Breather and Soliton Gases for the Focusing Nonlinear Schrodinger Equation: Theoretical and Applied Aspects

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The most familiar manifestation of nonlinear dispersive waves is perhaps that of a breaking ocean wave. Nonlinear dispersive waves are ubiquitous in nature, appearing in many fields ranging from water waves to optics, acoustics, condensed matter, cosmology and beyond. Nonlinear dispersive waves have been an area of intensive research interest in applied mathematics and physics. The integrable Nonlinear Schrödinger (NLS) equation is generally accepted as the universal model for waves propagating in nonlinear dispersive media. The main idea of this project is to model random nonlinear waves, which are frequently encountered in natural phenomena, with the so-called soliton or breather gases – random ensembles of the well-known solutions of the equation. Ultimately, the project aims to enhance our ability to predict and, in some cases, to control random nonlinear waves. This research employs ideas of pure and applied mathematics that are enhanced by the collaboration with experimental physicists. The project will also serve as a vehicle for training PhD students.The main goals of this project can be divided in the two categories: (a) development of the rigorous spectral theory for the focusing NLS (fNLS) gases and construction of their semiclassical limit realizations; (b) statistical characterization of breather and soliton gases in terms of their spectral characteristics together with applications in random nonlinear wave problems. The work in part (a) requires rigorous derivation and analysis of integro-differential equations describing spectral characteristics of the gases. Calculation of important statistical characteristics (probability density function, power spectrum, kurtosis, etc.) of the fNLS gases in part (b) contains both analytical and numerical components. The obtained results are expected to lead to lab experiments in collaboration with European experimentalists. In general, the project will have an impact on a broad area of nonlinear wave theory, including integrable turbulence. In fiber optics, results of the project may help to examine, and ultimately to control the evolution of noise in nonlinear optical fibers. The project is also expected to advance our general knowledge of random nonlinear waves, including the rogue waves, and to improve methods of their prediction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.