Classification of minimal-mass blow-up solutions on one edge

Classify every minimal-mass solution of the focusing quintic nonlinear Schrödinger equation on the two-edge star graph with delta interaction that blows up on one edge, determining whether it converges, up to a constant phase, to one of the two-parameter profiles in the terminal family and whether the associated concentration scale and renormalized center are unique.

Background

The paper constructs asymmetric minimal-mass finite-time blow-up solutions for the focusing quintic nonlinear Schrödinger equation on a star graph with a delta interaction. The blow-up is localized at an interior point of one edge, and the construction yields a two-parameter family indexed by a concentration scale and a renormalized center; distinct parameter values produce distinct solutions modulo constant phase.

The authors explicitly identify the classification of all minimal-mass solutions that blow up on one edge as unresolved. The proposed classification would establish whether every such solution is asymptotic, modulo constant phase, to one of the constructed terminal profiles and would also settle uniqueness of the scale and renormalized-center parameters. The paper mentions rigidity arguments for inhomogeneous mass-critical NLS without exact pseudoconformal symmetry as a possible strategy.