348 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
The three wave interaction. The quadratic resonant three wave equations
are the following ode's on IC^3 :
dq1. -
dt = ZS1/1q2q3
dq2.
dt = is212q1q3
dq 3. -
dt = ZS3/3q1q2.Here, q 1 ,q 2 ,q 3 E IC, i =A, the overbar means complex conjugate, and 11,/2
and 13 are nonzero real numbers with 11 +12 + /3 = 0. The choice (s1, s2, s3) =
(1,1,-1); gives the decay interaction, while (s 1 ,s 2 ,s3) = (-1,1,1) gives the
explosive interaction.
Resonant wave interactions describe energy exchange among nonlinear modes in
contexts involving nonlinear waves (the Benjamin-Feir instability, etc. ) in fluid me-
chanics, plasma physics and other areas. There are other versions of the equations
in which coupling associated with phase modulations appears through linear and
cubic terms. Much of our motivation comes from nonlinear optics (optical trans-
mission and switching). The three wave equations are discussed in, for example,
Whitham [1974] and its dynamical systems aspects are explored in Guckenheimer
and Mahalov [1992].
The methods we develop work rather generally for resonances-the rigid body
is well known to be intimately connected with the 1 :1 resonance (see, for example,
Cushman and Rod [1982], Churchill, Kummer and Rod [1983]). The three wave
interaction has an interesting Hamiltonian and integrable structure. We shall use
a standard Hamiltonian structure and the technique of invariants to understand
it. The decay system is Lie-Poisson for the Lie algebra .su(3) - this is the one
of notable interest for phases (the explosive case is associated with .su(2, 1)). This
is related to the Lax representation of the equations-the n-wave interaction is
likewise related to .su(n). The general picture developed is useful for many other
purposes, such as polarization control (building on work of David, Holm and Tratnik
[1989] and David and Holm [1990]) and perturbations of Hamiltonian normal forms
(see Kirk, Marsden and Silber [1996]).
The canonical Hamiltonian structure. We describe how the three wave sys-
tem is Hamiltonian relative to a canonical Poisson bracket. We choose (primarily a
matter of convenience) a Ii-weighted canonical bracket on IC^3. This bracket has the
real and imaginary parts of each complex dynamical variable qi as conjugate vari-
ables. Correspondingly, we will use a cubic Hamiltonian. The scaled canonical
Poisson bracket on IC^3 may be written in complex notation as
3
(aF ac ac aF)
{F, G} = -2i L sk{k 8 k ~ - 8 k ~.
k=l q qk q qkThe corresponding symplectic structure can be written