Friday, July 5th, 14h30, Room 454 A, Condorcet Building.

PhD defense of Filip Novkoski

Nonlinear Waves on a Torus of Fluid and Integrability.

supervised by Eric Falcon and  Chi-Tuong Pham

Hydrodynamic surface waves can admit dispersive properties under certain conditions as well as nonlinear phenomena for sufficiently high amplitudes. These properties can depend on the geometry in which these waves propagate. In this thesis, we study how dispersion and nonlinearity of surface waves are affected by a toroidal geometry. We present a novel technique for the creation of a stable and stationary torus of fluid on which we generate propagating waves. In the linear regime, these azimuthal waves are measured and a rich multi-branch dispersion relation is observed for the first time on the torus, such as the presence of gravity-capillary, sloshing, and center-of-mass modes. We describe them with a simple model including the effects of curvature. By increasing the amplitude of the monochromatic forcing, we demonstrate that, due to nonlinearity, an instability occurs and gives rise to two daughter waves. It is established due to a triadic resonant instability, i.e., a resonant three-wave interaction. We show that this interaction occurs between two different dispersion branches, a mechanism not reported so far. Further investigating nonlinear effects, by applying a pulse forcing on the torus, we observe the propagation of solitons along its outer border. We demonstrate that due to the periodicity of the torus, new types of solitons can be observed, notably subsonic elevation solitons, which can be modeled using the periodic Korteweg–de Vries equation.