On the existence and stability of standing solitary waves in Faraday resonance

[摘要] A rectangular tank of high-aspect ratio contains a liquid of moderate depth. The tank is subjected to vertical, sinusoidal oscillations. When the frequency offorcing 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 standingsoliton 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 foundthat represent the solitary waves, and regions of formal existence are discussed. Next, we investigate the stability of these solitary-wave solutions. A linear-stabilityanalysis 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, singulareigenvalue 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 localperrturbations about known solutions and numericallycontinuing the branches, we find the bound-mode eigenvalues. For the kink soliton, continuous-spectrum branches are also found, and their dependence onthe 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 nonlinearproof of stability when dissipation is neglected. We compute numerical solutions of the nonlinear Schrodinger equation directly and compare the results to the previouslocal 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 theenvelope equation when bound-mode instabilities are present.

[发布机构] University:California Institute of Technology;Department:Engineering and Applied Science

[关键词] Engineering