NSF Award
2406626
This project offers an in-depth investigation of the mathematical properties and physical applications of a special class of nonlinear waves. Over the past 50 years, a large body of knowledge has been accumulated on the nonlinear wave equations known as “integrable systems”, which are used to model a wide variety of physically interesting phenomena, ranging from fluid dynamics and nonlinear optics, to low temperature physics and Bose-Einstein condensation (BEC). Several integrable systems are studied as part of the project: scalar and multicomponent nonlinear Schrodinger (NLS) type equations, Maxwell-Bloch equations describing the interaction of light with an active optical medium, discrete integrable and non-integral lattices such as the Ablowitz-Ladik equations, the discrete NLS, the Salerno model, etc. The project has concrete applications in nonlinear optics and BEC, and the investigator also collaborates with physicists to seek experimental validation of the results of obtained by the principal investigator (PI). It is therefore anticipated that the outcomes of the project will also provide practical information that will help scientists and engineers more broadly, thus potentially benefiting the society at large. As part of the project, the investigator is involved in the organization of several conferences and special sessions at larger professional meetings in the US and abroad, e.g., a 4 week-long program on “Emergent Phenomena in Nonlinear Dispersive Waves”, to be held at Northumbria University, Newcastle, UK, in July-August 2024. Notably, the investigator organizes an event to showcase the research of female scientists. The training of graduate students is an integral component of the project, and two graduate students and a post-doctoral fellow are expected to work with the PI on research problems arising from this project.<br/> <br/>This project is aimed at advancing our theoretical and practical understanding of physically relevant integrable systems, as well as non-integrable systems in regimes that are not too far from the integrable ones, and it collects a suite of problems combined into a cohesive and coherent research effort, whose results will fundamentally further our knowledge of nonlinear waves and solitons, and their applications in various settings. Specifically, the investigator and her team pursue the following objectives: (i) formulation of a rigorous perturbation theory for dark and dark-bright solitons; (ii) development of a numerical inverse scattering transform on a nontrivial background; (iii) study of rogue waves, solitons and soliton interactions in scalar and coupled integrable and nonintegrable equations; (iv) investigation of the effect of radiation on the norming constants associated with the defocusing NLS equation, and their renormalization for solitonic models of condensates; (v) concrete applications to specific problems in BEC and nonlinear optics; (vi) study of Maxwell-Bloch systems both with rapidly decaying optical pulses and pulses with a constant background, and long-time asymptotics of their solutions. The project is carried out by developing and using a combination of: (a) exact methods, such as the inverse scattering transform, Backlund and Darboux transformations, and other direct methods; (b) asymptotic techniques, multiple scales and other perturbative tools; (c) state-of-the-art numerical simulations; (d) comparison with experiments in BEC and nonlinear optics.<br/><br/>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.
