Under construction.

In this paper, we investigate the stability and nonlinear local dynamics of spectrally-stable wave trains in reaction-diffusion systems. For each \(N\in\mathbb{N}\), such \(T\)-periodic traveling waves are easily seen to be nonlinearly asymptotically stable (with asymptotic phase) with exponential rates of decay when subject to \(NT\)-periodic, i.e., subharmonic, perturbations. However, both the allowable size of perturbations and the exponential rates of decay depend on \(N\); in particular, they tend to zero as \(N\to\infty\), leading to a degeneracy in such subharmonic stability results. In this work, we build on recent work by the authors (in particular, the paper on Lugiato-Lefever periodic waves below) and introduce a methodology by which a stability result for subharmonic perturbations which is uniform in \(N\) may be achieved at the nonlinear level. Our work is motivated by the dynamics of such waves when subject to perturbations which are localized, i.e., integrable on the line, which has recently received considerable attention by many authors.

In this paper, we study the linear dynamics of spectrally stable \(T\)-periodic stationary wave solutions of the Lugiato-Lefever equation (LLE), which is a damped, forced NLS-type equation that is widely used to investigate the dynamical properties of laser fields confined in nonlinear optical resonators. It has been established that such \(T\)-periodic solutions are nonlinearly stable to \(NT\)-periodic, i.e., subharmonic, perturbations for each \(N\in\mathbb{N}\) and that these perturbations experience exponential rates of decay of the form \(e^{-\delta_N t}\). Unfortunately, both the exponential rates of decay \(\delta_N\) and the allowable size of initial perturbations tend to \(0\) as \(N\to\infty\) so that this result is empty in the limit \(N=\infty\).

We introduce a methodology, in the context of the LLE, by which a uniform (in \(N\)) stability result may be achieved, at least at the linear level, thereby circumventing the degeneracy of the decay rates and the allowable size of initial perturbations. The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e., integrable on the line, perturbations. In this work, we unify and expand on several existing works concerning the stability and dynamics of such waves, and we set forth a general methodology for studying such problems in other contexts.

In this paper, we use rigorous spectral perturbation theory to establish the validity of the predictions made by Whitham's theory of wave modulations in the context of the conduit equation, a nonlinear dispersive PDE governing the evolution of the circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. In particular, we rigorously demonstrate that the Whitham modulation equations may be used to obtain a sufficient condition for the instability of periodic wave trains to modulational perturbations and a necessary condition for stability.

Under construction...

In this paper, we investigate how circles may be determined in \(\mathbb{F}_q^2\), the two-dimensional vector space over the finite field \(\mathbb{F}_q\). This investigation is complicated by the fact that two distinct points may have zero distance between them. Nonetheless, we demonstrate that a circle of nonzero radius may be uniquely determined by three distinct, noncollinear points, which pairwise have nonzero distance between them.