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\TITLE GRAVITY WAVES ON THE SURFACE OF THE SPHERE
%
\ENDTITLE
\AUTHOR
Rafael de la Llave
\FROM
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712-1802
\AUTHOR
Panayotis Panayotaros
\FROM
Department of Physics
The University of Texas at Austin
Austin, TX 78712-1081
\ENDTITLE
\vskip 5 em
\hbox{\hfill \bf Dedicated to the memory of J.C. Sim\'o}
\ABSTRACT
We propose a Hamiltonian model for
gravity waves on the surface of a fluid layer
surrounding a gravitating sphere. The general
equations of motion are non-local and can be used as a
starting point for simpler model equations,
which can be derived systematically by expanding the
Hamiltonian functional. In this paper,
we study in detail
the small wave amplitude regime.
The first order non-linear terms can be eliminated
by a formal canonical transformation. Similarly, many of the
second order terms can be eliminated. The resulting model
has the feature that it leaves invariant several finite dimensional
subspaces on which the motion is integrable.
\ENDABSTRACT
\SECTION Introduction
The goal of this paper is to describe a Hamiltonian formulation
for waves on the surface of a fluid layer
surrounding
a gravitating spherical body. The fluid satisfies hydrodynamic
equations inside the layer and
the surface of the
layer is moving consistently with the motions
of the of the fluid.
The resulting system is
a free boundary problem in which the region where the
hydrodynamic equations of motion hold is
also part of the unknowns.
We will assume that the fluid in the layer is inviscid, incompressible
and is moving irrotationally.
The assumption that the flow is irrotational (potential flow)
is very restrictive and in many cases inappropriate.
Our model is, however, useful for isolating and studying
hydrodynamic wave phenomena
where the restoring force is gravitational,
for instance sea waves and atmospheric tides.
In the case where the potential gravity waves take place
over a plane, a very elegant
Hamiltonian formulation was first introduced by Zakharov [Z]
(see also [M]).
More recently it has been shown that the formulation
leads to a very systematic
discussion of approximate gravity wave equations (see [CG])
and to efficient
numerical methods [CS]. Using the formalism presented here,
similar algorithms could be derived for the sphere.
The theory of Zakharov has also been extended to general
inviscid, incompressible free boundary flows in [LMMR].
In that work the Hamiltonian formulation of the potential flow
case is put in the context of the canonical theory of
the Euler equations (see [MG], [MW]).
In the first three sections we show how to use the
potential flow formalism on the
sphere and discuss some physical applications and limitations
of our model.
The main result of this section is that indeed
the free boundary problem can be written in
a Hamiltomian form.
The Poisson
bracket does not have the standard form(is non-canonical), but
we also find a change of variables that reduces it to standard form.
Note that there already several well-known examples
of non-canonical Poisson brackets in hydrodynamics, plasmas,
magnetism and
other areas.
The canonical variables in the Hamiltonian formulation
of free surface potential flow are
the wave amplitude (a function giving the shape of the surface)
and the hydrodynamic potential
at the surface. These two functions,
defined on the sphere, completely determine the velocity inside the
layer. This reduction in the dimension of the problem
is one of the benefits of the potential flow assumption.
On the other hand,
the equations of motion for wave amplitude and surface potential
are non-local. We
describe a method that
allows us to write the Hamiltonian and the equations of
motion in terms of Fourier multiplier operators in the fourth section.
The Hamiltonian formalism is particularly convenient for
deriving approximate gravity wave equations
for it suffices
to consider approximations of the Hamiltonian.
These approximations are based on dimensional analysis and are discussed in
section five, where we identify the dimesionless parameters of the
problem and indicate interesting asymtotic regimes.
In the present work we will be concerned with a small amplitude
regime and we focus on ``intermediate depth'' waves.
Another
advantage of Hamiltonian systems is that is easier
to simplify them by performing canonical
transformations.
The canonical perturbation theory is based on
systematic changes of variables that
try reduce the perturbed system to the unperturbed one.
One advantage of this procedure is that
the range of validity of the perturbation theory is limited
by the size of the solutions
but which does not depend explicitly in time.
In the Hamiltonian formalism,
the theory of small amplitude waves is analogous to the theory
of motions of a Hamiltonian system near an elliptic fixed point.
The completely quiescent state is the fixed point and
the linear waves
correspond to the normal modes of the linearised system.
The motion of the waves in the linear approximation is
governed by the quadratic terms in the Hamiltonian and
the cubic terms in the Hamiltonian can be
interpreted as describing the interactions of three waves,
the quartic as describing the interactions of four waves and
so on.
In Hamiltonian mechanics the motion around
elliptic fixed points is often analysed using the so-called
Birkhoff normal forms.
These are simplified equations of motion obtained after
a succession of canonical transformations.
The goal of each such transformation is to eliminate the lowest order
non-linear terms
in the Hamiltonian.
In section six we show
that for water waves on the surface
of the sphere a) the cubic (first order in the non-linearity)
terms of the Hamiltonian
can be eliminated and that b) the only quartic
(second order in the nonlinearity) terms that cannot be
eliminated are those of a very reduced class.
The canonical transformations are given explicitly
as formal time-1 maps of suitable
Hamiltonian flows. This method of constructing
canonical transformations is also known as
``Lie-series'' perturbation theory.
We will remark however that these normal form calculations
are more relevant to intermediate and large depth water waves
since in the shallow water limit we expect that the domain of convergence
of the appropriate
canonical tranformation shrinks to the origin.
As an application of these calculations we
find some invariant finite dimensional manifolds in which the
system is integrable. In these finite dimensional manifolds,
we can use the standard methods of dynamical
systems to, in particular, identify periodic
orbits of
the second order normal form system. These approximate
solutions of the full water wave system
are travelling and standing waves
with amplitude dependent frequency.
Even if at this point these transformations are purely formal, we hope
that they can be made rigorous and at least be used to prove
good lower bounds for the time for which solutions with small initial
data are well defined.
It also seems possible that
some of the periodic orbits and quasi-periodic orbits
of the linear problem
persist in the full non-linear system. For finite dimensional
systems, the persistence of families periodic orbits in the
full non-linear system was proved first by Lyapunov.
More general results of this
type are in \cite{Mo}, \cite{We}. The persistence of
some of the quasi-periodic orbits is also proved in finite dimensions
using KAM theory.
Since this paper is concerned with developing
the formalism we will postpone these analytic questions to future
work.
We also point out that our formalism is well adapted to the developement
of numerical methods for the problem. For example , to develop
finite dimensional approximations we truncate the Hamiltonian. The
truncation will automatically be Hamiltonian. We plan to discuss the
numerical implementations of that scheme in a forthcoming paper.
\SECTION Equations of motion
We consider a sphere
of radius $b$ and on top of the sphere a layer of fluid (``sea'' or
``atmosphere'')
of thickness (``depth'') $h$. Using the
standard spherical coordinates
$ r = $ radius, $ \vt = $ polar, $ \vf $ = azimuth,
the surface of the sea will be at
$ r(\vt,\vf) = \rho + \eft $ with $ \rho = b + h $.
The amplitude of the water waves is described by the
single valued function $ \eft $.
The dynamical problem we want to consider is that of free
surface potential flow of the layer of water
under the influence of gravity. For such a flow,
since the spherical shell is homotopically trivial,
there exists a velocity potential $\phi$ and
the velocity is given by
$ {\vec u} = \nabla \phi $. The conservation of mass
for an incompressible fluid is $\nabla \cdot {\vec u} = 0$
Hence, we should have
$$ \Delta \phi = 0
\EQ(1.1)
$$
in the region occupied by the fluid.
On the surface we have
$$ \ett = \fr - \oners \fth \eth - \sinrs \ff \ef ,
\EQ(1.2) $$
and
$$ \phi_t = - {1 \over 2} |\nabla \phi|^2 + { K \over {\rho+\eta}} - p
\EQ(1.3) $$
and at the bottom $ r = b \quad $ we have
$$ \fr = 0 .\EQ(1.4) $$
The equations of motion \equ(1.1)-\equ(1.4)
are obtained from the Euclidean Euler equations (see [L]) by a change of
variables:
\equ(1.1) is the conservation of mass for an incompressible
fluid,
\equ(1.4) is the rigid wall boundary condition at the
bottom of the sea and \equ(1.2) is the condition, in polar coordinates,
that the surface is transported by the flow, or
$$ \left[ { \partial \over \partial t }+{\vec u} \cdot \nabla \right] F = 0
\quad , \EQ(1.5) $$
where $ F({\vec r} ,t ) = r - \rho - \eft =0 $ is the implicit representation
of the surface and $\vec u$ is the velocity at the surface.
The dynamical boundary condition \equ(1.3) follows from
Euler's equation at the surface
$$ \nabla \left({{\partial \phi} \over {\partial t }} + {1 \over 2}
| \vec u |^2 + V(\eta) + p \right) =0 \quad . \EQ(1.6) $$
The term
$V(\eta) = { K \over {\rho + \eta}} = { K \over \rho} - g\eta +\cdots $
is the gravitational
potential due to the solid sphere of radius $b$.
Physically, $K = GM $ with $M$ the mass of the
planet, $G$ the gravitational constant and $g$ is the
acceleration of gravity at $r=b$.
Also we require that the pressure $ p $ be constant in space.
($ \nabla p \neq 0 $ would correspond to the presence of
an additional external force.)
Note that in \equ(1.3) quantities with no spatial dependence (e.g $p $ ,
$ K \over \rho $) do not play any role and can be set to zero.
The equations of motion suggest that if at any instant $ t_0$ we know
the function $ \eft $ and the potential $ \phi $ on the
surface, i.e. the function
$\Phi ( \vf , \vt ) = \phi (\vf , \vt , \rho + \eft ) $,
we can determine $ \phi $ at $t_0$ in the whole
region occupied by the fluid
by solving the boundary value
problem: $\triangle \phi = 0 $ inside the fluid with
$ \phi = \Phi $ at the surface $ r = \rho + \eft $ and
$ {{\partial \phi} \over {\partial r}} = 0 $
at $ r = b $. Thus we are essentially interested in the
evolution of $\eta$ and $\Phi$ , which is given by \equ(1.2) and \equ(1.3).
