On dispersive blow-ups for the nonlinear Schr\"odinger equation

TL;DR

This paper introduces a simplified method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation, leveraging dispersive estimates to reduce regularity requirements and expand examples of blow-up phenomena.

Contribution

It presents a new, simplified approach using dispersive estimates to construct dispersive blow-ups, including solutions blowing up on lines and spheres, with reduced regularity assumptions.

Findings

Simplified construction method for dispersive blow-ups

Reduced regularity requirements for solutions

Examples of blow-up on lines and spheres

Abstract

In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schr\"odinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schr\"odinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow up on a straight line and on a sphere.