Coherent structures are identifiable localized characteristics in a wave field, which play a central role as carriers of energy, information etc in many physical systems. Several examples are: solitary water waves at the water / air interface, aerodynamic shock waves and optical beams or pulses used in communication networks. This research concerns the study of nonlinear partial differential equations (PDEs) governing wave phenomena, in particular coherent structures, in optics, electromagnetics, hydrodynamics and quantum physics. Coherent structure solutions of PDEs often arise due to a combination of nonlinear and dispersive effects. Nonlinearity tends to concentrate energy and dispersion tends to spread it out; their balance often results in persistent or long-lived coherent structures. We will study the dynamics of coherent structures - their stability and scattering interactions. We shall also consider the interaction of such coherent structures with spatially inhomogeneous media. Inhomogeneities (spatially varying coefficients in the PDEs) introduce another class of localized states called "defect modes". We will study the nonlinear dynamics and energy exchange between coherent structures with defect modes and radiation modes. Finally, we will apply our previous work on energy transfer between discrete (bound state) and continuum (radiation) modes in energy conserving PDEs to the question of control of wave propagation, e.g. designing a photonic structure to maximize the lifetime of a designated optical state. Our mathematical approaches range from rigorous analytical methods to formal asymptotic methods to numerical simulation of PDEs and optimization.
Advances in the design and fabrication of micro- and nano-structured media are driving mathematical research to determine their effective properties and those of waves propagating in such structures. Such microstructures are important components in current and future communication and information processing technologies. For example, they enable the manipulation of light (photons) in a manner analogous to the way electricity (electrons) has been manipulated in solid state computer chips for many years. Advantages come through greater speed and virtually dissipation free propagation in photonic media. Due to the size, multi-scale character and complexity of these problems, relying on full computer simulations of the governing partial differential equations is not practical for the problem of device design. The mathematical problems explored in this project are aimed at the development of systematic approaches to characterization of such microstructured media and an understanding the properties of waves traveling through them. These will be used to develop hybrid analytical / computational approaches to the problems requiring the control of coherent structures.
