Amplification of pulses in nonlinear geometric optics

Abstract

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.