Kink stability of self-similar solutions of scalar field in 2 + 1 gravity

The kink stability of self-similar solutions of a massless scalar field with circular symmetry in 2 + 1 gravity is studied, and found that such solutions are unstable against the kink perturbations along the sonic line (self-similar horizon). However, when perturbations outside the sonic line are considered, and taking the ones along the sonic line as their boundary conditions, we find that non-trivial perturbations do not exist. In other words, the consideration of perturbations in the whole spacetime limits the unstable mode of the perturbations found along the sonic line, and the kink instability rises because of the incomplete treatment of the problem. As a result, the critical solution for the scalar collapse remains critical even after the kink perturbations are taken into account.