Note that $ \Ft = \ft + \et \fr|_{r=\rho+\eft} $.
Both equations are non-local since they contain the term
$ {{\partial \phi} \over {\partial r}} $.
To evaluate $ {{\partial \phi} \over {\partial r}} $
we need information about the solution of the
boundary value problem. A method for writing
the equations of motion in terms of $\eta $ and $ \Phi $ alone is given
in section three.
\SECTION Hamiltonian formulation
We will show that the equations of motion \equ(1.2) , \equ(1.3)
for $ \eta$ and $\Phi $ form a Hamiltonian system.
More precisely they can be written as
$$ \eta_t = [ \eta , H ] \quad , \quad \Phi_t = [ \Phi , H ] \EQ(2.1) $$
where $ [\,,\,] $ is an appropriate Poisson bracket and
$H $ is the Hamiltonian of the system.
To express the equations of motion in Hamiltonian form, it is
natural to guess that the Hamiltonian should be the
physical energy and then try to find the Poisson
bracket. In particular, here
the Hamiltonian $H$
will be the total energy $ K + U $ (kinetic $+$ potential) of the
water mass
$$ H = {1\over2} \int_{S^2}\!\int_{r=b}^{r=\rho+\eft}
|\nabla \phi|^2 dV + \int_{S^2}\!\int_{r=b}^{r=\rho+\eft}
\left( - { K \over r } \right) dV \EQ(2.2) $$
Integrating by parts the first term,
$$ H = {1\over2}\int_{S^2} \left. \Phi R \fn\right|_{r=\rho+\eft} dA_r +
{1\over2}\int_{S^2} (-K) dA_r \quad , \EQ(2.3) $$
where $ R = [1 + (\oner \eth)^2 + ( \sinr \ef )^2 ]^{1 \over 2 } $ and
$ {{ \partial \phi } \over { \partial n }}|_{r=\rho+\eft} $
is the normal
derivative at the surface. We also use the notation
$ dA_r = r^2 \sin \theta d\theta d\varphi $
with $ r = \rho + \eft $.
The quantity $ RdA_r $ is the area element of the water surface.
We introduce the Dirichlet - Neumann
(`` flux '') operator $ G(\eta) $ by
$$ G(\eta)\Phi = R \fn|_{r=\rho+\eft} \EQ(2.4) $$
and we can write the Hamiltonian as
$$ H = {1\over2}\int_{S^2} \Phi G(\eta) \Phi dA_r +
{1\over2}\int_{S^2} (-K) dA_r \quad. \EQ(2.5) $$
The Poisson bracket $ [\,,\,] $ of \equ(2.1) is defined by
$$ [F , G ] = \int_{S^2} ( {{ \rho + \eta } \over \rho } )^{-2}
\left( {{\delta F} \over {\delta \eta }} {{\delta G} \over {\delta \Phi }}
- {{\delta G} \over {\delta \eta }} {{\delta F} \over {\delta \Phi }}
\right) \rho^2 dA_1 \quad . \EQ(2.6) $$
The variational derivative $ {{\delta F} \over {\delta x}} $ is defined
by $ < {{\delta F} \over {\delta x }} , \delta x > = F^\prime $, where
$ F^\prime $ is the Frechet derivative of F with resp. to x, and
$ <,> $ is the
$ L^2 $ inner product on $S^2$ (with radius $\rho$).
{}From \equ(2.6) the equations of motion \equ(2.1) become
$$ \eta_t = ( {{ \rho + \eta } \over \rho } )^{-2}
{{\delta H } \over {\delta \Phi }} , \quad
\Phi_t = - ( {{ \rho + \eta } \over \rho } )^{-2}
{{\delta H } \over {\delta \eta }} \EQ(2.7) $$
It is easy to see that the above bracket indeed satisfies the
axioms of Poisson brackets (for the formalism of
Poisson brackets for fields see [D], ch1).
The only axiom that is not-immediate
is the Jacobi identity,
which can be verified by a direct computation.
{ \bf Proposition 3.1 : }
Hamilton's equations \equ(2.1) and the original equations of motion
\equ(1.2) , \equ(1.3) are equivalent.
{ \it Proof } We calculate : for \equ(1.2)
$ {{ \delta U } \over { \delta \Phi }} = 0 $ and
$$ K(\Phi + \df ) = {\aot} \int_{S^2} \Phi G(\eta) \df dA_r +
{\aot} \int_{S^2} \df G(\eta) \Phi dA_r + K(\Phi) + o(\delta^2 \Phi ) $$
so that
${{\delta K } \over {\df}} = (1 + {\eta \over \rho })^2 G(\eta)\Phi $
and therefore
$$ \eta_t = \geta \Phi = R \fn|_{r=\rho+\eta(x,y)} =
\nabla \phi \cdot
[1, - \oner \eth , - \sinr \ef ] \EQ(2.8) $$
which is \equ(1.2).
For \equ(1.3),
$${{\delta U } \over {\de}} = - (1 + {\eta \over \rho })^2
{ {K \over {\rho + \eta} } }$$
and
$$ K(\eta + \de) =
{\aot} \int_{S^2} |\nabla \phi|^2 \de dA +
{\aot} \int_{S^2}\!\int_{r=b}^{r=\rho+\eft}|\nabla \phi(\eta + \de)|^2dV
\quad . \EQ(2.9) $$
The second integral is
$$
- \int_{S^2}\!\int_{r=b}^{r=\rho+\eft} \nabla \phi
\nabla(\fr \de) dV + K(\eta) + o(\delta^2 \eta ) \EQ(2.8) $$
$$= - \int_{S^2} \fr \de \fn R dA_r + K(\eta) + o(\delta^2 \eta )
= - \int_{S^2} \fr \ett \de dA_r + K(\eta) + o(\delta^2 \eta) $$
using Green's identity and the eq. for $ \eta_t $. From
\equ(2.6), \equ(2.7), \equ(2.8) we obtain
$$ - ( {{ \rho + \eta } \over \rho } )^{-2} {{\delta H } \over {\de } } =
- {1 \over 2} |\nabla \phi|^2 - g\eta + \eta_t \fr \EQ(2.9) $$
But we also have $ \Ft = \ft + \et \fr|_{r=\rho+\eft} $ so that by \equ(2.9),
$ \Ft = - ( {{ \rho + \eta } \over \rho } )^{-2}
{{\delta H } \over {\de } } $ is equivalent to \equ(1.3). \QED
{\bf Remark 3.1 } A comparison of the above derivation of the Hamiltonian
structure of water wave equations on the sphere with
the derivations of the analogous result in Euclidean
space ([Z] , also [BO]) will reveal that
the adoption of the non-standard bracket was needed because of
the form of the surface element in polar coordinates.
The departure of our bracket from the standard one is small
- in a sense that will be made precise later - but not
small enough to be discarded.
It is possible to make a change of variables in such a
way that the bracket becomes the standard one. (We will
refer to this process as ``diagonalising'' the bracket.)
In particular, choosing a transformation $f(\eta,\Phi) = (\teta,\tPhi)$
defined by
$$ \teta = \eta ( 1 + { \eta \over {2 \rho}}) , \quad
\tPhi = \Phi(1 + { \eta \over { \rho}}) \EQ(2.12) $$
the bracket is diagonalised :
$ Df \cdot A \cdot Df^T = J $ where $A$ , $J$ are the
respective cosymplectic
forms of our bracket and the standard bracket.
The transformation of \equ(2.12) is invertible in the
range of interest $ \hbox{sup} |\eta| < h $.
\SECTION Applicability of the model to physical problems
The main physical limitation of our model
is the potential flow assumption.
The absence of vorticity and angular momentum in the
rest frame has the consequence that
the model can not describe phenomena such as Rossby waves
or the bulging of the equator observed in rotating
liquids.
Thus our model isolates the effects of gravitation.
We can also add to our formalism the apparent motions
due to the rotation of the observer's reference frame,
as well as another gravitational effect, namely the tides.
Also the model can be useful for
studying phenomena involving short waves, in particular energy
transport by wind generated ocean waves.
First we consider the water wave equations
in a frame rotating with angular velocity $\Omega$ around the
z-axis ($ \vt = 0 $).
We introducing the
rotating frame canonical variables
$ u_{\Omega} = ( \eta_{\Omega} , \Phi_{\Omega} ) $
related to the rest
frame variables $ u = ( \eta , \Phi ) $
by $ u_{\Omega} = A_{\Omega}^{-1}(t) u $, where
$ A_{\Omega}(t) $ acts on functions defined on the sphere by
$$ A_{\Omega}(t) f(\vt , \vf ) = f( \vt , \vf + \Omega t ) \quad . $$
The flow $A_{\Omega}(t)$ is generated by
$$ f_t = [f , \Omega L_z] \quad , $$
where $ [\, , \, ] $ is the Poisson bracket
introduced previously and
$L_z$, the angular momentum, is
$$ L_z = \int_{S^2}\!\int_{r=b}^{r=\rho+\eft}
( \vec r \times \nabla \phi ) \cdot \hat z dV =
\int_{S^2} \Phi { { \partial \eta } \over { \partial \vf } }
dA_r \quad . \EQ(2.10) $$
Note that $ [ H , L_z ] = 0 $ so that if the rest frame variables
evolve under $ u_t = [ u , H ] $ the rotating
frame observables evolve under
$$ { { \partial \eta_{\Omega}} \over { \partial t } } =
[ \eta_{\Omega} , H - \Omega L_z ] \quad , \quad { { \partial \Phi_{\Omega}} \over { \partial t } } =
[ \Phi_{\Omega} , H - \Omega L_z ] \quad . \EQ(2.11) $$
Also note that the rotating frame velocity $ u_{\Omega} $
is given by $ u_{\Omega} = \nabla \phi_{\Omega} + U_{\Omega} $
where $ \phi_{\Omega} $ is the solution of the Neumann problem
(i.e \equ(1.1), \equ(1.4) for $ \phi_{\Omega} $)
with $ \Phi_{\Omega} $ as the boundary value at the
surface, and
$ U_{\Omega} $ is the velocity of rigid rotation with
angular velocity $ \Omega $.
