发表机构
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究弱二维模型中KdV-Burgers方程前沿解的全局渐近稳定性，通过扩展能量方法证明在窄渠道及特定色散参数条件下，一维前沿为全局吸引子且唯一。
AI 中文摘要
我们研究了非线性色散-耗散偏微分方程的前沿型解，这些方程模拟了二维渠道中起伏水跃的传播。该系统通过引入弱横向运动扩展了Korteweg-de Vries-Burgers (KdVB)方程，并将KdVB前沿作为一维解。我们研究了它们在一般二维扰动下的稳定性。我们证明，当渠道在横向方向上足够窄且相对色散参数处于一维扰动稳定的范围内时，一维前沿在弱二维设置中是全局渐近吸引子。特别地，该前沿在空间平移意义下是唯一的。证明将先前在一维设置中发展的用于时间调制扰动解的能量方法扩展到适应横向动力学。
英文摘要
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.
Comments24 pages, 1 figure
