Axisymmetric capillary water waves on cylindrical fluid jets

Abstract: In their paper, Vanden-Broeck, Miloh and Spivack (Wave Motion 27, 1998) describe two limiting behaviours of irrotational, axisymmetric capillary water waves using numerical methods. They found a two-parameter family of nontrivial solutions. Some solutions of small amplitude approach a uniform stream, while others approach a static configuration. Our main interest is with the branch of static configurations, and most importantly, a bifurcating curve which occurs at each point of the static branch. The static branch begins at a cylinder-like solution and then smoothly varies with wave steepness until it terminates at a solution corresponding to spheres (droplets). In this work, we wish to show analytically that for each point of the static branch, there exists a connecting curve of non-static solutions. We do this by first identifying the static configuration, and then performing a continuation analysis at an arbitrary point of the static branch. Our methods include nonlinear functional analysis, as well as classical theory of existence and regularity of solutions through classical Schauder estimates for elliptic partial differential equations.