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PreprintCC BY 4.0arXiv

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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Cite

BibTeXRIS
BibTeX
@misc{bronski2026global,
  title = {{Global asymptotic stability of KdV-Burgers fronts in a weakly two-dimensional model}},
  author = {Jared C. Bronski and Olivia Clifton and Vera Mikyoung Hur},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05697},
  eprint = {2609.05697},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05697},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Global asymptotic stability of KdV-Burgers fronts in a weakly two-dimensional model
AU  - Jared C. Bronski
AU  - Olivia Clifton
AU  - Vera Mikyoung Hur
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05697
UR  - https://arxiv.org/abs/2609.05697
AB  - 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.
ER  -