Similarity Solutions of the Cubic Schroedinger Equation and Their Role in the Development of Wave-Collapse

摘要

Similarity transformations of the Cubic Schroedinger Equation (CSE) are investigated. The similarity transformations are used to remove the explicit time variation in the CSE and transform it into differential equations in the spatial variables only. Different methods for similarity reduction are employed and compared. The main purpose of this investigation is to study the significance of the similarity solutions in the evolution of a collapsing wave packet. Numerical solutions of the CSE in radial symmetry demonstrate that the similarity behaviour is local in space and time, and the similarity solutions are classified by invoking the concept of proper and improper solutions. The nature of the collapsing singularity is reexamined and finally, soliton solutions to the CSE are considered. (Atomindex citation 15:070622)