Self-Similar Solutions for Nonlinear Schrodinger Equations

We study the self-similar solutions for nonlinear Schrodinger type equations of higher order with nonlinear term vertical bar u vertical bar(alpha)u by a scaling technique and the contractive mapping method. For some admissible value alpha, we establish the global well-posedness of the Cauchy problem for nonlinear Schrodinger 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 Schrodinger equations with small data admit a class of self-similar solutions. Copyright (C) 2009 Yaojun Ye.