Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2306.13788

Asymptotic study of critical wave fronts for parameter-dependent Born-Infeld models: physically predicted behaviors and new phenomena

Maurizio Garrione
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study some parameter-dependent reaction-diffusion models governed by the Born-Infeld (or Minkowski) operator. In dependence on two parameters $a, b > 0$, related to the field strength and to the diffusivity, we investigate the limit critical speed for traveling fronts, together with the limit behavior of the associated critical profiles. As the two main results, on the one hand we rigorously show that for arbitrarily large electric fields the critical speed and the critical profiles converge to the ones of the linear diffusion problem, agreeing with the well-known physical prediction that, in this case, Born's law and Maxwell's law coincide. Such a result is accompanied by a counterpart for arbitrarily small electric fields and a complete analysis for vanishing/large diffusion. On the other hand, we prove the onset of a new and unexpected phenomenon for the singular perturbation problem: the critical speed of propagation does not converge to zero and, for KPP or combustion-type reactions, the limit front profile becomes sharp on one side only. In fact, this latter coincides with the $C^1$-gluing of a piecewise linear profile having slope equal to 1 when nonconstant, and of a regular inviscid profile propagating at the limit speed. The gluing point and the value of the limit speed are determined explicitly. The outcomes of the whole discussion, based on a careful analysis of the first-order reduction associated with the original equation, show substantial differences and peculiarities of the Born- Infeld operator with respect to linear or saturating diffusions, and are supported by several numerical experiments.

Explore related subjects

math.APmath.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maurizio Garrione. 2023-06-23. Asymptotic study of critical wave fronts for parameter-dependent Born-Infeld models: physically predicted behaviors and new phenomena. https://arxiv.org/abs/2306.13788

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Around the Quantum Lenard-Balescu equation

In the mean-field regime, a gas of quantum particles with Boltzmann statistics can be described by the Hartree-Fock equation. This dynamics becomes trivial if the initial distribution of particle is invariant by translation. However, the first correction is given on time of order $O(N)$ by the quantum Lenard--Balescu equation. In the first part of the present article, we justify this equation until time of order $O((\log N)^{1-δ})$ (for any $δ\in(0,1)$). A similar phenomenon exists in the classical setting (with a similar validity time obtained by Duerinckx \cite{Duerinckx}). In a second time, we prove the convergence for dimension $d\geq 2$ of the solutions of the quantum Lenard--Balescu equation to the solutions of its classical counterpart in the semi-classical limit. This problem can be interpreted as a grazing collision limit: the quantum Lenard--Balescu equation looks like a cut-off Boltzmann equation, when the classical one looks like the Landau equation.

math.AP↗
arXiv2026

Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schrödinger Equation

This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schrödinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency conditions, the Cauchy problem admits a solution that is almost periodic in both space and time, and that this solution is unique among solutions locally bounded in a suitable sense. The analysis combines direct and inverse spectral theory. In the inverse spectral theory part, we prove existence, almost periodicity, and uniqueness for solutions with initial data whose associated Dirac operator has purely a.c.\ spectrum that is not too thin. This resolves novel challenges presented by the NLS hierarchy, such as an additional degree of freedom and an additional commuting flow. In the direct spectral theory part, for Dirac operators with small analytic quasiperiodic potentials with Diophantine frequency conditions, we prove pure a.c.\ spectrum, exponentially decaying spectral gaps, and spectral thickness conditions (homogeneity and Craig-type conditions).

math.AP↗
arXiv2026

Large-Amplitude Steady Solitary Water Waves with General Vorticity

We study two-dimensional steady solitary gravity water waves with general vorticity, allowing for overhanging free-surface profiles. The main challenges arise from the free boundary, the unbounded fluid domain, and the inherent complexity of the general vorticity setting. To address these, we introduce a conformal reformulation that reduces the problem to an equivalent system on a fixed strip, consisting of an overdetermined elliptic problem coupled with an elliptic boundary-value problem. This framework enables a local analysis without imposing restrictive assumptions on the vorticity. Using the center-manifold construction of Chen, Walsh and Wheeler \cite{chennonlinearity}, we first establish the existence of small-amplitude solitary waves for such non-trivial vorticity distributions. Subsequently, via a global analytic bifurcation argument, we prove the existence of continuous branches of large-amplitude solitary waves for analytic vorticity functions. Along these global solution curves, the free surfaces may develop overhanging profiles and need not remain graphs over the horizontal coordinate. Our results provide a new constructive framework for large-amplitude solitary waves with general vorticity, extending the existing theory beyond the classical settings of constant vorticity \cite{susannaarma} and non-overhanging profiles \cite{milesjma}.

math.AP↗