Small data scattering and soliton stability in $\dot{H}^{-\frac16}$ for the quartic KdV Equation

TL;DR

This paper proves scattering for perturbations of solitons in the critical $ ext{dot}H^{-rac{1}{6}}$ space for the quartic KdV equation, using refined linearized estimates without extra regularity assumptions.

Contribution

It introduces a method to establish scattering in the critical space for quartic KdV, improving upon previous work by removing regularity constraints.

Findings

Proves scattering for soliton perturbations in $ ext{dot}H^{-rac{1}{6}}$ space.

Establishes existence of inverse wave operators in the critical space.

Develops refined linearized estimates for the KdV equation.

Abstract

In this note we prove scattering for perturbations of solitons in the scaling space appropriate for the quartic nonlinearity, namely $\dot{H}^{-\frac16}$. The article relies strongly on refined estimates for a KdV equation linearized at the soliton. In contrast to the work of Tao (2006), we are able to work purely in the scaling space without additional regularity assumptions, allowing us to prove some results on the existence of inverse wave operators.