Self-similar and Solitary Wave Solutions with Ring Profiles of Two-component Nonlinear Schrödinger Systems

Blowup ring profiles have been investigated by finding self-similar non-vortex solutions of nonlinear Schrödinger equations (NLSE) (cf. [4] and [5]). However, those solutions have infinite L norm so one may not maintain the ring profile all the way up to the singularity. To find selfsimilar H non-vortex solutions with ring profiles, we study self-similar solutions of two-component systems of NLSE with nonlinear coefficients β and νj , j = 1, 2. When β < 0 and ν1 ν2 > 0, the two-component system can be transformed into a multi-scale system with fast and slow variables which may produce self-similar H solutions with non-vortex ring profiles. We use the localized energy method with symmetry reduction to construct these solutions rigorously. On the other hand, these solutions may describe steady non-vortex bright ring solitons. Various types of ring profiles including m-ring and ring-ring profiles are presented by numerical solutions.