A Hamiltonian formalism for general Euler free boundary
flows is given in [LMMR]. There the authors use the unique
decomposition of divergenceless velocity field $u$ to
$ w + \nabla \phi $, with $w$ divergenceless and tangent to
the boundary, to write the canonical theory for
$ w$, the surface potential and the shape of the
boundary. However the description of the fluid motion by surface
quantities is obviously lost.
Tides result from the gravitational attraction of another body,
in particular, the
acceleration of points on the surface of the
sphere relative to the acceleration of the center of the
sphere. The effect of tides is described by a tidal
potential, which can be added to the equation for the evolution of the
surface hydrodynamic potential. The tidal potential $ W $
at the surface of
the fluid layer has, to lowest order in the distance to
the attracting body, the form (see [P])
$$ W = A P_2(\zeta) \EQ(2.12) $$
where $ P_2(\zeta) $ is the second Legendre function and $A$ is a
physical constant.
If we consider a point $x $ on the
surface of the layer, the point $O$ at the center of the
sphere and the point $S$ at the center of the distant star,
$ \zeta $ is the angle between the lines $Ox$ and $OS$.
The tidal potential can be expressed as a function of the polar and
azimuth angles $ \vt$ and $\vf$ of the point $x $ and of the
polar and azimuth angles $ \delta' $ and $T'$ of the star:
$$ W = A [ P_2(\vt)P_2(\delta') +
{3\over4}\sin2\vf\sin2\delta'\cos(T'+\vf) +
{3\over4} \cos^2\vf \cos^2 \delta' \cos2(T'+\vf) ] \quad . \EQ(2.13) $$
If the sphere rotates with angular velocity $ \Omega $ then
$ T' = \Omega t $, and similarly we can consider variations in time of
$ \delta' $. We can then use the rotating frame formalism
of equations \equ(2.11)
and the tidal potential to model the effects of the periodic variation
of the tidal forces. Notice that in \equ(2.12) we can
write $ P_2(\zeta) = Y^0_2(\zeta, \varpi) $ where $\varpi $ is the
angle around the axis $ OS $. In the second
form, \equ(2.13), we are using the
terrestrial system of axes and angles. Since the two systems are related by
a time-dependent rotation the tidal potential in the terrestrial system
can be written as a time-dependent linear combination of
the spherical harmonics $ Y^m_l $ with $l = 2$, $ m = -2, -1, \cdots2 $.
Thus the potential tide can be added to the spectral form
of the equations of motion in a straightforward way, provided that
we have the time dependence of the coefficient of each
spherical harmonic.
Since the harmonics are eigenfunctions of the linearised problem,
a simple way to mimic the varying tidal forces is
parametric excitation (variations of one of the parameters of
the problem)
at frequencies that are at resonance with the
$ l = 2 $ harmonics. In fact, the phenomenon of
parametric excitation of non-linear waves
(see for instance [FS] for water waves) is of independent interest.
One other area of possible application of our model is
in the the trasport of energy by wind-driven sea waves in the
ocean. The typical wavelength of these waves is very small
compared to the global scales and the effect of the Coriolis force
is minimal. We can include the effect of wind by adding to
the equations of motion a pressure (gradient) term computed from
given velocity profiles for the motion of the air masses above the sea.
Since the approximations used in the followiong chapters are uniform in
spatial wavenumber of the quantities involved, we expect that
the response of short waves will yield rather reliable information
on the long-range transport of wind energy by the sea waves.
\SECTION The flux operator
We now describe the Dirichlet-Neumann
operator $G(\eta)$ as a function of $ \eta $.
It will be assumed that $G(\eta)$ can be expanded around
$\eta = 0 $ so that we can write
$G(\eta) = \sum_{i=0}^{\infty}G_i(\eta) $, the $G_i(\eta)$ being
homogeneous of order $i$ in $\eta$. Our task will be to
calculate the $G_i(\eta)$. Clearly, from \equ(2.5), the expansion
of $G(\eta)$ will give us an expansion of the Hamiltonian
$ H = H_0 + H_1 + \cdots $ with $ H_0 = {1\over2}\int \Phi G_0(\eta) \Phi +
{1\over2}\int g \eta^2 $ , $ H_1 = {1\over2}\int \Phi G_1(\eta) \Phi$
and so forth.
As we will show, it is possible to compute recursively
the $G_i(\eta)$ in a way very similar to that in \cite{CG}, \cite{CS}
and [GAS].
Once the
$ G_i(\eta) $ are computed explicitly, the Hamiltonian of \equ(2.5)
can be systematically expanded in powers of $\eta$. If
we use some truncation of this expansion as a Hamiltonian, the resulting
model can be considered as an small amplitude approximation of
the water waves problem. This formalism, therefore, provides us
with a systematic way to produce increasingly more refined small
amplitude expansions that can be computed rather effectively.
We remark, however, that the expansion here is formal. We leave the problem of
convergence and the choice of appropriate spaces for $ \eta $
and $ \Phi $ open. For waves on the plane
the question has been addressed (in a more general context)
in [CM].
To calculate the $ G_i(\eta) $
we first look for harmonic
functions $ \phig $ of the form
$$ \phig(r, \vt , \vf ) = \ug (r) \yg(\vt , \vf ) \quad . $$
The functions $ \yg(\vt , \vf ) $ are the spherical harmonics,
indexed by
$ \gamma = [l,m] $ with $ l $ a positive integer being
the total angular momentum number,
and
$ m = -l , -l+1 , \cdots , l $ the azimuthal angular momentum
number.
The condition
$ \triangle \phig = 0 $ implies that
$$ r^2 \ug'' + 2r\ug' - l(l+1)\ug = 0 \quad . \EQ(3.1) $$
The only solution of \equ(3.1)
(up to a multiplicative constant)
that also satisfies $ \ug'(b) = 0 $ is
$$ \ug(r) = (l+1) \left({r \over b } \right)^l +
l \left( { b \over r } \right)^{l+1} \quad . \EQ(3.2) $$
Since the $ \phig = \ug \yg $ as above are harmonic and satisfy
$ {{ \partial \phig } \over { \partial r }} = 0 $ at
$ r = b $ , from the definition of $ \geta $ , \equ(2.4), we have that
for every index $ \gamma $
$$ \geta \phig(\vt,\vf,\rho+\eta) =
\nabla \phig(\vt,\vf,\rho+\eta) \cdot
\left[1, - \oner \eth, - \sinr \ef\right] \quad \EQ(3.3) $$
or
$$ \sum_{i=0}^{\infty}G_i(\eta) \ug (\rho + \eta) \yg =
\ug'(\rho + \eta) \yg \qquad \qquad \qquad \qquad \EQ(3.4)$$
$$ \qquad \qquad - \eth {{\ug(\rho + \eta)} \over {(\rho + \eta)^2}}
\yth - \ef {{\ug(\rho + \eta)} \over {(\rho + \eta)^2 \sin^2\vt}}
\yf $$
Expanding $ \ug(r) $ and $ 1 \over r^2 $ around $ \eta = 0 $
(i.e $ r = \rho) $ and matching powers in $ \eta $ we obtain,
at order $0$:
$$
\upR \yg = G_0(\eta) \uR \yg \quad $$
hence
$$ G_0(\eta) \yg = \ouR \upR \yg \quad . \EQ(3.5) $$
%
At order $1$ we have:
$$
\eta \u2pR \yg - \ovRs \uR \left[ \etth \yth + \osins \etf \yf \right] = $$
$$ \left[ G_0(\eta) \eta \upR + G_1(\eta) \uR \right] \yg \quad $$
hence
$$ \EQ(3.6) \quad G_1(\eta) \yg = \ouR [ \eta \u2pR \yg -
\ovRs \uR \left[ \etth \yth + \osins \etf \yf \right] $$
$$ - G_0 (\eta) \eta \ug'(\rho) \yg ] $$
%
In general, at order $k$ we have
$$
\eqalign{
& { 1 \over k!} \eta^k \ug^{(k+1)}(\rho) \yg -
{ 1 \over (k-1)!} \eta^{k-1} ({ \ug \over \ r^2 })^{(k-1)}(\rho)
\left[ \etth \yth + \osins \etf \yf \right]
\cr =
& \big[G_0(\eta) { 1 \over k!} \eta^k \ug^{(k)}(\rho)
+ G_1(\eta) { 1 \over (k-1)!} \eta^{k-1} \ug^{(k-1)}(\rho)
+ \cdots
+ G_k(\eta) \ug(\rho) ] \yg
}
\EQ(3.7) $$
($\ug^{(k)} = k\hbox{-th derivative} $)
and we can obtain $ G_k(\eta) $ recursively in terms of the
$ G_i(\eta)$ with $ i < k $.
%
The above formulas determine the $G_i(\eta)$ as an operator
in a, for the moment unspecified, space of functions in the
sphere. Note that the $G_i(\eta)$ are given as Fourier multipliers
and it is clear that they are not local operators.
\SECTION Scales and dimensionless variables
It will be advantageous to introduce several dimensionless
quantities that take into account the scales of the
problem.
The relevant dimensionless quantities are
i) the ratio $ \epsilon = { A \over h } $ of the typical
amplitude $A$ of the waves to the depth $h$ of the fluid layer
ii) the ratio $ \beta = { h \over b } $ of the depth over the
radius $ b$ of the planet, and in the rotating frame we also
have iii) $ R = { \Omega \over { \omega_0 } } $ where $ \Omega $
is the angular velocity of the frame and $ \omega_0 $ is
a typical frequency of linearised gravity waves in the rest
frame. Here we take the reference frequency to be
$ \omega_0 = { \sqrt{gh} \over b } $, which turns out to be
the phase velocity of linearised shallow water waves.
