Abstract

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.

Degree Date

Summer 8-4-2021

Document Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

Advisor

Benno Rumpf

Subject Area

Mathematics, Applied

Number of Pages

80

Format

.pdf

Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License

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