  In this companion paper to our study of amplification of wavetrains, we study
weakly stable semilinear hyperbolic boundary value problems with pulse data.
Here weak stability means that exponentially growing modes are absent, but the
so-called uniform Lopatinskii condition fails at some boundary frequency in the
hyperbolic region. As a consequence of this degeneracy there is again an
amplification phenomenon: outgoing pulses of amplitude $O(\eps^2)$ and
wavelength $\eps$ give rise to reflected pulses of amplitude $O(\eps)$, so the
overall solution has amplitude $O(\eps)$. Moreover, the reflecting pulses
emanate from a radiating pulse that propagates in the boundary along a
characteristic of the Lopatinskii determinant. In the case of N*N systems
considered here, a single outgoing pulse produces on reflection a family of
incoming pulses traveling at different group velocities. Unlike wavetrains,
pulses do not interact to produce resonances that affect the leading order
profiles. However, pulse interactions do affect lower order profiles and so
these interactions have to be estimated carefully in the error analysis.
Whereas the error analysis in the wavetrain case dealt with small divisor
problems by approximating periodic profiles by trigonometric polynomials (which
amounts to using a high frequency cutoff), in the pulse case we approximate
decaying profiles with nonzero moments by profiles with zero moments (a low
frequency cutoff). Unlike the wavetrain case, we are now able to obtain a rate
of convergence in the limit describing convergence of approximate to exact
solutions.