The dimensionless variables $ \eta^* $ , $ \Phi^*$ , $ t^* $ are
introduced by
$$ \eta^* = { \eta \over A } , \quad
\Phi^* = {{ \omega_0 \Phi} \over { g A } } , \quad
t^* = \omega_0 t \quad . \EQ(4.1) $$
For the Dirichlet-Neumann operator,
we observe from the formulas in the previous section that in
each $ G_k(\eta) $ we can factor out a term
$ 1 \over { b^{k+1} } $
arising from the $\ug^{(n)} $ so that
$$ G_k(\eta) \yg = {1 \over { b^{k+1}} } G^*_k(\epsilon h \eta^*) \yg =
{ 1 \over b } ( \epsilon \beta )^k G^*_k(\eta^*) \yg \quad . \EQ(4.2) $$
The $ G^*_k(\eta^*) $ defined by the first equality
in \equ(4.2) are dimensionless:
they depend only on $ \eta^* $ and $ \beta $.
The Hamiltonian can be written as
$$ H = ( \epsilon \beta )^2 b^2 { 1 \over 2 } \int_{S^2}
\left[ g \Phi^* { 1\over \beta}
[ \sum_k (\epsilon \beta )^k G^*_k(\eta^*)\Phi^*]
+ g ( \eta^* )^2 \right] dA_\rho \quad . \EQ(4.3) $$
%
Note that $g$ is the acceleration of gravity at $r = \rho $.
%
Defining $ \hat H $ by
$$ \hat H = { 1 \over 2 } \int_{S^2}
\left[ \Phi^* { 1\over \beta}
[ \sum_k (\epsilon \beta )^k G^*_k(\eta^*)\Phi^*]
+ ( \eta^* )^2 \right] \rho^2 dA_1 \quad , $$
%
Hamilton's equations \equ(3.7) become
$$ \eta^*_{ t^* } = (1 + \epsilon \beta \eta^* )^{-2}
{{\delta {\hat H} } \over {\delta \Phi^* }} \quad , \EQ(4.3) $$
$$ \Phi^*_{t^*} = - (1 + \epsilon \beta \eta^* )^{-2}
{{\delta {\hat H} } \over {\delta \eta^* }}
\quad . \EQ(4.4)$$
In the rotating frame we can
use the time-scale $ t^{\dagger} = R t^* $ and
Hamilton's equations become
$$ \eta^*_{ t^\dagger } = (1 + \epsilon \beta \eta^* )^{-2}
[ - {{\delta {\hat L} } \over {\delta \Phi^* }} + { 1 \over R }
{{\delta {\hat H} } \over {\delta \Phi^* }} ] \EQ(4.5) $$
$$ \Phi^*_{t^\dagger} = - (1 + \epsilon \beta \eta^* )^{-2}
[ - {{\delta {\hat L} } \over {\delta \eta^* }} + { 1 \over R }
{{\delta {\hat H} } \over {\delta \eta^* }} ]
\EQ(4.6)$$
with $ {\hat L_z } = \int_{S^2} \Phi^*
{ { \partial \eta^* } \over { \partial \vt } } dA_{\rho} $.
%The dimensionless form of the
%$ ( {{ \rho + \eta } \over \rho } )^{-2} $
%term of the bracket is approximate but it is clear that the
%departure of our bracket from the standard one
%is of first order in $ \epsilon $ and $ \beta $.
%Therefore the corrections to the standard bracket
%have to be included.
Note that the factor
$ (1 + \epsilon \beta \eta^* )^{-2}$ which appears in the
non-standard bracket differs from unity in terms which are first
order in the dimensionless variables $\epsilon$ and $\beta$.
Hence, the difference between the standard bracket and the
standard one has to be considered in expansions in the
dimensionless quantities of this order.
The parameter $ \beta $ is the analog of the depth to
wavelength ratio in Euclidean space. The small $ \beta$
regime gives us an analog of the ``shallow water''
regime(e.g see [Wi]),
in which the $ G^*_k(\eta^*) $ can be expanded in
$ \beta $. Note that the $ G^*_k(\eta^*) $ are already
multiplied by $ \beta^k $ in the Hamiltonian so that,
for instance , in the $ \beta \rightarrow 0 $ limit
(with $ \epsilon $ non-zero) the evolution becomes linear.
It is also possible to prescribe a relation between
$ \epsilon $ and $ \beta $ . For example setting
$ \epsilon = \beta^2 = \mu $
would lead to an analog of the Bousinesq
regime in $ \real^2 $.
In what follows we will be concerned with the small $ \epsilon $
regime with $\beta$ arbitrary, in particular we will
use the $ G_i(\eta)$ calculated previously to write the equations of motion
to second order in $ \epsilon $. From the dimensional analysis
we have to take into account terms arising from the
non-standard bracket. It is advantageous to present the Hamiltonian
in the variables $ \tilde \eta $, $ \tilde \Phi $ in which the
bracket is diagonal . Note that since $ \epsilon $ is small
the normalisations of the new variables can be taken to be the same
as that of the original variables. The o($\epsilon^2$) Hamiltonian
$ H = H_0 + H_1 + H_2 $ in
$ \tilde \eta $ , $ \tilde \Phi $ (unnormalised ) is
$$ H_0 = {1\over2}\int_{S^2} \tPhi G_0(\teta) \tPhi \rho^2dA_1 +
{1\over2}\int_{S^2} (\teta)^2 \rho^2 dA_1 \EQ(4.5) $$
$$ H_1 = -{1\over2} \int_{S^2} \tPhi G_0(\teta) (\tPhi \teta) \rho dA_1 +
{1\over2} \int_{S^2} \tPhi [G_0(\teta) \tPhi] \teta \rho dA_1 +
{1\over2} \int_{S^2} \tPhi G_1(\teta) \tPhi \rho^2 dA_1 \EQ(4.5) $$
$$ H_2 = {3\over4}\int_{S^2} \tPhi G_0(\teta)(\tPhi \teta^2 ) dA_1 -
{1\over2}\int_{S^2} \tPhi [G_0(\teta)(\Phi \teta)]\teta dA_1 \EQ(4.6) $$
$$ - {1\over4}\int_{S^2} \tPhi [G_0(\teta)\tPhi] \teta^2 dA_1
- {1\over2}\int_{S^2} \tPhi G_1(\teta)(\tPhi \teta)\rho dA_1 -
{1\over4}\int_{S^2} \tPhi G_1(\teta^2)\tPhi \rho dA_1 $$
$$ + {1\over2}\int_{S^2} \tPhi [G_1(\teta)\tPhi] \teta \rho dA_1
+ {1\over2}\int_{S^2} \tPhi G_2(\teta)\tPhi \rho^2 dA_1 \quad . $$
If we use the spectral variables
$ \eta_\gamma $, $\Phi_\gamma$
(we drop the tilde from the notation)
defined by $\eta = \sum_\gamma \eta_\gamma \yg$,
$\Phi = \sum_\gamma \Phi_\gamma \yg $ with
$ \eta^*_{ [l,m]} = \eta_{[l,-m]} $,
$ \Phi^*_{ [l,m]} = \Phi_{[l,-m]} $,
the Poisson bracket (which is now diagonalised) becomes
$$ [ f,g ] = \sum_\gamma \left(
{ {\partial f} \over {\partial {\eta_\gamma}} }
{ {\partial g} \over {\partial {\Phi^*_\gamma}} } -
{ {\partial f} \over {\partial {\Phi_\gamma}} }
{ {\partial g} \over {\partial {\eta^*_\gamma}} }
\right) \EQ(3.81) $$
and Hamilton's equations are
$$ \dot \eta_\gamma ={{\partial H }
\over{\partial \Phi^*_\gamma}}, \quad
\dot \Phi_\gamma = - {{\partial H } \over {\partial \eta^*_\gamma }}
\quad . \EQ(3.9) $$
Using the $G_i(\eta)$ from section 4, the 4-wave Hamiltonian
$ H = H_0 + H_1 + H_2 $ is
$$ H_0 = {1\over2}{\rho^2} \sum_\gamma {{\ug'(\rho)} \over {\ug(\rho)}}
\Phi_\gamma \Phi^*_\gamma +
{1\over2} {\rho^2}\sum_\gamma g \eta_\gamma \eta^*_\gamma \EQ(3.10) $$
$$ H_1 = {1 \over 2}{\rho^2} \sum_{\gamma_1 ,\gamma_2 , \gamma_3}
\left({{ u_{\gamma_2}''(\rho) } \over { u_{\gamma_2}(\rho)} } -
{{ u_{\gamma_1}'(\rho) } \over { u_{\gamma_1}(\rho)} }
{{ u_{\gamma_2}'(\rho) } \over { u_{\gamma_2}(\rho)} } \right)
\Phi_{\gamma_1} \Phi_{\gamma_2} \eta_{\gamma_3}
\int Y_{\gamma_1} Y_{\gamma_2} Y_{\gamma_3} \EQ(3.11) $$
$$ + {1 \over 2} \sum_{\gamma_1 ,\gamma_2 , \gamma_3}
\Phi_{\gamma_1} \Phi_{\gamma_2} \eta_{\gamma_3}
\int Y_{\gamma_1} \nabla Y_{\gamma_2} \cdot \nabla Y_{\gamma_3} $$
% this line should be ignored
$$ H_2 = {1 \over 2} \sum_{\gamma_1 ,\gamma_2 , \gamma_3, \gamma_4 }
\Phi_{\gamma_1} \Phi_{\gamma_2} \eta_{\gamma_3} \eta_{\gamma_4}
\cdot
[\sum_\gamma
\int Y_{\gamma_1} Y_{\gamma_4} Y_{\gamma}
\int Y_{\gamma_2} Y_{\gamma_3} Y^*_{\gamma} \cdot
\EQ(3.12)$$
$$
\bigg[ -{\rho} {{ u_{\gamma}''(\rho) } \over { u_{\gamma}(\rho)} }
%
+ {{\rho} \over 2} { { u_{\gamma_2}''(\rho) } \over { u_{\gamma_2}(\rho)} }
%
+{{\rho} \over 2} { {u_{\gamma_2}'(\rho)} \over {u_{\gamma_2}(\rho)} }
{ {u_{\gamma_1}'(\rho) } \over {u_{\gamma_1}(\rho)} }
%
+{{\rho^2} \over 2}\left( { {u_{\gamma_2}'''(\rho)} \over {u_{\gamma_2}(\rho)} }
-{{ u_{\gamma_2}''(\rho) } \over { u_{\gamma_2}(\rho)} }
{{ u_{\gamma_1}'(\rho) } \over { u_{\gamma_1}(\rho)} } \right) $$
%
$$ -{{\rho^2} \over 2}{{ u_{\gamma_2}'(\rho) } \over { u_{\gamma_2}(\rho)} }
{{ u_{\gamma}''(\rho) } \over { u_{\gamma}(\rho)} }
%
+ {{\rho^2} \over 2} {{ u_{\gamma_2}'(\rho) } \over { u_{\gamma_2}(\rho)} }
{{ u_{\gamma_1}'(\rho) } \over { u_{\gamma_1}(\rho)} }
{{ u_{\gamma}'(\rho) } \over { u_{\gamma}(\rho)} } \bigg] \quad + $$
%
% angular terms
$$ ( {1\over{\rho}}+{{ u_{\gamma_1}'(\rho) }
\over { u_{\gamma_1}(\rho)} } )
\sum_{\gamma}{\int Y_{\gamma_1} \nabla Y_{\gamma_4} \cdot \nabla Y_{\gamma}}
{\int Y_{\gamma_2} Y_{\gamma_3} Y^*_{\gamma} }
%
+ ( {2\over{\rho}}-{{ u_{\gamma_2}'(\rho) }
\over { u_{\gamma_1}(\rho)} } )
\int Y_{\gamma_1} \nabla Y_{\gamma_2} \cdot \nabla Y_{\gamma_3}Y_{\gamma_4}].
