Communications
in
Commun. Math. Phys.
77,
289-297
(1980)
MathβΓΠStίCSl
Physics
© by
Springer-Verlag
1980
A
Commutativity
Theorem
of
Partial Differential Operators
Gui-zhang
Tu
Computing
Centre
of
Chinese Academy
of
Sciences,
Beijing,
China
Abstract.
Let u =
u(x,t)
be a
function
of x and
ί,
and
u^&u,
@
=
d/dx,
ί
=
0,1,2,...,
be its
derivatives with respect
to x.
Denote
by
W
n
the set
[f\f
=
f(
u
,
u
v
.
.
.,
u
n
),
(d/du
n
)f
=(=
0},
where
f(u
9
.
.
.,
u
n
)
are
polynomials
of
u
t
with
oo
constant
coefficients.
To any
/eW=
(J
W
w
,
we
relate
it
with
an
operator
n=2
<%(f)=
Σ
(2
l
f)d/du
t
.
In
this paper
we
prove
that:
*(/)
commutes with
<%(g)
if
they
commute respectively with
<%(h),
provided
/,
g,
fteW.
Relating
to
this
commutativity theorem,
we
prove that,
if an
evolution equation
u
t
=
f(u,
.
.
.,
u
n
)
possesses nontrivial symmetries
(or
conservation laws
for a
class
of
poly-
nomials
/),
then
/ =
Cu
n
+/
1
(w,
.
.
.,
u
r
),
where
C
=
const,
and r < n. In the end of
this
paper,
we
state
a
related open problem whose solution would
be of
much
value
to the
theory
of
soliton.
1.
Introduction
The
soliton
[1],
being
a
particle-like solution
of the
nonlinear wave equation,
has
been
now
applied widely
in
various
fields
of
physics.
In the
recent years,
a
number
of
interesting mathematical problems have arisen
in the
study
of
soliton,
one of
them
is,
among other things,
the
commutativity
of
differential
operators
[2,3].
Let
sΛ
=
α
0
^
n
-f
...
+
a
n
_
1
Q)
+
a
n
,
a
0
Φ
0,
2>
=
d/dx
be a
differential
operator,
and
C(jf)
be the set of all
linear operators which commute with
jtf.
A
pronounced results
[4]
is
the
fact
that
C(efi/)
is a
commutative ring.
In
this paper
we
established
a
similar
result
concerning
the
partial
differential
operators
W(f)=
£
(&f)d/du
i9
where
i^O
/
=
f(u,...,
u
n
)
are
polynomials
of
u
{
=
&u
with constant
coefficients,
and u =
u(x
9
1)
is
a
sufficiently
smooth
function
of x and t. We
discuss
further
the
application
of
this result
to the
study
of
symmetries
and
conservation laws
of
nonlinear evolution
equations.
0010-3616/80/0077/0289/S01.80
290
Gui-zhang
Tu
2.
Notation
Let
u =
ίφc,
f)
be a
function
of
x
and
ί,
and
u^&u,
@
=
d/dx
be its
derivatives with respect
to x.
Throughout this
paper,
f,g,h
will stand
for
polynomials
of
u
t
with constant
coefficients
and
without constant term
[i.e.,
say,
/(O)
=
0].
The
small letters
i,
7, fc, /,
p,
g,
α,
fc,
c,
d
will stand
for
nonnegative integers,
and the
capital letter
C
solely
for
constants.
For
convenience,
the
binomial
coefficient
will
be
understood
as
k
}=kl/(il(k-i)l),
(Jtei^O);
(*)=0,
(otherwise).
Let
(2.1)
and
W=
y
W
n
.
For
convenience,
we
agree that
the
constant
CeW
0
.
To
any
/eW
fc
,
n=2
we
relate
it
with
the
following operators
[5,
6]
:
(2.2)
i
i
and
here
and
always below,
the
summation
in
]Γ
is
over
all
nonnegative integers
i, and
/
3,
=
0/011,,
(i^O);
3^
= 0,
(i<0).
