Blowup and solitary wave solutions with ring profiles of two-component nonlinear Schrödinger systems

Blowup ring profiles have been investigated by finding non-vortex blowup solutions of nonlinear Schrödinger equations (NLSE) (cf. [5] and [6]). However, those solutions have infinite L norm so one may not maintain the ring profile all the way up to the singularity. To find H non-vortex blowup solutions with ring profiles, we study blowup 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 H blowup 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.