arXiv subjects

Joannis Alexopoulos

Publications and source records attributed to Joannis Alexopoulos.

4 recordsLinked to original sources

arXiv2025

Nonlinear dynamics of reaction-diffusion wave trains under large and fully nonlocalized modulations

We study the dynamics of periodic wave trains in reaction-diffusion systems on the real line under large, fully nonlocalized modulations. We prove that solutions with nearby initial data converge, at an enhanced diffusive rate, to a modulated wave train whose leading-order phase and wavenumber dynamics are governed by an explicit solution to the viscous Hamilton-Jacobi equation. This constitutes a global stability result: such initial data are generally not close to the large-time modulated wave train. In contrast to previous modulational stability results, our analysis does not require that the initial data approach phase shifts of the wave train at spatial infinity. The central methodological advance is a nontrivial extension of the recently developed $L^\infty$-stability theory to accommodate large phase modulations. This framework, based entirely on $L^\infty$-estimates, removes all localization requirements as imposed in the previous literature, allowing us to treat the full range of bounded modulational initial data under minimal regularity assumptions. The main technical contributions include: the strategic use of interpolation inequalities to balance smallness and temporal decay, and a detailed analysis of the linear dynamics under fully nonlocalized modulational data.

math.AP↗

arXiv2025

Nonlinear dynamics of periodic Lugiato-Lefever waves against sums of co-periodic and localized perturbations

In recent years, essential progress has been made in the nonlinear stability analysis of periodic Lugiato-Lefever waves against co-periodic and localized perturbations. Inspired by considerations from fiber optics, we introduce a novel iteration scheme which allows to perturb against sums of co-periodic and localized functions. This unifies previous stability theories in a natural manner.

math.AP↗

arXiv2025

Uniformity in the Fourier inversion formula with applications to Laplace transforms

We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known results for Banach space-valued functions and, specifically, for C_0-semigroups.

math.FA↗

arXiv2024

Nonlinear stability of periodic wave trains in the FitzHugh-Nagumo system against fully nonlocalized perturbations

Recently, a nonlinear stability theory has been developed for wave trains in reaction-diffusion systems relying on pure $L^\infty$-estimates. In the absence of localization of perturbations, it exploits diffusive decay caused by smoothing together with spatio-temporal phase modulation. In this paper, we advance this theory beyond the parabolic setting and propose a scheme designed for general dissipative semilinear problems. We present our method in the context of the FitzHugh-Nagumo system. The lack of parabolicity and localization complicates mode filtration in $L^\infty$-spaces using the Floquet-Bloch transform. Instead, we employ the inverse Laplace representation of the semigroup generated by the linearization to uncover high-frequency damping, while leveraging a novel link to the Floquet-Bloch representation for the smoothing low-frequency part. Another challenge arises in controlling regularity in the quasilinear iteration scheme for the modulated perturbation. We address this by extending the method of nonlinear damping estimates to nonlocalized perturbations using uniformly local Sobolev norms.

math.AP↗