$$
In the rotating frame the quadratic Hamiltonian $ H_0 $ is
modified to
$$ H^{\Omega}_0 =
{1\over2R} {\rho^2}\sum_\gamma {{\ug'(\rho)} \over {\ug(\rho)}}
\Phi_\gamma \Phi^*_\gamma +
{1\over2R} {\rho^2} \sum_\gamma g \eta_\gamma \eta^*_\gamma
- i {\rho^2} \sum_\gamma m_\gamma \Phi_\gamma \eta^*_\gamma \EQ(3.13) $$
while the cubic and quartic Hamiltonians are the same as above.
The dispersion relations implied by $ H_0 $ and $ H^{\Omega}_0 $ are
$$ \omega^2 (\gamma) = { {\ug'(\rho)} \over {\ug(\rho)} }
\EQ(3.14) $$
and
$$ \omega_\Omega (\gamma) = m_\gamma
+ {1 \over R } \left({ {\ug'(\rho)} \over {\ug(\rho)} } \right)^{1\over2}
\EQ(3.15) $$
respectively.
{}From \equ(3.14) we can establish that $\omega (\gamma) =
\omega (l) $ viewed as a function of the positive real
variable $ l $ is smooth, monotonicaly increasing with
$ \omega (0) = 0 $. In addition, $ \omega ''(l) < 0 $. The
function $\omega (l) $ also depends on $ \beta $ and note
that in the limit $ \beta \rightarrow 0 $,
$ \omega'' (l) \rightarrow 0 $
(but not uniformly in $ l $). For fixed $ \beta $ it
is easy to see that for $ l $ large
$ \omega (l) \rightarrow \sqrt{l} $, which is the deep
water dispersion relation.
We now introduce some extra notation that will be useful in
the next section.
We define the variables $ a_\gamma $, $ a^*_\gamma $ by
$$ \eta_\gamma = {{\sqrt{2}} \over 2}
{\sqrt{\omega_\gamma \over g }} ( a_\gamma + a^*_{ \gamma^{-}} ) \quad ,
\quad \Phi_\gamma = -i {\sqrt{2} \over 2}{\sqrt{g \over \omega_\gamma } }
( a_\gamma - a^*_{ \gamma^{-}} ) \EQ(3.16)$$
where if $ \gamma = [l,m] $ then $ \gamma^{-} = [l,-m] $ .
Hamilton's equations become
$$ \dot a_\gamma = -i
{ {\partial H} \over {\partial a^*_\gamma} } \quad . $$
The Hamiltonian in these variables can be readily evaluated
using the formulas above:
the quadratic Hamiltonian is
$ H_0 = \sum_\gamma \omega_\gamma a_\gamma a^*_\gamma $
with $\omega_\gamma $ given by the dispersion relation.
If we write
$$ H_1 = \sum_{\gamma_1 ,\gamma_2 , \gamma_3}
\Phi_{\gamma_1} \Phi_{\gamma_2} \eta_{\gamma_3}
I_{\gamma_1 \gamma_2 \gamma_3} \quad \hbox{and} \quad
H_2 = \sum_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 }
\Phi_{\gamma_1} \Phi_{\gamma_2} \eta_{\gamma_3} \eta_{\gamma_4}
I_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 } $$
with the $I_{\gamma_1 \gamma_2 \gamma_3} $ ,
$I_{\gamma_1 ,\gamma_2 \gamma_3 \gamma_4 } $
given by \equ(3.11) and \equ(3.12) , we also write
$$ H_1 = \sum_{\gamma_1 \gamma_2 \gamma_3}
( A_{\gamma_1 \gamma_2 \gamma_3}
{ a_{\gamma_1} a_{\gamma_2} a_{\gamma_3} } +
A_{\gamma_1 \gamma_2 \gamma^{-}_3}
{ a_{\gamma_1} a_{\gamma_2} a^*_{\gamma_3} } ) + c.c. $$
$$ H_2 = \sum_{\gamma_1 \gamma_2 \gamma_3 \gamma_4}
( B_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 }
{ a_{\gamma_1} a_{\gamma_2} a_{\gamma_3} a_{\gamma_4} } +
B_{\gamma_1 \gamma_2 \gamma_3 \gamma^{-}_4 }
{ a_{\gamma_1} a_{\gamma_2} a_{\gamma_3} a^*_{\gamma_4} } + $$
$$ + B_{\gamma_1 \gamma_2 \gamma^{-}_3 \gamma^{-}_4 }
{ a_{\gamma_1} a_{\gamma_2} a^*_{\gamma_3} a^*_{\gamma_4} } ) + c.c . $$
with the coefficients given by
$$ A_{\gamma_1 \gamma_2 \gamma_3} =
N_{\gamma_1 \gamma_2 \gamma_3 }
I_{\gamma_1 \gamma_2 \gamma_3} \quad , $$
$$ A_{\gamma_1 \gamma_2 \gamma^{-}_3} =
N_{\gamma_1 \gamma_2 \gamma_3 }
( -I_{\gamma^{-}_3 \gamma_1 \gamma_2}
- I_{\gamma_1 \gamma^{-}_3 \gamma_2}
+ I_{\gamma_1 \gamma_2 \gamma^{-}_3} ) \quad , $$
$$ B_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 } =
N_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 }
I_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 } \quad , $$
$$ B_{\gamma_1 \gamma_2 \gamma_3 \gamma^{-}_4 } =
N_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 }
(- I_{\gamma^{-}_4 \gamma_1 \gamma_2 \gamma_3 }
- I_{\gamma_1 \gamma^{-}_4 \gamma_2 \gamma_3 }
- I_{\gamma_1 \gamma_2 \gamma^{-}_4 \gamma_3 }
+ I_{\gamma_1 \gamma_2 \gamma_3 \gamma^{-}_4 } ) \quad , $$
$$ B_{\gamma_1 \gamma_2 \gamma^{-}_3 \gamma^{-}_4 } =
N_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 }
(- I_{\gamma^{-}_3 \gamma_1 \gamma_2 \gamma^{-}_4 }
- I_{\gamma_1 \gamma^{-}_3 \gamma_2 \gamma^{-}_4 }
+ I_{\gamma_1 \gamma_2 \gamma^{-}_3 \gamma^{-}_4 } ) \quad , $$
$$ N_{\gamma_1 \gamma_2 \gamma_3 } =
-{ \sqrt{2} \over 2 }
{\sqrt{ {\omega_{\gamma_3}} \over {\omega_{\gamma_1} \omega_{\gamma_2}} } }
\quad , \quad
N_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 } =
-{ 1 \over 4 }
{ \sqrt{ {\omega_{\gamma_3} \omega_{\gamma_4}} \over
{\omega_{\gamma_1} \omega_{\gamma_2}} } } \quad . $$
Alternatively, define
$ z^k = a^{k_1}_1 a^{k_2}_2 \ldots $ ,
$ \overline z^{\overline k} = ({a^*_1})^{ \overline k_1}
({a^*_2})^{\overline k_2} \ldots $
where the $ k_i $ , $ \overline k_i $ are non-negative integers and
the subscripts $ i $ label (enumerate) the modes $ \gamma_i $.
Also let $ |k| = k_1 + k_2 + \ldots $,
$ | \overline k | = \overline k_1 + \overline k_2 + \ldots $.
We can write
$$ H_1 = \sum_{\scriptstyle k , \overline k \atop
\scriptstyle |k| + | \overline k | = 3 }
A_{k,\overline k} z^k \overline z^{\overline k}
\quad \hbox{and} \quad
H_2 = \sum_{ \scriptstyle k , \overline k \atop
\scriptstyle |k| + | \overline k | = 4 }
B_{k,\overline k} z^k \overline z^{\overline k} \quad . \EQ(3.161) $$
The coefficients $ A_{k,\overline k} $ are
$$ A_{ k_{\gamma_i} k_{\gamma_j} k_{\gamma_l} } =
\sum_{{ \rm perm.} i,j,l}
A_{ \gamma_i \gamma_j \gamma_l } \quad , \quad
A_{ k_{\gamma_i} k_{\gamma_j} {\overline k_{\gamma_l}} } =
\sum_{{ \rm perm.} i,j}
A_{ \gamma_i \gamma_j \gamma^{-}_l } \quad . \EQ(3.17) $$
Similar formulas define the $ B_{k,\overline k} $.
\SECTION Normal Forms
We will now see that the cubic (3-wave) terms in the Hamiltonian can be
eliminated by a formal canonical transformation. We will also try
to eliminate the quartic (4-wave) terms, but it will turn out that
some of them are resonant and can not be eliminated.
