Mathematical techniques for free boundary problems with multiple boundaries

In this thesis, we study six different free boundary problems arising in the field of fluid mechanics, and the mathematical methods used to solve them. The free boundary problems are all characterised by having more than one boundary and the solution of these problems requires special mathematical treatment. The challenge in each of these problems is to determine the shape of the multiple fluid interfaces making up the particular system under consideration. In each of the free boundary problems we employ aspects of complex function theory, conformal mapping between multiply connected domains, and specialist techniques devised in recent years by Crowdy and collaborators. At the heart of these techniques lies a special transcendental function known as the Schottky-Klein prime function. This thesis makes use of this function in a variety of novel contexts. We first examine a single row of so-called hollow vortices in free space. This problem has been solved before but we present a new methodology which is convenient in being extendible to the case of a double row, or von Karman vortex street, of hollow vortices. We find a concise formula for the conformal mapping describing the shapes of the free boundaries of two hollow vortices in a typical period window in the vortex street and thereby solve the free boundary problem. We next focus on the problem of a pair of hollow vortices in an infinite channel. This free boundary problem exhibits similar mathematical features to the vortex street problem but now involves the new ingredient of solid impenetrable walls. Again we solve the free boundary problem by finding a concise formula for the conformal mapping governing the hollow vortex shapes. We then extend this analysis to a single row of hollow vortices occupying the channel. The problem of a pair of hollow vortices of equal and opposite circulation positioned behind a circular cylinder, superposed with a uniform flow, is then considered. This system is a desingularisation of the so-called Foppl point vortex equilibrium. For this free boundary problem, we employ a hybrid analytical-numerical scheme and we are able to offer a Fourier-Laurent series expansion for the conformal mapping determining the shape of the hollow vortex boundaries. Finally, we investigate an asymmetric assembly of steadily translating bubbles in a Hele- Shaw channel. This free boundary problem can be formulated as a special Riemann-Hilbert problem solvable in terms of the Schottky-Klein prime function. Our method of solution can be used to determine the shapes of any finite number of bubbles in a given assembly

Planarizable Supersymmetric Quantum Toboggans

In supersymmetric quantum mechanics the emergence of a singularity may lead to the breakdown of isospectrality between partner potentials. One of the regularization recipes is based on a topologically nontrivial, multisheeted complex deformations of the line of coordinate $x$ giving the so called quantum toboggan models (QTM). The consistent theoretical background of this recipe is briefly reviewed. Then, certain supersymmetric QTM pairs are shown exceptional and reducible to doublets of non-singular ordinary differential equations a.k.a. Sturm-Schr\"odinger equations containing a weighted energy $E\to EW(x)$ and living in single complex plane