  A weakly nonlinear fluid wave propagating within a star can be unstable to
three-wave interactions. The resonant parametric instability is a well-known
form of three-wave interaction in which a primary wave of frequency $\omega_a$
excites a pair of secondary waves of frequency $\omega_b+\omega_c \simeq
\omega_a$. Here we consider a nonresonant form of three-wave interaction in
which a low-frequency primary wave excites a high-frequency p-mode and a
low-frequency g-mode such that $\omega_b+\omega_c >>\omega_a$. We show that a
p-mode can couple so strongly to a g-mode of similar radial wavelength that
this type of interaction is unstable even if the primary wave amplitude is
small. As an application, we analyze the stability of the tide in coalescing
neutron star binaries to p-g mode coupling. We find that the equilibrium tide
and dynamical tide are both p-g unstable at gravitational wave frequencies
f_{gw} > 20 Hz and drive p-g pairs to significant energies on very short
timescales (much less than the orbital decay time). Resonant parametric
coupling to the tide is, by contrast, either stable or drives modes at a much
smaller rate. We do not solve for the saturation of the instability and
therefore cannot say precisely how it influences neutron star binaries.
However, we show that if even a single daughter mode saturates near its wave
breaking amplitude, the p-g instability of the equilibrium tide: (i) induces
significant orbital phase errors ($\Delta \phi$ > 1 radian) that accumulate
primarily at low frequencies (f_{gw} < 50 Hz) and (ii) heats the neutron star
core to T~10^{10} K. Since there are >100 unstable daughters, $\Delta \phi$ and
T are potentially much larger than these values. Tides might therefore
significantly influence the gravitational wave signal and electromagnetic
emission from neutron star binaries at much larger orbital separations than
previously thought.
