  Some systems of nonlinear wave equations admit global solutions for all
sufficiently small initial data, while others do not. The (classical) null
condition guarantees that such a result holds, but it is too strong to capture
certain systems -- most famously the Einstein equations -- which nevertheless
admit global solutions for small initial data. The weak null condition has been
proposed as a sufficient condition for such a result to hold; it takes the form
of a condition on a related set of nonlinear ODEs known as the "asymptotic
system". Previous results in this direction have required certain structural
conditions on the asymptotic system in addition to the weak null condition. In
this work we show that, if the solutions to the asymptotic system are bounded
(given small initial data), and, in addition, if these solutions are stable
against rapidly decaying perturbations, then the corresponding system of
nonlinear wave equations admits global solutions for all sufficiently small
initial data. This avoids any direct assumptions on the structure of the
nonlinear terms. We also give an example of a class of systems obeying this
condition but not obeying previously identified structural conditions. For this
class the asymptotic system arises as a generalisation of the "Euler equations"
for rigid body motion, associated with a left-invariant Hamiltonian flow on a
finite dimensional Lie group. This work relies heavily on the author's previous
work, "The weak null condition and global existence using the p-weighted energy
method".
