Vilasi G. Hamiltonian Dynamics
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334
Kd
V
Equation
We can conclude that the eigenvalues of
L
are eigenvalues
of
and that
the associated eigenstates of
T
are the gradients
of
the eigenvalues
"
6X
6X
T-
=
4X-.
6%
6U
Let
us
now consider the Poisson bracket of two functionals
K
and
K',
(k,
k'}
=
fG, axG')
>
(12.25)
and
suppose that the function~s are in invo~uti~n; that
is,
their Poisson bracket
is vanishing:
(G,
O,,G')
=
0.
If
G,,
and
GLu
denote the derivatives at the point
u
of
G
and
G',
the derivative,
with respect to
u,
of
the above equation gives
(G*,(~%),
8,G')
+
fG,
OXG~,(6a))
=
0,
6%
E
T,M
.
The operator
G,
:
TM
-+
TM
is
symmetric with respect to the
La
scalar
product,
so
that
(&a,
G*O,G')
-
(G:,aZG,
6~)
z=
(621,
G,,O=G'
-
G',,a,G)
=
0,
or
equivalently
G,,O,G'
=
G',,azG.
(1
2.26)
If
G'
is
the gradient
of
&,
the left-hand side
of
Eq.
(12.26)
is
-G,
where
the dot denotes the derivatives
of
G
along integral curves given by the solutions
of
KdV, while the right hand
side
is
given by
FG,
where
F
=
a,,,
+
ua,.
Thus,
we
may write
G
=
-A~G,
(12.27)
where
A
=
-a,,,
-
ua,
-
u,,
the adjoint of the operator
F,
is
the derivative
of
dynamics.
The
"time derivative"
of
the sequence expressed by Eq.
(12.22)
gives
G,+I
=
+G,
-t
pGn,
so
that, by using
Eq.
(12.27),
we
have
TGn
=
[T,
At]G,,
Kd
V
as
a
Completely Integrable Hamiltonian Dynamics
335
where the bracket
[.
,
.f
denotes the usual commutator. The reader can
easily
check that
?'
=
[-At,
f']
.
(12.28)
The
analogy
between
L
and
can be resumed
as
folfows:
"
6X
6X
T-
=:
4X-
6U
624
L=
[B,L],
?'=
[-At,F],
.rl=BlCI,
G
=
-AtG.
~q~tion (12.28)
can
dso
be written in the fo~lowing form:
rf,
=
[A,T]
,
(12.29)
where the operator
f'
is
the adjoint, with respect
to
the
scalar
product
Lz,
of
f'
=
D-'Ef$
(12.30)
21
33
T*
=
a,,
.
f-u.
+-u,L?-1*
.
Chapter
13
General
Structures
In spite of its success
as
an integration algorithm,
a
compact
a
priori
criterion
of integrability in terms of
Lax
pairs is, to date, lacking.
On the other hand, the inverse scattering method being a transforma-
tion from generic coordinates (potentials) to action-angle variables,g2 makes
it only natural for us to state an integrability criterion, for soliton equa-
tions, by looking
at
them
as
dynamical system on an infinite-dimensional
also suggested by the occurrence in these remarkable systems of a peculiar
vant for the effectiveness of the methods, which naturally fits in this
geometrical setting
as
a mixed tensor field on the phase manifold
M.
phase m&fold.
137,138,103,179,168,81,82,144,78,80,147,73~100~166
This point of view is
Opera~or174,175,137,132,138,103,179,104,81,82,l44,78,80,147,73,100,20,162,98,99,ll7,67
rele-
13.1
Notation
and
Generalities
Many geometrical concepts, introduced in Part
11,
can be extended to infinite
dimensional manifolds, whose local model is an infinite-dimensional topolog-
ical vector space, if the necessary care, connected to the passage from finite-
dimensional case to the infinite-dimensional one,
is
taken.
Many properties of the finite-dimensional case, still hold, in the infinite
dimension, only if the considered manifold is a Banach manifold; that is, a
manifold locally homeomorphic to
a
Banach space. The reason is that the
337
338
General
Stmctwes
implicit functions theorem
does not hold in an arbitrary topological vector
space.
Given
a
nonlinear operator
A;
i.e. a function of
u
and its space derivatives,
the generic evolution equation
8th
-
=
A(u)
at
(13.1)
will be considered
as
a dynamical equation on the functional space
M
of field
functions
u(x,
t)
regarded as functions of the space coordinates only, defined
on
the whole real axis, and satisfying suitable boundary conditions.
The rate
of
change, along the solutions of
Eq.
(13.1),
of any functional
F[u]
will be given
by
(13.2)
6F
$00
dxA(u)
*
-
+"6F
8u
-dx
=
Lm
d
-F[u]
=
1,
-
dt
du
at
6u(x)
l
and then, by the action of the operator
6
+m
~xA(u)
*
-
s_,
6u(x)
A[u]
=
which
will
be called the
dynamical vector field.
