On the Existence and Stability of Standing Solitary Waves in Faraday Resonance

Citation

Guthart, Gary Steven (1992) On the Existence and Stability of Standing Solitary Waves in Faraday Resonance. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/fp2g-ds59. https://resolver.caltech.edu/CaltechTHESIS:08182011-080621226

Abstract

A rectangular tank of high-aspect ratio contains a liquid of moderate depth. The tank is subjected to vertical, sinusoidal oscillations. When the frequency of forcing is nearly twice the first natural frequency of the short side of the tank, waves are observed on the free surface of the liquid that slosh across the tank at a frequency equal to one half of the forcing frequency. These sloshing waves are modulated by a slowly varying envelope along the length of the tank. The envelope of the sloshing wave possesses two solitary-wave solutions, the standing soliton corresponding to a hyperbolic-secant solution and the standing kink wave corresponding to a hyperbolic-tangent solution. The depth and width of the tank determine which soliton is present. In the present work, we derive an analytical model for the envelope solitons by direct perturbation of the governing equations. This derivation is an extension of a previous perturbation approach to include forcing and dissipation. The envelope equation is the parametrically forced, damped, nonlinear Schrodinger equation. Solutions of the envelope equations are found that represent the solitary waves, and regions of formal existence are discussed. Next, we investigate the stability of these solitary-wave solutions. A linear-stability analysis is constructed for both the kink soliton and the standing soliton. In both cases, the linear-stability analysis leads to a fourth-order, nonself-adjoint, singular eigenvalue problem. For the hyperbolic-secant envelope, we find eigenvalues that correspond to the continuous and discrete spectrum of the linear operator. The dependence of the continuous-spectrum eigenvalues on the system parameters is found explicitly. By using local perturbations about known solutions and numerically continuing the branches, we find the bound-mode eigenvalues. For the kink soliton, continuous-spectrum branches are also found, and their dependence on the system parameters is determined. Bound-mode branches are found as well. In the case of the kink soliton, we extend the linear analysis by providing a nonlinear proof of stability when dissipation is neglected. We compute numerical solutions of the nonlinear Schrodinger equation directly and compare the results to the previous local analysis to verify the predicted behavior. Lastly, laboratory experiments were performed, examining the stability of the solitary waves, and comparisons are made with the foregoing work. In general, the agreement between the local analysis, the numerical simulations and the experiments is good. However, experiments and direct simulations show the existence of periodic solutions of the envelope equation when bound-mode instabilities are present.

Thesis Files