In this thesis, we study the response of nonlinear wave systems in bounded domains at or near resonance. Such circumstances arise in numerous physical regimes. In particular, the case of acoustic waves in various geometric configurations, the further understanding of which serves as a primary motivation for this work. There are typically two qualitatively distinct types of response which may be observed: continuous and shocked.
The main goal of this work is to understand the transition between these two regimes. To achieve this aim, we introduce a variety of nonlinear model problems at or near resonance and study the subsequent response. Then we explain how the features of this problem such as the nonlinearity form, and the nature of spectrum play a fundamental role in the response.
Further we aim to demonstrate an essential condition for the existence of the shocked solution, and discuss the methodology for approximating the solution in that regime. Furthermore, we analyze the interaction between the spectrum and the nonlinear features of the model problem. There is a correlation found between a continuous response and incommensurate eigenvalues and between a shocked response and commensurate eigenvalues. We carried out numerical simulations for each model problem with variation some features, such as the nonlinearity and boundary conditions, to study how this affects the outcome and the spectrum. The results of the numerical solutions are in a good quantitative agreement with the leading order approximations.
Moreover, we present wave model problems incorporating a damping effect to investigate how this impacts the qualitative nature of the response. However, for some cases of the damped model problem, multiple modes must be included to improve the comparison between the analytical solution and the numerical results. We carry out a weakly nonlinear analysis of higher modes. In order to better understand the role of higher modes, we provide amplitude-frequency comparisons.
Our results provide a better understanding for the transition between the two responses through the simple model problems, and this in turn will contribute to providing new insights into more relevant problems in acoustic and other related regimes.