Nonlinear Wave Motion

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Research will center on the construction of solutions and the investigation of the properties of a class of physically significant nonlinear wave equations. Applications include water waves and nonlinear optics. In water waves, when surface tension is relatively small, the solutions to be investigated describe interactions of non-decaying soliton waves that have X, Y and more complex structure. From results of prior NSF supported research by the PI, it is known that novel multi-lump solitons can occur in water waves with relatively large surface tension. In this case a complete characterization of the multi-lump class of solutions will be considered. Related nonlinear equations that have similar types of solutions will also be investigated. New classes of nonlocal equations that are solvable by the inverse scattering transform will be analyzed. Interactions of dispersive shock waves in a wide variety of physically interesting systems will be considered by the inverse scattering transform. Nonlinear wave propagation in optical materials with periodic lattice backgrounds will also be studied.Wave phenomena in oceans and optics have broad appeal. Numerous observations of shallow water wave interactions on flat beaches by the PI and a graduate student funded by NSF have led to mathematical descriptions that provide deeper understanding of such phenomena. These interactions frequently appear visually to be of X and Y structure, though sometimes they are more complex. Interestingly these waves also have application to tsunami propagation. As a result of the initial research, a journal article was recently published in Physical Review. The work was subsequently described in a focus article in Physics Today, the Society of Industrial and Applied Math News and was featured in a number of articles by popular news organizations. These articles included remarkable photos taken by the authors. In other directions, the mathematics also applies to nonlinear optics with applications that involve the dynamics, steering and manipulation of localized electromagnetic waves. The contemplated research will extend the mathematical understanding of the nonlinear equations to a variety of physically interesting systems. The PIs work has been widely referenced and used by researchers worldwide.