (6) is nonlinear if
at least one of the operators Di is nonlinear. Not all PDEs admit such repre-
sentation. And in positive cases such representation is not unique in general.
Note that as a matter of fact Di need not be the first-order differential
operators. So the composition procedure for nth order PDE, when n > 2 can
be as follows
(w) = u ,
(u) = 0 , (7)
where n1, n2 are integers and n1 + n2 = n, w = w(~x), u = u(~x), and (k ≤ j)
i (u) = Vi(~x, u,
, . . . ,
. . . ∂x
m |k1+···+km=k≤j
, . . . ,
The late representation allows us to carry out the PDE‘s decomposition or
order reduction gradually bit by bit.
We have to stress here that in general representations (5) and (7) may have
different meaning. For example, some PDEs do not admit representation (5)
but permit the form (7) with both solvable DEs.
2. Each step of the solving process for decomposed PDE is faced with the
necessity to solve differential equation Di(ui−1) = ui (or D
i (ui−1) = ui), so all
such DEs must be solvable. Note that only first step Dn(un−1) = 0 is free from
arbitrary functions.
So one of the PDEs solving strategies may be as follows. First of all we
try to decompose given PDE. In order to do so we have to solve corresponding
auxiliary nonlinear PDE system for unknown functions Vi, it is sufficient to
find a particular solution here. And, if it is successful, then, deciding between
the variants, try to solve each arising DE from the chain (5). Main obstacle
here, beginning at the second step is just mentioned necessity to solve DEs with
arbitrary functions. There are sufficiently narrow circle of solvable (in sense of
the general solutions) DEs with an arbitrary function as a parameter.
Another (classification) approach can be based on the usage of only solvable
DEs. That is, we can form a composition of successive solvable differential
operators and as a result obtain a families of solvable PDEs. Such a way leads to
extensive nontrivial families for different types of nonlinear PDEs which general
solutions can be expressed in closed form. But on this way we encounter a
difficulty to circumscribe such families integrally and are forced to consider
particular subfamilies. Nevertheless it yields extensive field of PDEs for methods
testing [15].
2.2 Decomposition algorithm for decomposable PDEs
For nth order PDE, when n > 2 there are some slightly different approaches
which are dictated by goals of the problem. If the goal is to decompose given
nonlinear operator then we have to use the scheme (7) with n1 = 1, n2 = n− 1.
And conversely we have to use the scheme (7) with n1 = n − 1, n2 = 1 if the
goal is to solve given PDE. The last procedure in some features resembles the
well-known technics of reducing ODEs order, e.g., by first integral method. Of
course, it is possible to use intermediate cases.
All above cases can be treated by the same way as we consider below but
each of them leads to auxiliary PDEs systems of different order, viz n2+1, with
corresponding calculation complexity.
In sequel we will consider for shortness only the case with n1 = n−1, n2 = 1,
as more practical for PDEs solving.
Let us consider the decomposition of type (7) with Dn−1
(w) as a solution
of the following equation with respect of u = u(~x)
J(u, ~x, w,
, . . . ,
. . . ∂x
m |k1+···+km=k≤n−1
, . . . ,
∂n−1w
) = 0 (8)
D2(u) = V (~x, u,
, . . . ,
) . (9)
If substitute u = Dn−1
(w) into (9) we obtain decomposable n-th order PDE
V (~x, U0, Ux1 , . . . , Uxm) = 0 , (10)
where (we use below the following notation w = W0 and
...∂x
= Wk1,...,km)
(w) = U0 , (11)
∂Wk1,...,km
,...,k∗
= Uxi (i = 1, . . . ,m) , (12)
where k∗j = kj + 1 if j = i and k
j = kj otherwise, and it is supposed that
differentiations in sum are carried out on all indexed W ‘s which are involved in
Here we can introduce U0 and Ux1 , . . . , Uxm as new independent variables
if express m variables from the set {Wk1,...,km} with k1 + · · ·+ km = n using
linear system (12).
Assuming that given PDE of order n
F (~x, w,
, . . . ,
. . . ∂x
m |k1+···+km=k≤n
, . . . ,
) = 0 (13)
is decomposable, we receive, that after substitution of the new variables, left-
hand side of given PDE must turn into (10) with some V .
Left-hand side of given PDE expressed in new variables is the first-order
differential expression with respect to
J(U0, ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
and must not depend on all indexed W ‘s, that is derivatives of F expressed in
new variables with respect to all indexed W ‘s are equal to zero. Sequence of
such derivatives of F equated to zero form a second-order PDE system for J .
So a solution (particular as well) the PDE system gives possible expression of
differential operator Dn−1
(w) through (8) and differential operator D2(u) by
substituting the solution of J into left-hand side of given PDE expressed in new
variables.
Of course, there are problems where a operator decomposition is required
only. But in most cases obtained decomposition is intended for finding solutions
for given PDE. If in obtained decomposition the corresponding PDE D2(u) = 0
is solvable, then substitution of obtained u into J expressed in original variables
gives us a first integral (see its definition in the next subsection) of given PDE.
It is easy to see that for decomposable PDEs the first integral is a differential
equation, so we can try to solve it or to find a first integral for this new DE (or
decompose it) by the scheme described above until we come to the first-order
Remarkably that in the approach under consideration the finding of first
integrals can be done more directly and effectively.