In the functional space
M,
first order differential operators
S/6u(z)
consti-
tute
a
basis
for vector fields. The
dual basis
is given by the variations
6u(x)
and, as it
is
usual
where
6
is the
Dirac delta function.
Then, any vector field
X[u]
can be written in the following form:
6
dxX(u)-
6u(x)
l
and
dual vectors
or
covectors
a[u]
as follows:
m
a[.]
=
dya(u)Su(y).
The contraction
(a,
X)
between vectors and covectors will be given by
L
00
Notation
and
Generalities
339
The rate of change, along the solutions of Eq.
(13.1),
of
a vector field
X[u]
is given by
6
+m
[Xu
1
A(u)
-
A,
*
X(U)]-
dt
6u(z)
(13.3)
where the operators
Xu
and
A,,
defined by
d
d
d&
d&
X,C~
:=
-X(U
+
E(P)[~=o,
A,(P
:=
-A(u
+
E(P)[~=o,
are the
week derivatives,
or
Gateaux derivatives
of
X(u)
and
A(u),
respectively.
Let
ua
observe that Eqs.
(13.2)
and
(13.3)
correspond to the usual
Lie
derivatives,
with respect to
A[.],
of
F[u]
and
X[u],
respectively.
So
such
time derivatives
will be denoted* by
LA
F
and
LAX;
LA
just being
the Lie derivative operator with respect to
A.
We notice that Eq.
(13.3)
can be written in the form
where the bracket
[a,
a]
denotes the usual commutator between differential
op
erators.
The tangent space and the cotangent space of
M
in
u,
will be denoted by
7,M
and
CM,
respectively.
In the continuous (formal) frame
6/6u(z)
and coframe
6u(z),
the evolution
equation can be regarded
&s
an ordinary differential equation,
du
dt
-
=
A[u]
In order to simplify notations and formulae, in the following a vector field
X[u]
will be identified with its components
X(u)
and a mixed tensor field
T
with its associated endomorphisms
T,
or
defined by
T(a,
X)
=
(a,
PX)
=
(Pa,
X)
.
These endomorphisms will be, in general, represented
as
operators acting
on vector fields
or
their dual.
*Henceforth, to avoid confusion with the
Lax
operator
L,
the Lie derivative with respect
to
a
vector field
X
will
be denoted
by
Cx.
340
General
Structures
Thus, with abuse of saying, the Lie derivative
LAX
of
a
vector field
X
with
respect to
A
will be identified with
Xu.
A(u)
-
A,
-
X(u),
and the symmetries
X
of
a
given dynamics
At
will be given by the solutions
of
the following linear
differential equation:
Xu
.
A(u)
-
A,
.
X(U)
=
0.
The Lie derivative, with respect to
A,
of
an operator
T;
that
is,
of an
endomorphism on vector fields associated with
a
mixed tensor field, will be
given by the operator or endomorphism
CAT
given by
,CATV
=
t(A,
'P)
-
[&,T]'P,
-
where
(13.4)
Therefore, the
invariance
with respect to the dynamics of
such
a
tensor
field will be expressed as*
13.1.1
Backward
to
KdV
In the case of the KdV equation,
M
is
the manifold
of
C"
field functions
u,
considered
as
functions only of
x,
and vanishing at the infinity together with
its
space derivatives.
The
dynamics
is given by the vector field
so
that the solutions
of
KdV correspond to the
integral
curves
of
A.
Let
us
observe that Eq.
(12.29)
is simply the expression, in local coordi-
nates,
of
the invariance, under the
KdV
flow,
of the mixed tensor field
T,
defined by
tA
very general and fundamental approach to the analysis
of
symmetries
of
nonlinear
partial differential equations,
is
described in Refs.
182
and
30.
tEquation
(13.4),
in spite
of
its
form,
does not correspond, generally, to the
Lax
represen-
tation.
A
possible tensorial version of this has been given by several authors,
some
of
them
in the context of alternative Lagrangian~~~','~
or
in reading
it
has the vanishing, along the
dynamics,
of
the covariant derivative of a section
of
an M-based bundle.81
Notation
and
Generalities
341
T(a,X)
=
(PX,a)
=
(X,?Q),
Q
E
T*M,
X
E
TM.
(13.5)
Indeed,
&.
(12.29)
can be written, in geometrical terms,
as
follows:
CAT=O,
(13.6)
where
&A
is the Lie derivative with respect to
A.
The tensor
T,
which in local coordinates can be written in the form
(13.7)
satisfies the condition
TU(PX,
Y)
-
PU(PX,
Y)
=
P[PU(X,
Y)
-
TU(X,
Y)I
,
(13.8)
which
is
the analogue, in this infinit~d~ensional setting, of
Eq.