To construct canonical transformations we will use the
``Lie series''
method, which we now briefly review (See for example \cite{DF},
\cite{Ca},\cite{Th},\cite{Fa}).
\def\Ad{\mathop{\rm Ad}\nolimits}
Consider a manifold $ M $ with a Poisson bracket $ J $ defined
on $C^\infty (M) $. To every function $ g \in C^\infty (M) $
we associate a map $ \Ad_g $:
$C^\infty (M) \rightarrow C^{\infty} (M) $
defined by
$ \Ad_g f = [g, f] $ . We also formally define $ \exp \Ad_g $ by
$$ (\exp \Ad_g)f = f + \sum_k {1\over k!} ( \Ad_g)^k =
f + [g,f] + {1\over2}[g,[g,f]] + \ldots
\quad . \EQ(5.1) $$
%
We can check that for every function $ g $
for which the series makes sense, the map $ \exp \Ad_g $
preserves the Poisson bracket structure on $ C^{\infty} (M) $, i.e.
$ (\exp \Ad_g)[f,h] = [(\exp \Ad_g)f, (\exp \Ad_g)h] $ and therefore
defines a (local) canonical transformation by acting on the
components of the coordinate charts of $ M $.
The above computations can be given some meanings beyond
formal manipulation,
depending on the interpretation of the series of
\equ(5.1). For example, the series converges for analytic
functions and then, sometimes it may be extended to the whole
space of smooth functions. Note also that, when this
can be done, $\exp(\Ad_g) f $ is the time-1 map of the
Hamiltonian vector field of $g$
acting on the function $f$. Alternatively, if
$g$ contains a small parameter, we may consider the
series \equ(5.1) as an asymptotic expansion in the small parameter.
These considerations pertain to the finite dimensional case.
Additional problems arise in infinite diemensional systems.
As we mentioned before, in this paper we will
present only the formal calculations and postpone the
analytic discussion to future work.
In the present application, $ M $ is the span of the $ \eta_\gamma $,
$ \Phi_\gamma $ , and $ J $ is the Poisson bracket given in \equ(3.81).
We want a function $ g $ such that $ \exp \Ad_g $ eliminates the
cubic terms from the Hamiltonian. In particular,
let $g = \epsilon S_0 $,
$ H = H_0 + \epsilon H_1 + \epsilon^2 H_2 $, then
$$ (\exp\epsilon \Ad_{S_0}) H =
H_0 + \epsilon ( H_1 + [S_0 , H_0 ]) +
\epsilon^2 ( H_2 + [S_0 , H_1 ] + {1\over2} [S_0,[S_0,H_0]] ) +
o(\epsilon^3) \EQ(5.3) $$
i.e. we want $ S_0 $ such that
%
$$ H_1 + [S_0 , H_0 ] = 0 \quad . \EQ(5.4) $$
%
If we can solve \equ(5.4) the new Hamiltonian becomes
$$ H_{new} = H_0 +
\epsilon^2( H_2 + {1\over2}[S_0 , H_1]) + o(\epsilon^3) \quad . \EQ(5.5) $$
%
To solve \equ(5.4) and calculate $ [S_0 , H_1 ] $
we will use the spectral variables $ a_\gamma $, $a_\gamma^*$
and the notation developed at the end of the previous section.
{ \bf Proposition 6.1 } Equation \equ(5.4) can be formally solved
or, equivalently
the cubic terms of the Hamiltonian can be eliminated by
a formal canonical transformation.
{ \it Proof }
The cubic Hamiltonian $ \epsilon H_1 $ is a sum of terms
$ \epsilon A_{k,\overline k} z^k \overline z^{\overline k} $
with $ |k| + | \overline k | = 3 $.
Let $ S_0 = \sigma_{k,\overline k} z^k \overline z^{\overline k}$,
$ |k| + | \overline k | = 3 $ . Using the derivation
property of the bracket and induction we have the formula
%
$$ [ z^\mu \overline z^{\overline \mu} ,
z^k \overline z^{\overline k} ] = i \sum_j
(\overline \mu_j k_j - \overline k_j \mu_j )
{ 1 \over {z_j \overline z_j } }
z^{ \mu + k } \overline z^{\overline \mu + \overline k} \EQ(5.61) $$
%
for arbitrary $ \mu $, $ \overline \mu $,
$ k $, $ \overline k $.
In particular,
$ [S_0 , H_0 ] = -i \sigma_{k,\overline k}
(\sum_i \omega_i(k_i - \overline k_i )) z^k \overline z^{\overline k} $.
If we set
$$ \sigma_{k,\overline k} = {{1} \over {i}}
{ { A_{k,\overline k} z^k \overline z^{\overline k}} \over
{\sum_i \omega_i(k_i - \overline k_i ) } } \EQ(5.7) $$
%
the term $ \epsilon A_{k,\overline k} z^k \overline z^{\overline k} $
is eliminated, provided that the 3-wave resonance condition
$$ A_{k,\overline k} \not= 0 \quad , \quad
{\sum_i \omega_i(k_i - \overline k_i ) } = 0
\EQ(5.8) $$
is not satisfied.
Letting
$ S_0 = -i \sum_{ k , \overline k }
{ { A_{k,\overline k} } \over
{\sum_i \omega_i (k_i - \overline k_i )} }
z^k {\overline z^{\overline k} } $ all the non-resonant
terms are thus eliminated. Therefore it is enough to show that
\equ(5.8) is never satisfied.
{}From \equ(3.161) and \equ(3.17)
the resonances \equ(5.8) occur if and only if
$$ I_{\gamma_1 \gamma_2 \gamma_3} \not= 0 \quad, \quad
\omega_{\gamma_1} + \omega_{\gamma_2} + \omega_{\gamma_3} = 0
\quad , \EQ(5.9) $$
$$ I_{\gamma_1 \gamma_2 \gamma^{-}_3} \not= 0 \quad, \quad
\omega_{\gamma_1} + \omega_{\gamma_2} - \omega_{\gamma_3} = 0
\quad . \EQ(5.10) $$
To show that none of the above equations are satisfied we examine the
dispersion $ \omega_\gamma $ and the terms
$ I_{\gamma_1 \gamma_2 \gamma_3} $.
First, since $ \omega_\gamma > 0 $ , \equ(5.9) can not be satisfied.
Also,
$ {{ d \omega } \over { dl } } > 0 $ and
$ {{ d^2 \omega } \over { dl^2 } } < 0 $ , so that
$$ \omega(l_3) = \omega(l_1) + \omega(l_2)
\Rightarrow l_3 > l_1 + l_2 \EQ(5.13) $$
%
On the other hand, from \equ(3.11)
$$ I_{\gamma_1 \gamma_2 \gamma_3} =
b_{\gamma_1 \gamma_2 \gamma_3} \int Y_{\gamma_1} Y_{\gamma_2} Y_{\gamma_3}
+ c_{\gamma_1 \gamma_2 \gamma_3}
\int Y_{\gamma_1} \nabla Y_{\gamma_2} \cdot \nabla Y_{\gamma_3}
\EQ(5.12)$$
with the $ b_{\gamma_1 \gamma_2 \gamma_3} $,
$c_{\gamma_1 \gamma_2 \gamma_3}$ determined by \equ(3.11) .
%
Now, $Y_{\gamma} = Y^m_l(\vt,\vf) =
e^{im\vf} P^{|m|}_l(\mu) $ with $ \mu = \cos \vt $ and $ P^m_l(\mu) $
the associated Legendre functions (we are using the
conventions of [J, ch1]). We have
$$ \int Y_{\gamma_1} Y_{\gamma_2} Y_{\gamma_3} =
\int^{2 \pi}_0 e^{i ( \sum_{i=1}^3 m_i ) \vf } d\vf
\int^1_{-1} P_{\gamma_1} P_{\gamma_2} P_{\gamma_3} d\mu $$
and (see [J], appendix)
$$ \int^1_{-1} P_{\gamma_1} P_{\gamma_2} P_{\gamma_3} d\mu \neq 0
\quad \hbox{only if} \quad |l_1 - l_2 | \le l_3 \le l_1 + l_2
\quad \hbox{and} \quad
l_1 + l_2 + l_3 = { \rm even } . $$
Hence, when
the frequency addition rule holds
$ \int Y_{\gamma_1} Y_{\gamma_2} Y_{\gamma_3} $ vanishes.
For
the second integral of \equ(5.12)
we have
$$ \int Y_{\gamma_1} \nabla Y_{\gamma_2} \cdot \nabla Y_{\gamma_3} =
\delta_{123}
\int^1_{-1} [ (1-\mu^2)P_{\gamma_1} { {d P_{\gamma_2}} \over d \mu }
{ {d P_{\gamma_3}} \over d \mu } -
{ { m_2 m_3} \over { 1 - \mu^2 } }
P_{\gamma_1} P_{\gamma_2} P_{\gamma_3} ] d\mu $$
with $ \delta_{123} =
\int^{2 \pi}_0 e^{i (\sum_{i=1}^3 m_i) \vf } d\vf $.
