Existence and uniqueness of solutions for a quasilinear KdV equation with degenerate dispersion

TL;DR

This paper proves the existence and uniqueness of solutions for a quasilinear KdV equation with degenerate dispersion, which models phenomena like sedimentation and shallow water waves, for smooth localized initial data.

Contribution

It establishes the first rigorous proof of existence and uniqueness for solutions to a quasilinear KdV with degenerate dispersion.

Findings

Solutions exist and are unique for smooth localized initial data.

The model captures physical phenomena such as sedimentation and magma dynamics.

The analysis advances understanding of degenerate dispersive equations.

Abstract

We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data.