Propagation of regularity and decay of solutions to the $k$-generalized Korteweg-de Vries equation

TL;DR

This paper investigates how regularity and decay properties of solutions to the $k$-generalized KdV equation propagate over time, showing that certain regularity features spread infinitely fast to the left as the solution evolves.

Contribution

It proves that regularity in the initial data propagates infinitely fast to the left for solutions of the $k$-generalized KdV equation, extending previous understanding of solution behavior.

Findings

Regularity propagates infinitely fast to the left.

Solutions gain regularity in regions where initial data has regularity.

Decay properties are preserved and propagated over time.

Abstract

We study special regularity and decay properties of solutions to the IVP associated to the $k$-generalized KdV equations. In particular, for datum $u_0\in H^{3/4^+}(\mathbb R)$ whose restriction belongs to $H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$ we prove that the restriction of the corresponding solution $u(\cdot,t)$ belongs to $H^l((\beta,\infty))$ for any $\beta \in \mathbb R$ and any $t\in (0,T)$. Thus, this type of regularity propagates with infinite speed to its left as time evolves.