2.3 Differential equation for first integral of decomposable
The first integral I of the PDE is an expression, involving one arbitrary func-
tion, which is equivalent in some sense to the given PDE. The first integral
vanishes on the set of solutions of given PDE. And (in accordance with [4]) all
differential consequences of the equation I = 0 coincide with respective differen-
tial consequences of given PDE (e.g., elimination of the arbitrary function leads
to the given PDE).
Our goal here is to find PDE for first integral of a decomposable PDE. To
do so we first of all have to take into account that u(~x) is the solution of the
corresponding PDE
V (~x, u,
, . . . ,
) = 0 ,
so u(~x) depends only on ~x but in no way on indexed W ‘s. Secondly, the depen-
dent variable in this case, namely
J(u(~x), ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
of given PDE (13) expressed in new variables do not to depend on Ux1 , . . . , Uxm
and is a first integral of given PDE.
If now consider u(~x) as an unknown function, we can denote the first integral
I(~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1) =
J(u(~x), ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
and instead of (12) in the form
∂Wk1,...,km
,...,k∗
= −Uxi
(i = 1, . . . ,m)
we arrive to the following system
∂Wk1,...,km
,...,k∗
= 0 (i = 1, . . . ,m) . (14)
If express m variables from the set {Wk1,...,km} with k1 + · · ·+ km = n (at
least one of which is actual for given PDE - note that there are some variants
here as a rule, so we can obtain some consistent PDEs on this stage) using linear
system (14) and substitute them into given PDE (13) we receive a first-order
(even linear if PDE (13) is linear in its highest derivatives) PDE with respect to
first integral I. And it remains only to solve this PDE(s) to find a first integral
of given PDE.
Note, given PDE is decomposable iff exists a solution of such first-order
PDE(s).
3 Examples
To facilitate necessary calculations in the process of finding first integrals I
have implemented above described method in prototype of Maple procedure
reduce PDE order (see Appendix). The input data of the procedure are given
PDE of any order and dependent variable of the PDE with any number of
independent variables. The procedure tries to find first integral(s) of the input
linear or nonlinear PDE.
The Maple built-in procedure pdsolve is used inside my procedure to solve
the first-order PDE for first integral. As different Maple versions have different
PDE solving abilities so the output depends on Maple version. In the following
examples I refer to Maple 11.
The procedure reduce PDE order is able to find first integrals for many
known and unknown linear and nonlinear PDEs. Here we give examples of
PDEs for which it is possible to find finally their general solutions. More exam-
ples one can find in collection of solvable nonlinear PDEs [15].
3.1 Second-order PDE with two independent variables
For PDE (w = w(t, x))
−kw− bc
= 0 (15)
with a 6= 0 and 4ak − b2 6= 0 the procedure reduce PDE order outputs the
following first integral
I = F1
4ak − b2 − 2 arctan
c+ 2a∂w
4ak − b2
4ak − b2
with arbitrary function F1.
The ODE I = 0 can be solved and one obtains (after some hand simplifica-
tions and edition) the following general solution to (15)
w(t, x) =
exp(t
b2 − 4ak)F (x)(b +
b2 − 4ak)−
b2 − 4ak + b
1 + exp(t
b2 − 4ak)F (x)
G(t)} exp
exp(t
b2 − 4ak)F (x)(b +
b2 − 4ak)−
b2 − 4ak + b
1 + exp(t
b2 − 4ak)F (x)
where F (x) and G(t) are arbitrary functions.
3.2 Second-order PDE with four independent variables
For PDE
∂x1∂x4
∂x2∂x4
∂x3∂x4
+ C0 +B1
C1(A1
+B1w +B0)+
C2(A1
+B1w +B0)
2 = 0 , (16)
where w = w(x1, x2, x3, x4) and Ai, Bi, Ci are constants, the procedure re-
duce PDE order outputs the following first integral
I = F1
x1, x2, x3, x4 +
2 arctan
2C2(A1
+B1w +B0) + C1
4C0C2 − C21
4C0C2 − C21
with arbitrary function F1.
The PDE I = 0 can be solved and one obtains the following general solution
to (16)
w(x1, x2, x3, x4) =
2A1C2
exp(−B1x1
)(2B0C2 + C1 + tan[
4C0C2 − C21
+G(ξ, (A2ξ +A1x2 −A2x1), (A3ξ +A1x3 −A3x1))]
4C0C2 − C21 )dξ
+ exp(−B1x1
)F [(A1x2 − A2x1), (A1x3 −A3x1), x4] ,
where F (t1, t2, t3) and G(t1, t2, t3) are arbitrary functions, c is arbitrary con-
stant.
3.3 Third order PDE with two independent variables
For PDE (w = w(t, x))
∂t∂x2
− 2w ∂
− w∂w
− aw3 = 0 (17)
the procedure reduce PDE order outputs the following first integrals
I1 = F1
− axw2), 1
ax2w2 + 2w(
− x ∂
) + 2x
I2 = F1
− atw2 −
with arbitrary function F1.
We can form some PDEs from I1 and to solve them we can repeat the process
of order reduction with the procedure reduce PDE order. The ODE I2 = 0 can
be solved directly and one obtains in any way the following general solution to
w(t, x) = F (t) exp
− xH(t) + x
G(x)dx −
xG(x)dx
where F (t), H(t) and G(x) are arbitrary functions.
