Diffusive stability against nonlocalized perturbations of planar wave trains in reaction–diffusion systems

Abstract

Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in R 2 and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems.

Linear modulational and subharmonic dynamics of spectrally stable Lugiato-Lefever periodic waves

“…Our resolution is to combine the strategies in the works by Johnson et al with a recent method developed by Sandstede & de Rijk in [4] to establish nonlinear stability of periodic traveling waves in planar reaction-diffusion systems against perturbations which are bounded along a line in R 2 and decay in the distance from this line. Since such non-integrable perturbations prohibit the use of L 2 -estimates (and thus, in particular, nonlinear damping estimates), the nonlinear analysis in [4] is based on pointwise estimates and their approach to controlling regularity is to incorporate the "unmodulated perturbation" ṽ(x, t) = ψ(x, t) − φ(x), into the nonlinear iteration scheme, which, in our case, satisfies the semilinear equation obtained by setting γ ≡ 0 in (1.5). While the Duhamel's principle based iteration scheme associated to ṽ does not experience a loss of derivatives, the associated decay rates of ṽ are too slow to close an independent iteration scheme; see Remark 4.3.…”

“…In addition, as mentioned in Remark 1.4, nonlinear damping estimates, which provide control over higher-order derivatives in terms of lower-order derivatives, are unavailable for the LLE. Instead, we address this loss of derivatives by following the approach developed in [4].…”

“…The approach in [4] relies on four crucial observations. The first is that no loss of derivatives arises in the semilinear equations (4.1) and (4.5) for the unmodulated perturbation, as N ( v) does not contain any derivatives of v. The second is that, using the mean value theorem, the derivatives v x and v xx of the modulated perturbation can be bounded in terms of the phase modulation γ, the unmodulated perturbation v and their derivatives.…”

Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems

“…Rearranging slightly as in [1] to remove temporal derivatives of the perturbation v in present in N in (4.3) yields the following.…”

Global existence and decay in nonlinearly coupled reaction-diffusion-advection equations with different velocities

“…This leads to the natural question whether the additional decay can be exploited to relax the localization assumption on the initial data. In the setting of planar traveling waves, it is shown in [11] that one can allow for non-localized perturbations. We expect that these results transfer to the current setting in this paper.…”