2008/01/03 by Gideon Simpson, Michael I. Weinstein, Simpson, Gideon +1 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.0801.0463
openalex publication_date 2008/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Coherent structures, such as solitary waves, appear in many physical problems, including fluid mechanics, optics, quantum physics, and plasma physics. A less studied setting is found in geophysics, where highly viscous fluids couple to evolving material parameters to model partially molten rock, magma, in the Earth's interior. Solitary waves are also found here, but the equations lack useful mathematical structures such as an inverse scattering transform or even a variational formulation. A common question in all of these applications is whether or not these structures are stable to perturbation. We prove that the solitary waves in this Earth science setting are asymptotically stable and accomplish this without any pre-exisiting Lyapunov stability. This holds true for a family of equations, extending beyond the physical parameter space. Furthermore, this extends existing results on well-posedness to data in a neighborhood of the solitary waves.