3.4 Fourth order PDE with two independent variables
For PDE (w = w(t, x))
∂t2∂x2
− 2w2
∂t2∂x
∂t∂x2
− 2∂w
= 0 (18)
the procedure reduce PDE order outputs the following first integrals
I1 = F1(t,
∂t2∂x
− 2w∂w
− x ∂
∂t2∂x
w − 2x∂w
I2 = F1(x,
∂t∂x2
w + 2
− t ∂
∂t∂x2
w − 2t∂w
with arbitrary function F1.
The wealth of first integrals here allows us to operate with them in many
different ways. Apart from aforesaid subsequent order reduction we can, for
example, from
∂t∂x2
− 2w∂w
w + 2
= F (x)
∂t∂x2
] = G(x) ,
where F (x) and G(x) are arbitrary functions, algebraically eliminate mixed
derivative and obtain the following ODE
+ [tF (x)−G(x)]w2 = 0 ,
which gives the general solution to (18)
w(t, x) = H(t) exp
xF (x) dx − tx
F (x) dx+
G(x) dx −
xG(x) dx + xK(t)
where F (x), H(t), G(x) and K(t) are arbitrary functions.
4 Conclusion
The method have considered above is efficient enough for solving decomposable
PDEs of relatively high order with many independent variables. The main
limitation here is concerned with abilities to solve corresponding auxiliary first-
order PDEs for first integrals.
An adaptability of the method to PDEs which are not decomposable but
which general solutions can be expressed in closed form remains unsolved yet.
But it can be shown on examples that there are some ways to extend the method
for some types of such PDEs. These approaches deserve further thorough study
in another publication.
References
[1] G. Darboux, Sur les equations aux derivees partieles du second ordre. Ann.
Sci. Ecole Norm. Sup. 1870, v. 7, pp. 163-173.
[2] G. Darboux, Lecons sur la theorie generale des surfaces. v.II. Paris: Her-
mann, 1915.
[3] E. Goursat, Lecons sur l’integration des equations aux derivees partieles
du second ordre a deux variables independantes. V.I,II. Paris: Hermann,
1896, 1898.
[4] A.R. Forsyth, Theory of differential equations. Part 4. Partial differential
equations, vol. 6, Dover Press, New York, 1959.
[5] M. Juras, Generalized Laplace invariants and classical integration meth-
ods for second-order scalar hyperbolic partial differential equations in the
plane, Differential Geometry and Applications: Proc., Conf. Brno (Czech
Republic), 28 Aug.-1 Sept. 1995, Brno: Masaric Univ., 1966, pp. 275-284.
[6] M. Juras, Geometric aspects of second-order scalar hyperbolic partial dif-
ferential equations in the plane, Ph.D. thesis, 1997, Utah State University,
[7] V.V. Sokolov, A.V. Zhiber, On the Darboux integrable hyperbolic equa-
tions. Phys. Lett. A, v. 208, pp. 303-308, 1995.
[8] A.V. Zhiber, V.V. Sokolov, Exact integrable Liouville type hyperbolic equa-
tions [in Russian], Uspekhi Mat. Nauk, Vol. 56, No. 1, pp. 64-104, 2001.
[9] S.P.Tsarev, On Darboux integrable nonlinear partial differential equations,
Proc. Steklov Institute of Mathematics, v. 225, p. 372-381, 1999.
[10] J. Le Roux. Extensions de la methode de Laplace aux equations lin-
eaires aux derivees partielles dordre superieur au second. Bull. Soc. Math.
de France, v. 27, p. 237262, 1899. A digitized copy is obtainable from
http://www.numdam.org/
[11] U. Dini, Sopra una classe di equazioni a derivate parziali di secondordine
con un numero qualunque di variabili. Atti Acc. Naz. dei Lincei. Mem.
Classe fis., mat., nat. (ser. 5) v. 4, 1901, p. 121178. Also Opere v. III, p.
489566.
[12] U. Dini, Sopra una classe di equazioni a derivate parziali di secondordine.
Atti Acc. Naz. dei Lincei. Mem. Classe fis., mat., nat. (ser. 5) v. 4, 1902,
p. 431467. Also Opere v. III, p. 613660.
[13] S.P. Tsarev, On factorization and solution of multidimensional linear par-
tial differential equations. http://arxiv.org/abs/cs.SC/0609075, 2006.
[14] V.M. Boyko, W.I. Fushchych, Lowering of order and general solutions of
some classes of partial differential equations, Reports on Math. Phys., V.
41, No. 3, pp. 311-318, 1998.
[15] Yu.N. Kosovtsov, The general solutions of some nonlinear second
and third order PDEs with constant and nonconstant parameters.
http://arxiv.org/abs/math-ph/0609003 , 2006.
