In this paper we study self-similar solutions for nonlinear Schrodinger equations using a scaling technique and the partly contractive mapping method. We establish the small global well-posedness of the Cauchy problem for nonlinear Schrodinger equations in some non-reflexive Banach spaces which contain many homogeneous functions. This we do by establishing some a priori nonlinear estimates in Besov spaces, employing the mean difference characterization and multiplication in Besov spaces. These new global solutions to nonlinear Schrodinger equations with small data admit a class of self-similar solutions. Our results improve and extend the well-known results of Planchon , Cazenave and Weissler [4, 5] and Ribaud and Youssfi .
Changxing Miao. Bo Zhang. Xiaoyi Zhang. "Self-Similar Solutions for Nonlinear Schrodinger Equations." Methods Appl. Anal. 10 (1) 119 - 136, March 2003.