Interrelation between various branches of stable solitons in dissipative systems - Conjecture for stability criterion

We show that the complex cubic-quintic Ginzburg-Landau equation has a multiplicity of soliton solutions for the same set of equation parameters. They can either be stable or unstable. We show that the branches of stable solitons can be interrelated, i.e. stable solitons of one branch can be transformed into stable solitons of another branch when the parameters of the system are changed. This connection occurs via some branches of unstable solutions. The transformation occurs at the points of bifurcation. Based on these results, we propose a conjecture for a stability criterion for solitons in dissipative systems.