Using
$$ { {d P^{|m|}_l} \over {d\mu} } =
(1-\mu^2)^{1/2} P^{|m|+1}_l - |m|\mu(1-\mu^2)^{1/2} P^{|m|}_l $$
(note that $P^{|m|}_l = 0 $ for $|m| > l $) we have
$$ \int Y_{\gamma_1} \nabla Y_{\gamma_2} \cdot \nabla Y_{\gamma_3} = $$
$$ - \delta_{123} \int^1_{-1} { {\mu} \over { (1-\mu^2)^{1/2} } }
(|m_3| P^{|m_1|}_l P^{|m_2|+1}_l P^{|m_3|}_l
- |m_2|
P^{|m_1|}_l P^{|m_2|}_l P^{|m_3|+1}_l ) d\mu $$
$$ + \delta_{123} \int^1_{-1}
( P^{|m_1|}_l P^{|m_2|+1}_l P^{|m_3|+1}_l
- m_2 m_3
P^{|m_1|}_l P^{|m_2|}_l P^{|m_3|}_l ) d\mu $$
$$ - \delta_{123} \int^1_{-1} { {\mu} \over { (1-\mu^2)^{1/2} } }
( m_2 m_3
P^{|m_1|}_l P^{|m_2|}_l P^{|m_3|}_l
- |m_2 m_3 |
P^{|m_1|}_l P^{|m_2|}_l P^{|m_3|}_l ) d\mu $$
The terms $ { {\mu} \over { (1-\mu^2)^{1/2} } } P^{|m_{a_i}|+1}_{l_i} $
in the integral can be written
as a linear combination of
$ P^{|m_{a_i}|}_{l_i} $ and $P^{|m_{a_i}|+2}_{l_i} $
(see [McR, p115]). Likewise, for the the terms of
the last integral , unless one of $ m_2$, $m_3$ are zero, in
which case the integral is zero. Therefore,
$\int Y_{\gamma_1} \nabla Y_{\gamma_2} \cdot \nabla Y_{\gamma_3}$
can be expressed as a sum of
$$\int^1_{-1} P_{l_1}^{m_{a_1}} P_{l_2}^{m_{a_2}} P_{l_3}^{m_{a_3}} d\mu $$
and vanishes when the frequency addition rule holds. \QED
Note that the bounds on the size of the denominators of $ S_0 $
depend on $ \beta $ and since the $ \beta \rightarrow 0 $
limit is dispersionless we expect that the domain of
convergence of the transformation will shrink to zero in that case.
Thus the proposition concerns more the intermediate ($ \beta \sim o(1) $)
and deep water regimes.
In the rotating frame we can obtain a similar result for the
Hamiltonian $ H^\Omega = H_{0}^{\Omega} + { 1 \over R }
( \epsilon H_1 + \epsilon^2 H_2 + \cdots ) $. Here the quadratic
hamiltonian $ H_{0}^{\Omega} $ involves the dispersion
$ \omega_\Omega $ of \equ(3.15).
{ \bf Corollary 6.2 } In the rotating frame, the cubic terms in the
Hamiltonian can be eliminated by a canonical transformation.
{ \it Proof }
The normal form calculation is as above and we have to check that there
are no resonances. From \equ(3.15), the resonance condition is now
$$ I_{\gamma_1 \gamma_2 \gamma_3} \not= 0 \quad, \quad
R m_{\gamma_1} + \omega_{\gamma_1} + R m_{\gamma_2} + \omega_{\gamma_2} + R m_{\gamma_3} + \omega_{\gamma_3} = 0 \quad , \EQ(5.13) $$
$$ I_{\gamma_1 \gamma_2 \gamma^{-}_3} \not= 0 \quad , \quad
R m_{\gamma_1} + \omega_{\gamma_1} + R m_{\gamma_3} + \omega_{\gamma_2} - R m_{\gamma_3} - \omega_{\gamma_3} = 0 \quad . \EQ(5.14) $$
The coefficients are as in \equ(5.12). From the above discussion
of the triple integrals of harmonics $ I_{\gamma_1 \gamma_2 \gamma_3} \not= 0
\Rightarrow m_{\gamma_1} + m_{\gamma_1} +m_{\gamma_1} = 0 $ and
$ I_{\gamma_1 \gamma_2 \gamma^{-}_3} \not= 0
\Rightarrow m_{\gamma_1} + m_{\gamma_1} - m_{\gamma_1} = 0 $ and therefore
the resonance conditions \equ(5.13), \equ(5.14) are equivalent to
the rest frame resonance conditions. \QED
We now consider the problem of eliminating the quartic terms.
However, this time there are resonances.
As previously we try to find a function $ S_1 $ such that
$ (\exp\epsilon^2 \Ad_{S_1}) H_{new} $ has no $ o(\epsilon^2 ) $
terms and we are led to the equation
$$ ( H_2 + {1\over2 } [S_0 , H_1] ) + [ S_1 , H_0 ] = 0 \quad . \EQ(5.16) $$
The contribution from $[S_0 , H_1] $ can be computed with the
aid of the formula \equ(5.61). The calculation is long but
straightforward but can be simplified with the use of
a diagramatic method that will appear elsewhwere.
We write the result as
$$ [S_0 , H_1] = K + L $$
with
$$ K = -i\sum_{ \gamma_1 \gamma_2 \gamma_3 \gamma_4 \gamma_q }
[ { a_{\gamma_1} a_{\gamma_2}a_{\gamma_3} a_{\gamma_4} }
( A'_{\gamma_q \gamma_1 \gamma_2}A_{\gamma_3 \gamma_4 \gamma^{-}_q } +
A'_{\gamma_1 \gamma_q \gamma_2}A_{\gamma_3 \gamma_4 \gamma^{-}_q } +
A'_{\gamma_1 \gamma_2 \gamma_q}A_{\gamma_3 \gamma_4 \gamma^{-}_q } + $$
$$ A_{\gamma_q \gamma_1 \gamma_2}A'_{\gamma_3 \gamma_4 \gamma^{-}_q } +
A_{\gamma_1 \gamma_q \gamma_2}A'_{\gamma_3 \gamma_4 \gamma^{-}_q } +
A_{\gamma_1 \gamma_2 \gamma_q}A'_{\gamma_3 \gamma_4 \gamma^{-}_q } ) + $$
$$ a_{\gamma_1} a_{\gamma_2}a_{\gamma_3} a^*_{\gamma_4}
[ A_{\gamma_3 \gamma^{-}_4 \gamma^{-}_q}
( A'_{\gamma_q \gamma_1 \gamma_2} +
A'_{\gamma_1 \gamma_q \gamma_2} +
A'_{\gamma_1 \gamma_2 \gamma_q} ) + $$
$$ A'_{\gamma_3 \gamma^{-}_q \gamma^{-}_4 }
( A_{\gamma_q \gamma_1 \gamma_2} +
A_{\gamma_1 \gamma_q \gamma_2} +
A_{\gamma_1 \gamma_2 \gamma_q} ) + $$
$$ A'_{\gamma_3 \gamma^{-}_4 \gamma^{-}_q}
( A_{\gamma_q \gamma_1 \gamma_2} +
A_{\gamma_1 \gamma_q \gamma_2} +
A_{\gamma_1 \gamma_2 \gamma_q} ) +
A_{\gamma_3 \gamma^{-}_q \gamma^{-}_4 }
( A'_{\gamma_q \gamma_1 \gamma_2} +
A'_{\gamma_1 \gamma_q \gamma_2} +
A'_{\gamma_1 \gamma_2 \gamma_q} ) + $$
$$ A_{\gamma_3 \gamma^{-}_4 \gamma^{-}_q}
( A_{\gamma_q \gamma_1 \gamma^{-}_2} -
A_{\gamma_1 \gamma_q \gamma^{-}_2 } ) +
A'_{\gamma_1 \gamma_2 \gamma^{-}_q }
( A_{\gamma_q \gamma_3 \gamma^{-}_4} -
A_{\gamma_3 \gamma_q \gamma^{-}_4} ) ] $$
where
$ A'_{\gamma_1 \gamma_2 \gamma_5} =
A_{\gamma_1 \gamma_2 \gamma_5}
(\omega_{\gamma_1}+ \omega_{\gamma_2}+\omega_{\gamma_5} )^{-1} $,
$ A'_{\gamma_1 \gamma_2 \gamma^{-}_5}=
A_{\gamma_1 \gamma_2 \gamma^{-}_5}
(\omega_{\gamma_1}+ \omega_{\gamma_2}- \omega_{\gamma_5} )^{-1} $
and so forth.
$$ L = i \sum_{ \gamma_1 \gamma_2 \gamma_3 \gamma_4 \gamma_q }
a_{\gamma_1} a_{\gamma_2} a^*_{\gamma_3} a^*_{\gamma_4}
C_{\gamma_1 \gamma_2 \gamma_3 \gamma_4}, $$
$$C_{\gamma_1 \gamma_2 \gamma_3 \gamma_4} =
\left( { 1 \over {\omega_{\gamma_1}+ \omega_{\gamma_2}+ \omega_{\gamma_q} } }
+{ 1\over {\omega_{\gamma_1}+\omega_{\gamma_2}+\omega_{\gamma_5} } }\right) $$
$$ (A_{\gamma_q \gamma_1 \gamma_2} +
A_{\gamma_1 \gamma_q \gamma_2} +
A_{\gamma_1 \gamma_2 \gamma_q} )
(A_{\gamma^{-}_q \gamma^{-}_3 \gamma^{-}_4} +
A_{\gamma^{-}_1 \gamma^{-}_q \gamma^{-}_4} +
A_{\gamma^{-}_3 \gamma^{-}_4 \gamma^{-}_q} ) + $$
$$\left( { 1 \over { \omega_{\gamma_1}+\omega_{\gamma_2}- \omega_{\gamma_q} } }
- { 1 \over {\omega_{\gamma_q}- \omega_{\gamma_3}- \omega_{\gamma_4} } }\right)
A_{\gamma_1 \gamma_2 \gamma^{-}_q}A_{\gamma_q \gamma^{-}_3 \gamma^{-}_4}-$$
$$\left ( { 1 \over {\omega_{\gamma_1}+ \omega_{\gamma_q}- \omega_{\gamma_3} } }
- { 1 \over {\omega_{\gamma_2}- \omega_{\gamma_q}- \omega_{\gamma_4} } }\right )
(A_{\gamma_1 \gamma_q \gamma^{-}_3} +
A_{\gamma_q \gamma_1 \gamma^{-}_3} )
(A_{\gamma_2 \gamma^{-}_q \gamma^{-}_4} +
A_{\gamma_2 \gamma^{-}_4 \gamma^{-}_q} ). $$
Writing $ H_2 + {1\over2 } [S_0 , H_1] $ as
$ \sum_{ k , \overline k }
\tilde B_{k,\overline k}
z^k \overline z^{\overline k} $ with
$ |k| + | \overline k | = 4 $ the resonance condition is now
$$ \tilde B_{\gamma_1 \gamma_2 \gamma_3 \gamma_4} \not= 0 \quad , \quad
\omega_{\gamma_1} + \omega_{\gamma_2} + \omega_{\gamma_3} +
\omega_{\gamma_4} = 0 \quad , \EQ(5.17) $$
$$ \tilde B_{\gamma_1 \gamma_2 \gamma_3 \gamma^{-}_4} \not= 0 \quad , \quad
\omega_{\gamma_1} + \omega_{\gamma_2} + \omega_{\gamma_3} -
\omega_{\gamma_4}= 0 \quad , \EQ(5.18) $$
$$ \tilde B_{\gamma_1 \gamma_2 \gamma^{-}_3 \gamma^{-}_4} \not= 0 \quad , \quad
\omega_{\gamma_1} + \omega_{\gamma_2} - \omega_{\gamma_3} -
\omega_{\gamma_4} = 0 \quad . \EQ(5.19) $$
{ \bf Proposition 6.3 } The resonance conditions \equ(5.17) and \equ(5.18)
are never satisfied while condition \equ(5.19)
is satisfied for suitable $ \gamma_1 $, $ \gamma_2 $,
$ \gamma_3 $, $ \gamma_4 $.
