Collaborative research: Nonlinear wave equations and inverse scattering

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This project concerns a class of nonlinear evolution equations (NLEEs) usually referred to as soliton equations or integrable systems. Over the past fifty years, a large body of knowledge has been accumulated on these systems, which continue to be extensively studied worldwide. Nonetheless, problems in which the boundary conditions play a crucial role still pose significant challenges. This project is aimed at undertaking a wide-ranging investigation of initial-boundary value problems for integrable NLEEs, both continuous and discrete, in both one and two spatial dimensions, including their applications in several concrete scientific and technological settings. This study will be carried out by developing and using exact methods, such as the inverse scattering transform and direct methods, in combination with appropriate asymptotic and numerical techniques. Specific problems that are being studied include: (i) nonlinear Schrodinger systems (scalar and vector, continuous and discrete) with non-zero boundary conditions at space infinity; (ii) boundary value problems for discrete and continuous NLEEs; (iii) characterization of soliton interactions for all of the above systems; (iv) classification of solitary wave structures in (2+1)-dimensional integrable systems such as the Davey-Stewartson system.Nonlinear wave equations are well-known to model a variety of physically interesting phenomena arising in areas ranging from fluid dynamics and nonlinear optics, to plasmas, cosmology and quantum field theory. The study of these equations is especially attractive because it offers a unique combination of interesting mathematics and concrete physical/technological applications. In particular, the systems that will be studied in this project model wave propagation in water waves, optical fibers, lasers and Bose-Einstein condensates. Hence the outcomes of this project will not only advance our mathematical understanding, but they will also provide practical information that will help scientists and engineers. The training of undergraduate and graduate students is also an integral component of this project. One of the PIs has co-authored a monograph on NLS systems that has been referenced extensively and is helping the dissemination of knowledge in the field of nonlinear waves. Both PIs have a long record of working with both graduate and undergraduate students, and will work both with undergraduate students and with Ph.D. students as part of this project.