http://www.numdam.org/
http://arxiv.org/abs/cs.SC/0609075
http://arxiv.org/abs/math-ph/0609003
5 Appendix.
Maple procedure reduce PDE order
reduce PDE order:=proc(pde,unk)
local B,W,N,NN,ARG,acargs,i,M,pde0,DN,IND,IND2,IND3,IND4,ARGS,SUB,SUB0,
Z0,Bargs,EQS,XXX,WW,BB,PP,pdeI,IV,s,AN;
option ‘Copyright (c) 2006-2007 by Yuri N. Kosovtsov. All rights reserved.‘;
N:=PDETools[difforder](op(1,[selectremove(has,indets(pde,function),unk)]));
NN:=op(1,[selectremove(has,op(1,[selectremove(has,indets(pde,function),unk)]),diff)]);
ARG:=[op(unk)];
acargs:={};
for i from 1 to nops(ARG) do
if PDETools[difforder](NN,op(i,ARG))=0 then else acargs:=acargs union {op(i,ARG)}
fi; od;
acargs:=convert(acargs,list);
M:=op(0,unk)(op(acargs));
if type(pde,equation)=true then
pde0:=lhs(subs(unk=M,pde))-rhs(subs(unk=M,pde)) else pde0:=subs(unk=M,pde)
DN:=[seq(seq(i,i=1..nops(acargs)),j=1..N)];
IND:=seq(op(combinat[choose](DN,i)),i=1..N);
IND2:=seq(op(combinat[choose](DN,i)),i=1..N-2);
IND3:=op(combinat[choose](DN,N-1));
IND4:=op(combinat[choose](DN,N));
ARGS:=op(unk),M,seq(convert(D[op(op(i,[IND2]))](op(0,unk))
(op(acargs)),diff),i=1..nops([IND2]));
SUB:={M=W[0],seq(convert(D[op(op(i,[IND]))](op(0,unk))
(op(acargs)),diff)=W[op(op(i,[IND]))],i=1..nops([IND]))};
SUB0:={W[0]=op(0,unk)(op(ARG)),
seq(W[op(op(i,[IND]))]=subs(M=op(0,unk)(op(ARG)),
convert(D[op(op(i,[IND]))](op(0,unk))(op(acargs)),diff)),i=1..nops([IND]))};
Z0:=B(ARGS,seq(convert(D[op(op(i,[IND3]))](op(0,unk))(op(acargs)),diff),
i=1..nops([IND3])));
Bargs:=op(indets(subs(SUB,Z0),name));
EQS:=convert(subs(SUB,{seq(diff(Z0,op(i,acargs))=0,i=1..nops(acargs))}),diff);
XXX:={seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
WW:=select(type,indets(subs(SUB,pde0)), ’name’) intersect
{seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
BB:=select(has,combinat[choose](XXX, nops(acargs)),WW);
PP:={};
pdeI:={seq({subs(subs(solve(EQS,op(i,BB)),subs(SUB,pde0)))},i=1..nops(BB))};
IV:={seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
for s from 1 to nops(pdeI) do
AN:=pdsolve(op(s,pdeI),{B},ivars=IV);
for i from 1 to nops(AN) do
if op(0,lhs(op(i,AN)))=B then
PP:=PP union {rhs(op(i,AN))}
catch:
end try;
PP:=subs(SUB0,PP);
RETURN(PP);
end proc:
Calling Sequence: reduce PDE order(PDE, f(~x));
PDE - partial differential equation;
f(~x) - indeterminate function with its arguments.
	Introduction
	Bases of the method
	Decomposable PDEs
	Decomposition algorithm for decomposable PDEs
	Differential equation for first integral of decomposable PDEs
	Examples
	Second-order PDE with two independent variables
	Second-order PDE with four independent variables
	Third order PDE with two independent variables
	Fourth order PDE with two independent variables
	Conclusion
	Appendix. Maple procedure reduce_PDE_order

  In present paper we propose seemingly new method for finding solutions of
some types of nonlinear PDEs in closed form. The method is based on
decomposition of nonlinear operators on sequence of operators of lower orders.
It is shown that decomposition process can be done by iterative procedure(s),
each step of which is reduced to solution of some auxiliary PDEs system(s) for
one dependent variable. Moreover, we find on this way the explicit expression
of the first-order PDE(s) for first integral of decomposable initial PDE.
Remarkably that this first-order PDE is linear if initial PDE is linear in its
highest derivatives.
  The developed method is implemented in Maple procedure, which can really
solve many of different order PDEs with different number of independent
variables. Examples of PDEs with calculated their general solutions demonstrate
a potential of the method for automatic solving of nonlinear PDEs.

Introduction
Nonlinear partial differential equations (PDEs) play very important role in many
fields of mathematics, physics, chemistry, and biology, and numerous applica-
tions. If for nonlinear ordinary differential equations (ODEs) one can observe
incontestable progress in their automatic solving, the situation for nonlinear
PDEs seems as nearly hopeless one.
Despite the fact that various methods for solving nonlinear PDEs have been
developed in 19-20 centuries as the suitable groups of transformations, such as
point or contact transformations, differential substitutions, and Backlund trans-
formations etc., the most powerful method for explicit integration of second-
order nonlinear PDEs in two independent variables remains the method of Dar-
boux [1]-[4]. The original Darboux method (as already Darboux stated in [1])
is extendable in principle to equations of all orders in an arbitrary number of
independent variables, even to systems of equations; however, in [1]-[2] and sub-
sequent papers by many authors, the detailed calculations were performed only
http://arxiv.org/abs/0704.0072v1
for a single second-order equation with one dependent and two independent
variables.
The Darboux method was refined in recent years into more precise and effi-
cient (although not completely algorithmic) form [5]-[8] and references therein.
Nevertheless this approaches suffer from high complexity and necessitate to use
some tricks.
There were some partially successful attempts to extend modern variants of
the Darboux method based on Laplace cascade method on higher-order PDEs
and PDEs in the space of more than two independent variables [10]-[13] but
they suffer from high complexity too.