It
may be
noted that
i^
0
(f)
=
i
r
(f)
and
W(f)g
=
Ϋ"(0)f.
We
introduce furthermore
the
operation
(2.4)
It
is
easy
to see
that
λ
n)=
Σ
Hence
an
operator
^(/)
commutes with
^(^f)
iff
[/,0]=0.
Consider
an
evolution equation
u
t
=
f(u,u
l9
...,Uj),
/eW
Λ
,
(2.6)
if
there exists
an
infinitesimal
transformation
u-*v
=
u +
eg
9
where
gfeWj
and
^
is an
infinitesimal
parameter, such that
d/de(v
t
-f(v))\
e=0
=
0
holds
for
solutions
u(x,
t) of Eq.
(2.6), then
g is
called
a
symmetry
of
order
/ of
(2.6).
Commutativity
Theorem
291
It
is
known
[5]
that
Proposition
1. In
order that
g
is
a
symmetry
of
(2.6),
it is
necessary
and
sufficient
that
[/,#]
=0,
or
equivalently
%(f)
commutes with
%(g).
Since
it
always holds that
[/,w
1
]
=
[/,/]=0,
for
arbitrary/,
thus
we
call
the
symmetries
g =
C
i
u
i
+
C
2
f
as
trivial.
Let
1
2
'
'
M
l
q
(Ί
Ί\
(A
')
be two
monomials
of
w
f
,
we
introduce lexicography order
<^
among monomials,
that
is,
M^wf
if fc
1
</
1
, or fe
1
=
/
1
but
a
ί
<b
i
,
or fe
1
=/
1
,
a
ί
=b
ί
but
k
2
</
2
,
and so
on.
For
convenience,
we
agree that
0
<^
w^.
To a
nonzero polynomial
/
=
Σ
C
AK
Uχ,
we
denote
by
M(f)
the
monomial which
is of the
highest order among
C
AK
u&
and
call
it
dominant,
e.g.,
M(3uu
ί
+2u
:
2
u
4
+ ul)
=
2ulu
4
.
It is
obvious
from
the de-
finition
that
f=g
implies
M(f)
=
M(g).
(2.8)
The
following commutative formula
[7]
will
be
used
frequently
in
this
paper
:
3. A
Necessary Condition
for the
Existence
of
Nontrivial Symmetries
Lemma
1
It
is
easy
to see by
(2.2)
and
(2.9) that
a
similar equation
for
d
k
(i^(g)f)
can be
also derived,
from
which together with (2.4)
and
(2.3), (3.1)
follows.
Corollary.
/eW
fc
,
0eWj,fc,/^l
imp/j;
[/,gf]eW
Γ
,
r<fe
+
/,
Proof.
From (3.1),
iί
is
easily seen that
d
k+l
U,9l=
Σ
Lemma
2.
//[/,^]=0/or/eW
k
,
(3
ίff
)*
=
C(5/)
1
,
CΦO.
(3.3)
Proo/
From (3.1)
we
have
292
Gui-zhang
Tu
hence
(3.3) then follows upon integrating.
Corrollary. Suppose that
/eW
k
,
0eW,,
fc,/^2
<wd
[/,0]=0,
then
f=C
1
u
k
+f
l
,
Proo/
It is a
immediate consequence
of
(3.3).
Lemma
3.
///
#eW
fc
,
fc^2,
and
[/0]=0
5
ίΛen
there
exists
a
nonzero constant
C
such
thatf-CgeW
k
,,
k
f
<k.
Proof.
By
Lemma
2, a
constant
C can be
chosen such that
d
k
(f—Cg)
=
0,
thus
Corollary.
If
an
evolution equation (2.6)
possesses
a
nontrivial symmetry
ge
W
fe
,
ίί
possesses
also
a
nontrivial
symmetry
/zeW
fe
,,
k' <k.
Proof.
Let C be the
constant
as
stated
in
Lemma
3, and
take
h
=
f—Cg,
then
it
holds obviously that
[/z,
/]
= 0, and the
fact
that
g is
nontrivial implies that
h is
also nontrivial.
