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

Abstract

Blowup ring profiles have been investigated by finding non-vortex blowup solutions of nonlinear Schrödinger equations (NLSEs) (cf. Fibich et al. (2005) [7] and Fibich et al. (2007) [8]). However, those solutions have infinite L 2 norm, so one may not maintain the ring profile all the way up to the singularity. To find H 1 non-vortex blowup solutions with ring profiles, we study the blowup solutions of two-component systems of NLSEs with nonlinear coefficients β and ν j , j = 1 , 2 . When β < 0 and ν 1 ≫ ν 2 > 0 , the two-component system can be transformed into a multi-scale system with fast and slow variables which may produce H 1 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.