There is an original approach to the problem, based on the special type
of local change of variables which leads to the order reduction of initial PDE,
proposed in [14], which is suitable for high dimensions problems but of very
special class though.
In present paper we propose seemingly new method for finding solutions
of some types of nonlinear PDEs in closed form. The method is based on
decomposition of nonlinear operators on sequence of operators of lower orders. It
is shown that decomposition process can be done by iterative procedure(s), each
step of which is reduced to solution of some auxiliary PDEs system(s) for one
dependent variable. Moreover, we find on this way the explicit expression of the
first-order PDE(s) for first integral of decomposable initial PDE. Remarkably
that this first-order PDE is linear if initial PDE is linear in its highest derivatives.
The developed method is implemented in Maple procedure, which can really
solve many of different order PDEs with different number of independent vari-
ables. Examples of PDEs with calculated their general solutions demonstrate a
potential of the method for automatic solving of nonlinear PDEs.
2 Bases of the method
2.1 Decomposable PDEs
The simplest second-order non-linear PDE for w = w(t, x)
can be easily transformed to the following decomposed form
) = 0 , (2)
from which we can without difficulty obtain the general solution to PDE (1) in
two steps. First step gives us
d ln(G(x))
, (3)
where G(x) is an arbitrary function. And then, solving the equation (3) on the
second step, we obtain
w(t, x) = F (t)G(x) , (4)
where F (t) is one more arbitrary function.
The main observations on analyzing the grounds of solvability of the PDE
(1) by the above method are that
1. The PDE (1) is ”decomposable”, i.e., it can be represented as a composi-
tion of successive differential operators of type (5) (not necessarily linear). It is
clear that such type of decomposition can be done for some PDEs of any order
and with any number of independent variables in the following manner
D1(w) = u1 ,
D2(u1) = u2 ,
. . . . . . , (5)
Dn(un−1) = 0 ,
where ~x = (x1, . . . , xm), w = w(~x), ui = ui(~x) and
Di(u) = Vi(~x, u,
, . . . ,
Assuming that Vi are arbitrary functions, and eliminating ui by successive sub-
stitutions in system (5), we get a family of PDEs for w of nth order
Dn(Dn−1(. . . D1(w) . . . )) = 0 . (6)
which are ”decomposable” and in principle their solutions general or particular
can be obtained by integration of split system (5). The PDE (6) is nonlinear if
at least one of the operators Di is nonlinear. Not all PDEs admit such repre-
sentation. And in positive cases such representation is not unique in general.
Note that as a matter of fact Di need not be the first-order differential
operators. So the composition procedure for nth order PDE, when n > 2 can
be as follows
(w) = u ,
(u) = 0 , (7)
where n1, n2 are integers and n1 + n2 = n, w = w(~x), u = u(~x), and (k ≤ j)
i (u) = Vi(~x, u,
, . . . ,
. . . ∂x
m |k1+···+km=k≤j
, . . . ,
The late representation allows us to carry out the PDE‘s decomposition or
order reduction gradually bit by bit.
We have to stress here that in general representations (5) and (7) may have
different meaning. For example, some PDEs do not admit representation (5)
but permit the form (7) with both solvable DEs.
2. Each step of the solving process for decomposed PDE is faced with the
necessity to solve differential equation Di(ui−1) = ui (or D
i (ui−1) = ui), so all
such DEs must be solvable. Note that only first step Dn(un−1) = 0 is free from
arbitrary functions.
So one of the PDEs solving strategies may be as follows. First of all we
try to decompose given PDE. In order to do so we have to solve corresponding
auxiliary nonlinear PDE system for unknown functions Vi, it is sufficient to
find a particular solution here. And, if it is successful, then, deciding between
the variants, try to solve each arising DE from the chain (5). Main obstacle
here, beginning at the second step is just mentioned necessity to solve DEs with
arbitrary functions. There are sufficiently narrow circle of solvable (in sense of
the general solutions) DEs with an arbitrary function as a parameter.
Another (classification) approach can be based on the usage of only solvable
DEs. That is, we can form a composition of successive solvable differential
operators and as a result obtain a families of solvable PDEs. Such a way leads to
extensive nontrivial families for different types of nonlinear PDEs which general
solutions can be expressed in closed form. But on this way we encounter a
difficulty to circumscribe such families integrally and are forced to consider
particular subfamilies. Nevertheless it yields extensive field of PDEs for methods
testing [15].
2.2 Decomposition algorithm for decomposable PDEs
For nth order PDE, when n > 2 there are some slightly different approaches
which are dictated by goals of the problem. If the goal is to decompose given
nonlinear operator then we have to use the scheme (7) with n1 = 1, n2 = n− 1.
And conversely we have to use the scheme (7) with n1 = n − 1, n2 = 1 if the
goal is to solve given PDE. The last procedure in some features resembles the
well-known technics of reducing ODEs order, e.g., by first integral method. Of
course, it is possible to use intermediate cases.
All above cases can be treated by the same way as we consider below but
each of them leads to auxiliary PDEs systems of different order, viz n2+1, with
corresponding calculation complexity.
In sequel we will consider for shortness only the case with n1 = n−1, n2 = 1,
as more practical for PDEs solving.
