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arXiv · 2609.05697

Global asymptotic stability of KdV-Burgers fronts in a weakly two-dimensional model

Jared C. Bronski·Olivia Clifton·Vera Mikyoung Hur
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Abstract

We study front-type solutions of nonlinear dispersive-dissipative PDEs modeling the propagation of undular bores in a channel in two dimensions. The system extends the Korteweg-de Vries-Burgers (KdVB) equation by incorporating weak transverse motion, and admits the KdVB fronts as one-dimensional solutions. We investigate their stability under general two-dimensional perturbations. We prove that a one-dimensional front is a global asymptotic attractor in the weakly two-dimensional setting when the channel is sufficiently narrow in the transverse direction and the relative dispersion parameter lies in a range for stability to one-dimensional perturbations. Particularly, the front is unique up to spatial translations. The proof extends the energy method for temporally-modulated perturbed solutions, developed previously in the one-dimensional setting, to accommodate the transverse dynamics.

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Jared C. Bronski, Olivia Clifton, Vera Mikyoung Hur. 2026-09-04. Global asymptotic stability of KdV-Burgers fronts in a weakly two-dimensional model. https://arxiv.org/abs/2609.05697

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