Sufficient and necessary conditions for self-similar motions of three point vortices in generalized fluid systems

In this paper, we obtained sufficient and necessary conditions for self-similar motions of three point vortices in generalized two-dimensional fluid systems by reducing of Hamiltonian and developing the theory of self-similar functions. The results are concise and clear, and the occurrence of both self-similar collapse and self-similar expansion only depend on the strengths of point vortices. We have provided an explicit and exact expression for each nontrivial self-similar solution. The application of theoretical results in Surface-Quasi-Geostrophic system is consistent with previous known numerical results in the symmetric case. We also have found that, in some asymmetric cases, both self-similar collapse and self-similar expansion occur if and only if the strength of the third point vortex is within an open interval.