The global bifurcation of periodic internal waves with point vortex and capillary effect

In this paper, we first construct two-dimensional periodic interface waves with point vortex and capillary effect and then obtain the global structure of the set of solutions. This is done using the local and global bifurcation argument. Especially, we establish a global continuation theorem by using the degree for $C^1$ Fredholm mappings. As far as we know, the global result is new and general, which is promising to deal with other new phenomena in water waves.