Let us consider the decomposition of type (7) with Dn−1
(w) as a solution
of the following equation with respect of u = u(~x)
J(u, ~x, w,
, . . . ,
. . . ∂x
m |k1+···+km=k≤n−1
, . . . ,
∂n−1w
) = 0 (8)
D2(u) = V (~x, u,
, . . . ,
) . (9)
If substitute u = Dn−1
(w) into (9) we obtain decomposable n-th order PDE
V (~x, U0, Ux1 , . . . , Uxm) = 0 , (10)
where (we use below the following notation w = W0 and
...∂x
= Wk1,...,km)
(w) = U0 , (11)
∂Wk1,...,km
,...,k∗
= Uxi (i = 1, . . . ,m) , (12)
where k∗j = kj + 1 if j = i and k
j = kj otherwise, and it is supposed that
differentiations in sum are carried out on all indexed W ‘s which are involved in
Here we can introduce U0 and Ux1 , . . . , Uxm as new independent variables
if express m variables from the set {Wk1,...,km} with k1 + · · ·+ km = n using
linear system (12).
Assuming that given PDE of order n
F (~x, w,
, . . . ,
. . . ∂x
m |k1+···+km=k≤n
, . . . ,
) = 0 (13)
is decomposable, we receive, that after substitution of the new variables, left-
hand side of given PDE must turn into (10) with some V .
Left-hand side of given PDE expressed in new variables is the first-order
differential expression with respect to
J(U0, ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
and must not depend on all indexed W ‘s, that is derivatives of F expressed in
new variables with respect to all indexed W ‘s are equal to zero. Sequence of
such derivatives of F equated to zero form a second-order PDE system for J .
So a solution (particular as well) the PDE system gives possible expression of
differential operator Dn−1
(w) through (8) and differential operator D2(u) by
substituting the solution of J into left-hand side of given PDE expressed in new
variables.
Of course, there are problems where a operator decomposition is required
only. But in most cases obtained decomposition is intended for finding solutions
for given PDE. If in obtained decomposition the corresponding PDE D2(u) = 0
is solvable, then substitution of obtained u into J expressed in original variables
gives us a first integral (see its definition in the next subsection) of given PDE.
It is easy to see that for decomposable PDEs the first integral is a differential
equation, so we can try to solve it or to find a first integral for this new DE (or
decompose it) by the scheme described above until we come to the first-order
Remarkably that in the approach under consideration the finding of first
integrals can be done more directly and effectively.
2.3 Differential equation for first integral of decomposable
The first integral I of the PDE is an expression, involving one arbitrary func-
tion, which is equivalent in some sense to the given PDE. The first integral
vanishes on the set of solutions of given PDE. And (in accordance with [4]) all
differential consequences of the equation I = 0 coincide with respective differen-
tial consequences of given PDE (e.g., elimination of the arbitrary function leads
to the given PDE).
Our goal here is to find PDE for first integral of a decomposable PDE. To
do so we first of all have to take into account that u(~x) is the solution of the
corresponding PDE
V (~x, u,
, . . . ,
) = 0 ,
so u(~x) depends only on ~x but in no way on indexed W ‘s. Secondly, the depen-
dent variable in this case, namely
J(u(~x), ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
of given PDE (13) expressed in new variables do not to depend on Ux1 , . . . , Uxm
and is a first integral of given PDE.
If now consider u(~x) as an unknown function, we can denote the first integral
I(~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1) =
J(u(~x), ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
and instead of (12) in the form
∂Wk1,...,km
,...,k∗
= −Uxi
(i = 1, . . . ,m)
we arrive to the following system
∂Wk1,...,km
,...,k∗
= 0 (i = 1, . . . ,m) . (14)
If express m variables from the set {Wk1,...,km} with k1 + · · ·+ km = n (at
least one of which is actual for given PDE - note that there are some variants
here as a rule, so we can obtain some consistent PDEs on this stage) using linear
system (14) and substitute them into given PDE (13) we receive a first-order
(even linear if PDE (13) is linear in its highest derivatives) PDE with respect to
first integral I. And it remains only to solve this PDE(s) to find a first integral
of given PDE.
Note, given PDE is decomposable iff exists a solution of such first-order
PDE(s).
3 Examples
To facilitate necessary calculations in the process of finding first integrals I
have implemented above described method in prototype of Maple procedure
reduce PDE order (see Appendix). The input data of the procedure are given
PDE of any order and dependent variable of the PDE with any number of
independent variables. The procedure tries to find first integral(s) of the input
linear or nonlinear PDE.
The Maple built-in procedure pdsolve is used inside my procedure to solve
the first-order PDE for first integral. As different Maple versions have different
PDE solving abilities so the output depends on Maple version. In the following
examples I refer to Maple 11.
The procedure reduce PDE order is able to find first integrals for many
known and unknown linear and nonlinear PDEs. Here we give examples of
PDEs for which it is possible to find finally their general solutions. More exam-
ples one can find in collection of solvable nonlinear PDEs [15].
3.1 Second-order PDE with two independent variables
For PDE (w = w(t, x))
−kw− bc
= 0 (15)
with a 6= 0 and 4ak − b2 6= 0 the procedure reduce PDE order outputs the
following first integral
I = F1
4ak − b2 − 2 arctan
c+ 2a∂w
4ak − b2
4ak − b2
with arbitrary function F1.
The ODE I = 0 can be solved and one obtains (after some hand simplifica-
tions and edition) the following general solution to (15)
w(t, x) =
exp(t
b2 − 4ak)F (x)(b +
b2 − 4ak)−
b2 − 4ak + b
1 + exp(t
b2 − 4ak)F (x)
G(t)} exp
exp(t
b2 − 4ak)F (x)(b +
b2 − 4ak)−
b2 − 4ak + b
1 + exp(t
b2 − 4ak)F (x)
where F (x) and G(t) are arbitrary functions.
