Physical phenomena such as wave propagation are often represented mathematically by nonlinear equations. Such equations can describe large amplitude behavior and large amplitude wave motion. While it is difficult to find solutions to most nonlinear equations, there is a subclass of equations that have deep mathematical structure and admit an important set of special wave solutions termed "solitons." Solitons are localized, stable waves that are of keen interest to physicists and engineers. They arise in diverse fields such as nonlinear optics, fluid dynamics, Bose-Einstein condensation, magnetic systems, and plasma physics, amongst many others. In this project new solutions and properties of a class of physically important nonlinear soliton equations will be investigated. The research will study new solutions and properties of multidimensional equations, including multi-lump solutions to the Kadomtsev-Petviashvili equation; investigate nonlinear equations that can be expressed in terms of modular forms, including new reductions of the self-dual Yang-Mills system; investigate boundary value problems for continuous and discrete scalar and vector nonlinear Schrodinger equations; and study dispersive shock wave phenomena in Bose-Einstein condensation and nonlinear optics.
Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics has demonstrated novel wave phenomena. Under suitable conditions a narrow soliton wave front with an associated modulated wave train are observed. The experiments illustrate what is termed dispersive blast waves or dispersive shock waves. This wave phenomenon is the dispersive, non-dissipative, wave equivalent of well-known atmospheric blast waves which are dissipative in nature. In this project a number of new and fundamental research directions associated with dispersive shock waves will be studied, including interactions of dispersive shock waves and dispersive rarefaction waves.
