Countable branches of radial self-similar singular solutions for the supercritical CGL equation

Let the supercritical complex Ginzburg–Landau equation be parametrized by $\epsilon$ and $\delta$, with the nonlinear radial self-similar profiles characterized by their monotone index, the number of monotone intervals of the profile. The nonlinear Schrödinger parameter values correspond to $\epsilon=\delta=0$.

Branch-counting conjecture. The supercritical CGL equation has a countable number of branches of nontrivial radial self-similar singular solutions. These branches can be labeled by the monotone index of their profiles and are born at the NLS parameter values

$$\epsilon=\delta=0.$$

This conjecture is motivated by numerical results indicating countably many multi-bump solutions for the NLS boundary value problem and branches that persist as the CGL parameter varies. The supplied passage does not report a proof or disproof of the conjecture.