3.2 Second-order PDE with four independent variables
For PDE
∂x1∂x4
∂x2∂x4
∂x3∂x4
+ C0 +B1
C1(A1
+B1w +B0)+
C2(A1
+B1w +B0)
2 = 0 , (16)
where w = w(x1, x2, x3, x4) and Ai, Bi, Ci are constants, the procedure re-
duce PDE order outputs the following first integral
I = F1
x1, x2, x3, x4 +
2 arctan
2C2(A1
+B1w +B0) + C1
4C0C2 − C21
4C0C2 − C21
with arbitrary function F1.
The PDE I = 0 can be solved and one obtains the following general solution
to (16)
w(x1, x2, x3, x4) =
2A1C2
exp(−B1x1
)(2B0C2 + C1 + tan[
4C0C2 − C21
+G(ξ, (A2ξ +A1x2 −A2x1), (A3ξ +A1x3 −A3x1))]
4C0C2 − C21 )dξ
+ exp(−B1x1
)F [(A1x2 − A2x1), (A1x3 −A3x1), x4] ,
where F (t1, t2, t3) and G(t1, t2, t3) are arbitrary functions, c is arbitrary con-
stant.
3.3 Third order PDE with two independent variables
For PDE (w = w(t, x))
∂t∂x2
− 2w ∂
− w∂w
− aw3 = 0 (17)
the procedure reduce PDE order outputs the following first integrals
I1 = F1
− axw2), 1
ax2w2 + 2w(
− x ∂
) + 2x
I2 = F1
− atw2 −
with arbitrary function F1.
We can form some PDEs from I1 and to solve them we can repeat the process
of order reduction with the procedure reduce PDE order. The ODE I2 = 0 can
be solved directly and one obtains in any way the following general solution to
w(t, x) = F (t) exp
− xH(t) + x
G(x)dx −
xG(x)dx
where F (t), H(t) and G(x) are arbitrary functions.
3.4 Fourth order PDE with two independent variables
For PDE (w = w(t, x))
∂t2∂x2
− 2w2
∂t2∂x
∂t∂x2
− 2∂w
= 0 (18)
the procedure reduce PDE order outputs the following first integrals
I1 = F1(t,
∂t2∂x
− 2w∂w
− x ∂
∂t2∂x
w − 2x∂w
I2 = F1(x,
∂t∂x2
w + 2
− t ∂
∂t∂x2
w − 2t∂w
with arbitrary function F1.
The wealth of first integrals here allows us to operate with them in many
different ways. Apart from aforesaid subsequent order reduction we can, for
example, from
∂t∂x2
− 2w∂w
w + 2
= F (x)
∂t∂x2
] = G(x) ,
where F (x) and G(x) are arbitrary functions, algebraically eliminate mixed
derivative and obtain the following ODE
+ [tF (x)−G(x)]w2 = 0 ,
which gives the general solution to (18)
w(t, x) = H(t) exp
xF (x) dx − tx
F (x) dx+
G(x) dx −
xG(x) dx + xK(t)
where F (x), H(t), G(x) and K(t) are arbitrary functions.
4 Conclusion
The method have considered above is efficient enough for solving decomposable
PDEs of relatively high order with many independent variables. The main
limitation here is concerned with abilities to solve corresponding auxiliary first-
order PDEs for first integrals.
An adaptability of the method to PDEs which are not decomposable but
which general solutions can be expressed in closed form remains unsolved yet.
But it can be shown on examples that there are some ways to extend the method
for some types of such PDEs. These approaches deserve further thorough study
in another publication.
References
[1] G. Darboux, Sur les equations aux derivees partieles du second ordre. Ann.
Sci. Ecole Norm. Sup. 1870, v. 7, pp. 163-173.
[2] G. Darboux, Lecons sur la theorie generale des surfaces. v.II. Paris: Her-
mann, 1915.
[3] E. Goursat, Lecons sur l’integration des equations aux derivees partieles
du second ordre a deux variables independantes. V.I,II. Paris: Hermann,
1896, 1898.
[4] A.R. Forsyth, Theory of differential equations. Part 4. Partial differential
equations, vol. 6, Dover Press, New York, 1959.
[5] M. Juras, Generalized Laplace invariants and classical integration meth-
ods for second-order scalar hyperbolic partial differential equations in the
plane, Differential Geometry and Applications: Proc., Conf. Brno (Czech
Republic), 28 Aug.-1 Sept. 1995, Brno: Masaric Univ., 1966, pp. 275-284.
[6] M. Juras, Geometric aspects of second-order scalar hyperbolic partial dif-
ferential equations in the plane, Ph.D. thesis, 1997, Utah State University,
[7] V.V. Sokolov, A.V. Zhiber, On the Darboux integrable hyperbolic equa-
tions. Phys. Lett. A, v. 208, pp. 303-308, 1995.
[8] A.V. Zhiber, V.V. Sokolov, Exact integrable Liouville type hyperbolic equa-
tions [in Russian], Uspekhi Mat. Nauk, Vol. 56, No. 1, pp. 64-104, 2001.
[9] S.P.Tsarev, On Darboux integrable nonlinear partial differential equations,
Proc. Steklov Institute of Mathematics, v. 225, p. 372-381, 1999.
