Dispersive blow up for nonlinear Schroedinger equations revisited

TL;DR

This paper revisits the phenomenon of dispersive blow up in nonlinear Schrödinger equations, extending existing results to higher dimensions, more general equations, and nonlinearities, with implications for rogue wave phenomena.

Contribution

It broadens the theory of dispersive blow up to include higher dimensions, additional equations, and nonlinearities, providing new mathematical insights and estimates.

Findings

Dispersive blow up occurs in higher-dimensional nonlinear Schrödinger equations.

The theory is extended to Davey-Stewartson and Gross-Pitaevskii equations.

A sharp global smoothing estimate for the Duhamel integral term is established.

Abstract

The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the possibility of dispersive blow up for nonlinear Schroedinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey-Stewartson and Gross-Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schroedinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.