[1] Laboratory assays of methanotroph activity in upland (i.e., well-drained, oxic) ecosystems alter soil physical structure and weaken inference about environmental controls of their natural behavior. To overcome these limitations, we developed a chamber-based approach to quantify methanotroph activity in situ on the basis of measures of soil diffusivity (from additions of an inert tracer gas to the chamber headspace), methane concentration change, and analysis of results with a reaction-diffusion model. The analytic solution to this model predicts that methane consumption rates are equally sensitive to changes in methanotroph activity and diffusivity, but that doubling either of these parameters leads to only a p 2 increase in consumption. With a series of simulations, we generate guidelines for field deployments and show that the approach is robust to plausible departures from assumptions. We applied the approach on a dry grassland in north central Colorado. Our model closely fit measured changes in methane concentrations, indicating that we had accurately characterized the biophysical processes underlying methane uptake. Field patterns showed that, over a 7-week period, soil moisture fell from 38% to 15% water-filled pore spaces, and diffusivity doubled as the larger soil pores drained of water. However, methane uptake rates fell by $40%, following a 90% decrease in methanotroph activity, suggesting that the decline in methanotroph activity resulted from water stress to methanotrophs. We anticipate that future application of this approach over longer timescales and on more diverse field sites has potential to provide important insights into the ecology of methanotrophs in upland soils.

An inverse problem for the identification of an unknown coefficient in a quasilinear parabolic partial differential equation is considered. We present an approach based on utilizing adjoint versions of the direct problem in order to derive equations explicitly relating changes in inputs (coefficients) to changes in outputs (measured data). Using these equations it is possible to show that the coefficient to data mappings are continuous, strictly monotone and injective. The equations are further exploited to construct an approximate solution to the inverse problem and to analyze the error in the approximation. Finally, results of some numerical experiments are displayed.

The two-dimensional cubic nonlinear Schrödinger equation admits a large family of onedimensional bounded traveling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant phase have been well studied previously. Some of these solutions were found to be stable with respect to one-dimensional perturbations. No such solutions are stable with respect to two-dimensional perturbations.Here we consider stability of the larger class of solutions whose phase is dependent on the spatial dimension of the one-dimensional wave form. We study the spectral stability of such nontrivialphase solutions numerically, using Hill's method. We present evidence which suggests that all such nontrivial-phase solutions are unstable with respect to both one-and two-dimensional perturbations. Instability occurs in all cases: for both the elliptic and hyperbolic nonlinear Schrödinger equations, and in the focusing and defocusing case.

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