[10] J. Le Roux. Extensions de la methode de Laplace aux equations lin-
eaires aux derivees partielles dordre superieur au second. Bull. Soc. Math.
de France, v. 27, p. 237262, 1899. A digitized copy is obtainable from
http://www.numdam.org/
[11] U. Dini, Sopra una classe di equazioni a derivate parziali di secondordine
con un numero qualunque di variabili. Atti Acc. Naz. dei Lincei. Mem.
Classe fis., mat., nat. (ser. 5) v. 4, 1901, p. 121178. Also Opere v. III, p.
489566.
[12] U. Dini, Sopra una classe di equazioni a derivate parziali di secondordine.
Atti Acc. Naz. dei Lincei. Mem. Classe fis., mat., nat. (ser. 5) v. 4, 1902,
p. 431467. Also Opere v. III, p. 613660.
[13] S.P. Tsarev, On factorization and solution of multidimensional linear par-
tial differential equations. http://arxiv.org/abs/cs.SC/0609075, 2006.
[14] V.M. Boyko, W.I. Fushchych, Lowering of order and general solutions of
some classes of partial differential equations, Reports on Math. Phys., V.
41, No. 3, pp. 311-318, 1998.
[15] Yu.N. Kosovtsov, The general solutions of some nonlinear second
and third order PDEs with constant and nonconstant parameters.
http://arxiv.org/abs/math-ph/0609003 , 2006.
http://www.numdam.org/
http://arxiv.org/abs/cs.SC/0609075
http://arxiv.org/abs/math-ph/0609003
5 Appendix.
Maple procedure reduce PDE order
reduce PDE order:=proc(pde,unk)
local B,W,N,NN,ARG,acargs,i,M,pde0,DN,IND,IND2,IND3,IND4,ARGS,SUB,SUB0,
Z0,Bargs,EQS,XXX,WW,BB,PP,pdeI,IV,s,AN;
option ‘Copyright (c) 2006-2007 by Yuri N. Kosovtsov. All rights reserved.‘;
N:=PDETools[difforder](op(1,[selectremove(has,indets(pde,function),unk)]));
NN:=op(1,[selectremove(has,op(1,[selectremove(has,indets(pde,function),unk)]),diff)]);
ARG:=[op(unk)];
acargs:={};
for i from 1 to nops(ARG) do
if PDETools[difforder](NN,op(i,ARG))=0 then else acargs:=acargs union {op(i,ARG)}
fi; od;
acargs:=convert(acargs,list);
M:=op(0,unk)(op(acargs));
if type(pde,equation)=true then
pde0:=lhs(subs(unk=M,pde))-rhs(subs(unk=M,pde)) else pde0:=subs(unk=M,pde)
DN:=[seq(seq(i,i=1..nops(acargs)),j=1..N)];
IND:=seq(op(combinat[choose](DN,i)),i=1..N);
IND2:=seq(op(combinat[choose](DN,i)),i=1..N-2);
IND3:=op(combinat[choose](DN,N-1));
IND4:=op(combinat[choose](DN,N));
ARGS:=op(unk),M,seq(convert(D[op(op(i,[IND2]))](op(0,unk))
(op(acargs)),diff),i=1..nops([IND2]));
SUB:={M=W[0],seq(convert(D[op(op(i,[IND]))](op(0,unk))
(op(acargs)),diff)=W[op(op(i,[IND]))],i=1..nops([IND]))};
SUB0:={W[0]=op(0,unk)(op(ARG)),
seq(W[op(op(i,[IND]))]=subs(M=op(0,unk)(op(ARG)),
convert(D[op(op(i,[IND]))](op(0,unk))(op(acargs)),diff)),i=1..nops([IND]))};
Z0:=B(ARGS,seq(convert(D[op(op(i,[IND3]))](op(0,unk))(op(acargs)),diff),
i=1..nops([IND3])));
Bargs:=op(indets(subs(SUB,Z0),name));
EQS:=convert(subs(SUB,{seq(diff(Z0,op(i,acargs))=0,i=1..nops(acargs))}),diff);
XXX:={seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
WW:=select(type,indets(subs(SUB,pde0)), ’name’) intersect
{seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
BB:=select(has,combinat[choose](XXX, nops(acargs)),WW);
PP:={};
pdeI:={seq({subs(subs(solve(EQS,op(i,BB)),subs(SUB,pde0)))},i=1..nops(BB))};
IV:={seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
for s from 1 to nops(pdeI) do
AN:=pdsolve(op(s,pdeI),{B},ivars=IV);
for i from 1 to nops(AN) do
if op(0,lhs(op(i,AN)))=B then
PP:=PP union {rhs(op(i,AN))}
catch:
end try;
PP:=subs(SUB0,PP);
RETURN(PP);
end proc:
Calling Sequence: reduce PDE order(PDE, f(~x));
PDE - partial differential equation;
f(~x) - indeterminate function with its arguments.
	Introduction
	Bases of the method
	Decomposable PDEs
	Decomposition algorithm for decomposable PDEs
	Differential equation for first integral of decomposable PDEs
	Examples
	Second-order PDE with two independent variables
	Second-order PDE with four independent variables
	Third order PDE with two independent variables
	Fourth order PDE with two independent variables
	Conclusion
	Appendix. Maple procedure reduce_PDE_order
