We carry out a systematic numerical stability analysis of ZND detonations of
Majda's model with Arrhenius-type ignition function, a simplified model for
reacting flow, as heat release and activation energy are varied. Our purpose
is, first, to answer a question of Majda whether oscillatory instabilities can
occur for high activation energies as in the full reacting Euler equations,
and, second, to test the efficiency of various versions of a numerical
eigenvalue-finding scheme suggested by Humpherys and Zumbrun against the
standard method of Lee and Stewart.

We establish one-dimensional spectral, or "normal modes", stability of ZND
detonations in the small heat release limit and the related high overdrive
limit with heat release and activation energy held fixed, verifying numerical
observations of Erpenbeck in the 1960s. The key technical points are a
strategic rescaling of parameters converting the infinite overdrive limit to a
finite, regular perturbation problem, and a careful high-frequency analysis
depending uniformly on model parameters.

In this paper, we complement recent results of Bronski and Johnson and of
Johnson and Zumbrun concerning the modulational stability of spatially periodic
traveling wave solutions of the generalized Korteweg-de Vries equation. In this
previous work it was shown by rigorous Evans function calculations that the
formal slow modulation approximation resulting in the Whitham system accurately
describes the spectral stability to long wavelength perturbations.

Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that
spectral stability implies linearized and nonlinear stability of spatially
periodic traveling-wave solutions of viscous systems of conservation laws for
systems of generic type, removing a restrictive assumption that wave speed be
constant to first order along the manifold of nearby periodic solutions.

In this paper, we consider the spectral stability of spatially periodic
traveling wave solutions of the generalized Korteweg-de Vries equation to
long-wavelength perturbations. Specifically, we extend the work of Bronski and
Johnson by demonstrating that the homogonized system describing the mean
behavior of a slow modulation (WKB) approximation of the solution correctly
describes the linearized dispersion relation near zero frequency of the
linearized equations about the background periodic wave.

We illustrate in a simple setting the instantaneous shock tracking approach
to stability of viscous conservation laws introduced by Howard, Mascia, and
Zumbrun. This involves a choice of the definition of instanteous location of a
viscous shock-- we show that this choice is time-asymptotically equivalent both
to the natural choice of least-squares fit pointed out by Goodman and to a
simple phase condition used by Gu\`es, M\'etivier, Williams, and Zumbrun in
other contexts. More generally, we show that it is asymptotically equivalent to
any location defined by a localized projection

In this paper, we investigate the spectral instability of periodic traveling
wave solutions of the generalized Korteweg-de Vries equation to long wavelength
transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By
analyzing high and low frequency limits of the appropriate periodic Evans
function, we derive an orientation index which yields sufficient conditions for
such an instability to occur. This index is geometric in nature and applies to
arbitrary periodic traveling waves with minor smoothness and convexity
assumptions on the nonlinearity.