Lemma
4.
///eW
Λ
,
,
_
lfl
f)
+
k
.
J)
+
(3
z
^
2
(δ
fc
/)
.
(3.4)
Proof.
From
(3.1)
from
which
(3.4)
follows.
Lemma
5. Let
f
=
uί+fι,
Λ«"ί;
0
=
w?
+
0ι,
^i^wf,
(3.5)
w^
and
uf
are
defined
as
(2.7),
and
&
1?
/
x
^2,
ίferc
[/,^]
=0
ί'm/7/y
Proof.
From
(3.3)
and
(2.8)
we
have
from
which
(3.6)
follows.
Now we
proceed
to
prove
the
following
Theorem
A. A
necessary condition
for an
evolution
equation
(2.6)
to
possess
a
nontrivial symmetry
isf=Cu
k
+f
ί9
where
f^^u
k
,
i.e.,
f
ί
=f
ί
(u,u
ί9
...,%),
k
<k.
Commutativity
Theorem
293
By
virtue
of
Proposition
1, the
Theorem
A can be
stated
equivalently
as
Theorem
A*.
Suppose that
/eW
fc
,
#eW
z
,
fc^2,
/^l,
^φC
1
/+C
2
w
1?
then
ίfte
necessary
conditions
for
%(f)
being commuted with
tfί(g) are
f=C
ί
u
k
+f
1
,
and
g
=
C
2
u
1
+g
1
,
where
f^u
k
and
g
l
<u
l
.
The
idea
of the
proof
of
this theorem
is
straightforward,
but the
whole
discussion
is
tedious, since
we
must
verify
carefully
all the
possible cases.
To
reduce
the
length
we
shall,
in
some minor cases, pass over
a
series
of
simple argument
and
purely
quote
the
conclusions.
Proof.
Let
/eW
fcl
,
#eW
Zl
,
since
C
1
C
2
[/,#]
=
[C
1
/,
C
2
#],
we may
assume
that/
and g
take
the
form
of
(3.5).
Now by
hypothesis that
[/,
g]
=
0 and the
Corollary
of
Lemma
3
we
need only
to
discuss
the
cases
(I).
fe
l5
1^2,
k
1
Φ/j
and
(II).
k
1
^2,
/!
^
1. (By
symmetry,
the
discussion
of the
case
/c
t
^
1,
/
t
^2
is
similar.)
Case
I.
k
i9
1^2,
k^l
v
(la)
a^b^2
In
this case,
the
(3.6)
implies
fc
1
=
/
1
,
which contradicts with
the
hypothesis
/c
1
Φ/
15
and
hence
is
impossible.
(Ib)
α
1
=
l,b
1
^2.
(By
symmetry,
the
discussion
of the
case
b
1
=
1,
α
x
^
2 is
similar.)
In
this case,
(3.6)
reads
U
k
2
hence
it
must hold that
(3.7)
(Iba)
Mi
^3
(i)
k
2
<k
ί
— ί,
I
2
<l
1
—
ί.
In
this case
we
have,
by
Lemma4
and
(2.8), that
M
((^)
(3
tl
/)®
2
(3
/lff
)j
=M
^
j
(δ,^)®
2
^/)),
(3.8)
from
which
it is
easy
to
deduce that
J
b
ί
(b
1
—
l)=
I
M6
1
α
2
?
by
means
of
(3.7)
that
(b
1
—
I)k
1
=a
2
l
l9
we get
fc.,^
=/
1?
which contradicts also with
the
hypothesis,
(ii)
k
2
<k
ί
— 1,
I
2
=
l
i
— 1. In
this case,
(3.4)
reads
1
[^
\
/
H
\
V
/2
^
fcl
\2/
h
^
Λl
But
it is
easy
to see
M(^
2
(d
h
g))^M(^(d
l2
g))
and by
virtue
of
u
k2
+
2
=
u
ll
+
2
,
hence
the Eq.
(3.8)
holds again,
the
case
is
therefore impossible
by the
same reason.