(6.483.
The
above condition can also be written
as
follows:
(LpxT)"Y
=
T(CxT)"Y
,
X,
Y
E
7,M.
(13.9)
We
recall that
Eq.
(13.9),
or
Eq.
(13.8),
is called the
Nijenhuis condition
or the
~~je~~Ui$ ~r~c~e~,
and that the tensor field
NT[U](a,x,
y)
=
(a,
(L.fxT)"y
-
P(LXT)^y>
Y
(13.10)
with
a
E
T:M,
X,Y
E
7,M,
is called the
Nijenhuis torsion
of
T.
Thus,
Nijenhuis' condition
(13.9)
is
expressed by
N;.=O.
(13.11)
A
consequence of
Eq.
(13.11)
is
that the vector fields
of
the sequence
An+,
=
TAn
,
(A,
-US),
Vn
2
1
dose
on an Abelian Lie algebra
of
sym~etr~~s
for
KdV, and KdV being
a
~amiltoni~ dynamics, the sequence
is
a
sequence of gradients
of
conserved functionals, In other words, Eq.
(13.11)
ensures that the endomorph is^
ih
generates
a
sequence
of
closed l-forms, in
the sense that
(6a=0,
6Ta=0)-6(Fna)=0,
QETM,
'dnZ1.
342
General
Stmctwes
In our case, the functional 1-forms are exacts; that is, they are exterior deriva-
tives of functionals which, since
T
is A-invariant, are first integraIs of KdV.
13.2
Strongly and Weakly Symplectic
Forms
At this point, it
is
advisable to spend some words about the definition of
symplectic form on
an
infinite-dimensional manifold, since in this case,
a
dis-
tinction between strongly symplectic forms and weakly symplectic forms must
be introduced.
We
say that a differential 2-form
w,
on an infinite-dimensional manifold
M,
is a
strongly symplectic structure,
if
(a)
w
is
closed, that
is
dw
=
0;
(b)
Vp
E
M,wp
:
7,M
x
7,M
-+
R
is
a
nondegenerate bilinear form;
i.e. the map
Z:7,M-+7,rM,
(13.12)
which with every vector
X
E
7,M
associates the differential 1-form
Z(X)
on
7,M,
defined
as
below
(I(X))(Y)
=
wp(X,Y)
VY
E
"&4
7
is
injective
and
surjective.
In other words,
Z
is an isomorphism between
the spaces
7,M
and
Tp*M
If
the map
{
13.12)
is
only injective, then the differential 2-form
w
is
said to
be a
weakly symplectic structure.
Such
a
distinction
has
not been done in finite dimensions, since an injective
map between two finite dimensional vector space, with the same dimension,
is
also
surjective.
In infinite dimensions, the distinction
is
instead important. Indeed, let
us
consider a locally Hamiltonian vector field
X
and a strongly symplectic form
w;
then
Cxw
=
6ixw
=
0.
If
ixw
is also an exact differential form; i.e.
ixw
=
-6H,
(13.13)
Invariant
Endomorphism
343
the vector field
X
is
a
globally Hamiltonian vector field and
H
is the Hamilton
function.
Vice versa, if
H
is
a
differentiable function on
M
H:M-+R,
there exists a vector field
X
on
M
such that Eq.
(13.13)
holds, since the map
(13.12)
is
an isomorphism; but, if
w
is only weakly symplectic, the vector field
X
cannot exist.
13.3
Invariant
Endomorphism
All
the evolution equations, introduced
at
the beginning (p.
267),
apart from
the Burgers' equation, are Hamiltonian systems with respect to a symplectic
structure. Actually, many
of
them are Hamiltonian dynamics with
respect to
two symplectic ~tr~~t~re~,~~~~~~~*~~~ namely
w1
and
w2.
For
instance,
0
in the case
of
KdV equation, we have
w(X,
Y)
=
(X(u),
D-'Y(u))
w2(X,
Y)
=
(X(u),
EL1Y(u))
with
2
1
3 3
,
Ek
=
ax,,
+
-.a,
+
-u,,
D-'
=
1
2
(
Ix
-a7
dx
-
I"
dz)
where the bracket
(a,.)
denotes the
Lz
scalar product. The Lenard
sequence
of
gradients
of
conserved functionals
is
established, in terms
of the operators
D
=
d/ax
and
Ek,
as
follows:
DGn+1
=
EkGn
;
in the case of the
sine-Gordon
equations
wxt
+
sinw
=
0,
we have
Wl(X,Y)
=
(Ww),
DY(4)
1
w2F,
Y)
=
(X(V),
K'Y(4)
7
§Here
x,
t
denote light-cone coordinates.