Hydrodynamic approach to constructing solutions of Hydrodynamic approach to constructing solutions of nonlinear Schr\"odinger equation in the critical case

TL;DR

This paper develops a hydrodynamic method to construct exact solutions of the nonlinear Schrödinger equation, capturing phenomena like collapse and scattering, and generalizing existing blow-up solutions.

Contribution

It introduces a hydrodynamic framework to generate solutions that describe both collapse and scattering in the critical nonlinear Schrödinger equation, extending known solutions.

Findings

Constructed solutions exhibit finite-time collapse and infinite-time scattering.

Generalized known blow-up solutions based on the ground state.

Provided a new approach to analyze nonlinear Schrödinger dynamics.

Abstract

Proceeding from the hydrodynamic approach, we construct exact solutions to nonlinear Schr\"odinger equation with special properties. The solutions describe collapse, in finite time, and scattering, over infinite time, of wave packets. They generalize known blow-up solutions based on the "ground state."