Nonlinear waves on a torus of fluid and integrability

Overview & Synopsis

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 is 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 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 with the effects of curvature included. By increasing the amplitude of a monochromatic wave, we demonstrate that, due to nonlinearity, this wave is unstable 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. Beyond the torus, we have also turned to nonlinear mechanical systems, in order to study solitons and nonlinear dynamics, the first of which is a periodic coupled chain of pendulums, which is governed by the sine-Gordon equation. We present a novel numerical technique to determine the soliton spectrum on the periodic line of this equation. Additionally, moving to discrete systems, we have studied experimentally and analytically the nonlinear dynamics of a hanging string with a pivoting tip mass, showing how nontrivial boundary conditions can significantly affect the resonant frequencies.