High-amplitude coherent structures have been observed in many nonlinear wave systems, ranging from fluids, plasmas to optical waves. In this dissertation, we explore the interaction of Rayleigh-Jeans distributed low-amplitude random waves with coherent solitary structures in a nonintegrable and non-collapsing version of the nonlinear Schr\"odinger equation. We try to understand if such an interaction enhances or erodes the coherent structures with the method of statistical mechanics. We find the threshold of the growth and decay of the coherent structure by equating the phase frequency of the coherent structure to the chemical potential of low-amplitude weakly nonlinear random waves. If the phase frequency exceeds the critical threshold value, the coherent structure accumulates wave action from random waves while transferring energy to random waves and then grows. Otherwise it decays. We also verify this finding with numerical simulations and the numerical results match our theoretical prediction.
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Chen, Yuanting, "Statistical Mechanics of Nonlinear Waves: Growth and Decay of Coherent Structures Interacting with Random Waves" (2021). Mathematics Theses and Dissertations. 16.