# Dispersive blow-up and persistence properties for the Schr\"odinger-Korteweg-de Vries system

## TL;DR

This paper investigates dispersive blow-up phenomena in the Schr"odinger-Korteweg-de Vries system, demonstrating the development of point singularities and establishing smoothing and persistence properties for solutions.

## Contribution

First analysis of dispersive blow-up in a nonlinear dispersive system, introducing new smoothing and persistence results for the Schr"odinger-Korteweg-de Vries system.

## Key findings

- Dispersive blow-up can occur due to linear dispersive effects.
- Duhamel term is smoother, preventing blow-up in certain components.
- Persistence properties are established in fractional weighted Sobolev spaces.

## Abstract

We study the dispersive blow-up phenomena for the Schr\"odinger-Korteweg-de Vries (S-KdV) system. Roughly, dispersive blow-up has being called to the development of point singularities due to the focussing of short or long waves. In mathematical terms, we show that the existence of this kind of singularities is provided by the linear dispersive solution by proving that the Duhamel term is smoother. It seems that this result is the first regarding systems of nonlinear dispersive equations. To obtain our results we use, in addition to smoothing properties, persistence properties for solutions of the IVP in fractional weighted Sobolev spaces which we establish here.