Self-Similar Solutions for Nonlinear Schrödinger Equations

We study the self-similar solutions for nonlinear Schrödinger type equations of higher order with nonlinear term|u|αuby a scaling technique and the contractive mapping method. For some admissible valueα, 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.