Asymptotic Stability of Ascending Solitary Magma Waves

TL;DR

This paper proves the asymptotic stability of solitary magma waves in geophysical models, extending stability results to a broader class of equations without relying on traditional mathematical structures.

Contribution

It establishes the asymptotic stability of solitary magma waves in a novel geophysical context, even without inverse scattering or variational frameworks.

Findings

Solitary magma waves are asymptotically stable in the studied models.

Stability holds for a family of equations beyond physical parameters.

Results extend well-posedness near solitary wave solutions.

Abstract

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.