On the regularity of solutions to the $k$-generalized Korteweg-de Vries equation

TL;DR

This paper investigates the regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation, extending previous results from integer to positive real Sobolev space indices.

Contribution

It generalizes existing regularity results for solutions to the $k$-gKdV equation from integer to positive real Sobolev spaces.

Findings

Solutions gain regularity in the spatial variable over time.

Extension of regularity results to non-integer Sobolev spaces.

Provides a broader understanding of solution smoothness for the $k$-gKdV equation.

Abstract

This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In \cite{IsazaLinaresPonce} it was established that if the initial datun $u_0\in H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$, then the corresponding solution $u(\cdot,t)$ belongs to $H^l((\beta,\infty))$ for any $\beta \in \mathbb R$ and any $t\in (0,T)$. Our goal here is to extend this result to the case where $\,l\in \mathbb R^+$.