The propagation of regularity and dispersive blow-up phenomenon to higher-order generalized KdV equations

TL;DR

This paper studies how regularity propagates and dispersive blow-up occurs in higher-order generalized KdV equations, revealing infinite speed regularity transfer and singularity formation at rational points.

Contribution

It demonstrates the propagation of regularity across the real line and constructs initial data leading to dispersive blow-up, highlighting the linear part's role in singularity formation.

Findings

Regularity propagates from right to left with infinite speed.

Dispersive blow-up occurs at rational points while solutions remain smooth at irrational times.

Blow-up is caused by the linear component due to wave focusing.

Abstract

Some special properties of smoothness and singularity concerning to the initial value problem associated with higher-order generalized KdV equations are investigated. On one hand, we show the propagation of regularity phenomena. More precisely, the regularity of initial data on the right-hand side of the real line is propagated to the left-hand side with infinite speed under the higher-order KdV flow. On the other hand, we show that the dispersive blow-up phenomenon will occur by constructing a class of smoothing initial data such that global solutions with the given initial data keep smooth at positive generic irrational times, while global solutions display singularity at each time-space positive rational point. The blow-up phenomenon is exclusively caused by the linear part of solutions due to the focusing of short or long waves.