Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 298980, 15 pages
Research Article

Self-Similar Solutions for Nonlinear Schrödinger Equations

Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received 19 March 2009; Accepted 22 August 2009

Academic Editor: Ben T. Nohara

Copyright © 2009 Yaojun Ye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the self-similar solutions for nonlinear Schrödinger type equations of higher order with nonlinear term |u|αu by a scaling technique and the contractive mapping method. For some admissible value α, we establish the global well-posedness of the Cauchy problem for nonlinear Schrödinger equations of higher order in some nonstandard function spaces which contain many homogeneous functions. we do this by establishing some nonlinear estimates in the Lorentz spaces or Besov spaces. These new global solutions to nonlinear Schrödinger equations with small data admit a class of self-similar solutions.