{ \it Proof }
Since $ \omega_\gamma > 0 $, \equ(5.17) is never satisfied. For
\equ(5.18), we have that
$ \tilde B_{\gamma_1 \gamma_2 \gamma_3 \gamma_4^{-}}$
is of the form
$$ \tilde B_{\gamma_1 \gamma_2 \gamma_3 \gamma_4^{-}} =
\sum_{Q,R}
b_{\gamma_1 \gamma_2 \gamma_3 \gamma_4^{-}}^{Q,R}
\sum_{\gamma} Q_{ \gamma_i \gamma_j \gamma }
R_{ \gamma_n \gamma_m^{-} \gamma^{-} } $$
where $ Q_{ \gamma_i \gamma_j \gamma } $ is
$ \int Y_{\gamma_i} Y_{\gamma_j} Y_{\gamma} $ or
$\int ( \nabla Y_{\gamma_i} \cdot \nabla Y_{\gamma_j} ) Y^*_{\gamma} $
and similarly for $ R_{ \gamma_n \gamma_m^{-} \gamma^{-} } $.
{}From our previous discussion of the triple integrals of spherical
harmonics, a necessary
condition for $ \tilde B_{\gamma_1 \gamma_2 \gamma_3 \gamma_4^{-}} $
not to vanish is that there exist indices $ \gamma = [l,m] $
such that
$$ | l_i - l_j | \le l \le | l_i + l_j | \quad \hbox{and} \quad
| l_n - l_m | \le l \le | l_n + l_m | \EQ(5.20) $$
(and permutations on the $ i, j, n, m $). However, since
$ {{ d \omega } \over { dl } } > 0 $ and
$ {{ d^2 \omega } \over { dl^2 } } < 0 $ we have that
$ \omega_{\gamma_i} = \omega_{\gamma_j} + \omega_{\gamma_n} +
\omega_{\gamma_m} \Rightarrow l_i > l_j + l_n + l_m $ or
$ | l_i - l_j | > | l_n + l_m | $
( and permutations ) so that \equ(5.20) is not
satisfied, and \equ(5.18) can not be satisfied either.
For \equ(5.19),
pick $ \gamma_1 = {\gamma_3 } $ and
$ \gamma_2 = { \gamma_4 } $, then the frequency
sum rule is satisfied, moreover
$ \tilde B_{\gamma_1 \gamma_2 \gamma_3 \gamma_4} $
is (generically) non-zero (a few of them have been computed). \QED
Therefore the resonant part of the Hamiltonian
is of the form
$$ H_{2, res} =
\sum_{\gamma_1 \gamma_2 \gamma_3 \gamma_4}
R_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 }
{ a_{\gamma_1} a_{\gamma_2} a^*_{\gamma_3} a^*_{\gamma_4} } $$
with
$$ R_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 } =
B_{\gamma_1 \gamma_2 \gamma^{-}_3 \gamma^{-}_4 }
+ { 1\over 2} C_{\gamma_1 \gamma_2 \gamma_3 \gamma_4 }
\EQ(5.21) $$
We note that $ H_{2, res}$
should include all the ``generic'' resonances corresponding to
$ l_3 = l_1 , l_4 = l_2 $ or $l_3 = l_2 , l_4 = l_1 $.
In addition the resonance condition
\equ(5.19) may be satisfied for other integers.
A preliminary numerical search has not found any so far, but
note that for large $ l $ we have $ \omega(l) \rightarrow
\sqrt{l} $ and since
$ \sqrt{a} + \sqrt{b} = \sqrt{c} + \sqrt{d} $ has
other types of solutions we expect that there are quartets of
integers that are arbitrarily close to resonance.
A more detailed study of this issue and its dynamical implications will
be considered in the future.
The equations arising from the
above second order normal form Hamiltonian
$ H = H_0 + H_{2,res} $ are easily seen to have
families of periodic orbits. To simplify the argument we will
assume that $ H_{2,res} $ contains only the
``generic'' 4-wave resonances, in which case
the subspaces
$ M_L = { a(l,m) = 0 \,\,\hbox{for}\,\, l \neq L } $ are invariant
under the flow.
On these subspaces we can find periodic orbits.
{ \bf Proposition 6.4 }
Let $ \phi_H $ be the flow generated by the
Hamiltonian vector field of $ H = H_0 + H_{2,res} $. Then,
(a) the subspaces
$ T_{(L,M)} = \{ a_{l,m} = 0 \quad \hbox{for} \quad l\neq L, m \neq M \} $
with $ L = 1, 2, 3 \cdots $, $ M = -L, -L+1, \cdots L $
are invariant under $ \phi_H $.
Moreover, the restriction of the flow $ \phi_H $ to $ T_{(L,M)} $
folliates $ T_{(L,M)} $ by periodic orbits.
(b) The subspaces
$ S_{(L,\tilde M)} = \{ a_{l,m} = 0
\quad \hbox{for} \quad l \neq L, m \neq \pm M \} $
with $ L = 1, 2, 3 \cdots $, $ M = -L, -L+1, \cdots L $ and
$ 3|M| > L $ are invariant under $ \phi_H $ and the
restriction of $ \phi_H $ to $ S_{(L,\tilde M)} $ folliates
$ S_{(L,\tilde M)} $ by periodic orbits.
{ \it Proof }
(a) From the form of $ H_{2,res} $ in \equ(5.21)
observe that in order for the coefficients of the terms
$ { a_{(l1,m1)} a_{(l2,m2)} a^*_{(l1,m3)} a^*_{(l2,m4)} } $
to be non-zero, the $ m_i $ must satisfy the relation
$$ m_1 + m_2 - m_3 - m_4 = 0. \EQ(5.22) $$
To show that
the $ T_{(L,M)} $ are invariant, first $ \dot a_{(l,m)} = 0 $
if $ l \neq L $; also for $ m \neq M $ we have
$ \dot a_{(L,m)} = i C(M,M,m,M) a_{(L,M)} a_{(L,M)} a^*_{(L,M)} $
with the coefficient $ C(M,M,m,M) $ given by \equ(5.21).
But $ C(M,M,m,M) \neq 0 $ and \equ(5.22) imply $ m = M $.
Thus the $ T_{(L,M)} $ are invariant and
$$ \dot a_{(L,M)} = i\omega_L a_{(L,M)} +
iC(M,M,M,M) a_{(L,M)} a_{(L,M)} a^*_{(L,M)} $$
This equation is a Hamiltonian system
generated by
$$ H_{L,M} = \omega_L a_{(L,M)} a^*_{(L,M)}
+ C_M a_{(L,M)} a_{(L,M)} a^*_{(L,M)} a^*_{(L,M)} $$
with $ C_M = C(M,M,M,M) $. The Hamiltonian
$ H_{L,M} $ depends only on the ``action''
$ J_{L,M} = a_{(L,M)} a^*_{(L,M)} $.
By a canonical transformation to the variables
$ J_{L,M} $ and $ \theta_{L,M} $ (`` angle '')
the equation becomes
$$ \dot J_{L,M} = 0 \quad ,
\quad \dot \theta_{L,M} = \omega_L + C_M J_{L,M}.$$
The solutions of this equation are all periodic or quasiperiodic and
correspond to travelling waves -- or non-linear superposition of
them -- with amplitude dependent
phase velocity. (b) Similarly, for
$ a_{(l,m)} \in S_{(L,\tilde M)} $,
$ \dot a_{(L,\pm M)} $ has terms of the form
$ C(\tilde M,\tilde M,m, \tilde M) a_{(L,M)} a_{(L,M)} a^*_{(L,M)} $
with $ \tilde M = \pm M $. From \equ(5.22) the
coefficients $ C(\tilde M,\tilde M,m, \tilde M) $ vanish
unless $ m = \pm M $ or $ m = \pm3M $. Thus for $ 3|M| > L $
the subspaces $ S_{(L,\tilde M)} $ is invariant. On $ S_{(L,\tilde M)} $
the equation of motion for the restricted flow is
generated by the Hamiltonian
$$ H_{L,\tilde M} = i \omega_L ( a^*_{(L,M)} a^*_{(L,M)}
+ a^*_{(L,-M)} a^*_{(L,-M)} )
+ C_M J_{L,M} J_{L,M} $$
$$ + 2C_{M,-M} J_{L,M} J_{L,-M} + C_{-M} J_{L,-M} J_{L,-M} $$
with $J_{L,\pm M } = a_{(L,\pm M)} a^*_{(L,\pm M)} $,
$ C_{M,-M} = C(M,-M,M,-M) $ and so on.
Using ``action - angle'' variables
we again have
$$ \dot J_{L,\pm M} = 0 \quad , \quad
\dot \theta_{L,\pm M} = \omega_L + 2 C_M J_{L,M} + 2 C_{-M} J_{L,-M}. $$
These periodic orbits are superpositions of the previous travelling waves
and if the amplitudes $ J_{L,\pm M} $ are equal the result is
standing waves. \QED
\SECTION Acknowledgements
It is a pleasure to thank W. Craig for getting us interested in this
work and supplying many references and correspondence.
This work has been supported by NSF and TARP grants. R.L. has also
been supported by the AMS Centennial Fellowship and an URI grant
of the University of Texas.
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\end