Gerard Killeen ()

Title: 

Abstract: 

Jerome Moloney ( University of Arizona, US)

Title: Ultrafast intense laser pulse propagation: Dispersion vs nonlinearity dominated physics.

Abstract: In this talk, I will briefly review the early development of the theory of ultrashort pulse (USP) filamentation. The Unidirectional Pulse Propagation equation (UPPE), a pseudo-spectral solver, will be introduced as a robust Maxwell-like full wave propagator that allows long distance propagation of arbitrary vectorial optical waveforms while resolving the underlying optical carrier wave. In limiting cases, propagating USPs, whether dispersion or nonlinearity dominated, can exhibit very different singular behaviors. In the former limit, the respective mathematical descriptions are typically the envelope NLSE-types which exhibit a blow-up singularity at a finite distance. Nonlinearity dominated interactions are best captured by the optical carrier resolved modified Kadomtsev-Petviashili (mKP)) equation, describing optical carrier shock-like singularities. Applications to be discussed will include both near-IR and long wave atmospheric propagation, multifilament generation seeded by a modulational instability, pulse splitting and X-wave generation, exotic Bessel and Airy beams, extended plasma string generation proposed for guiding high voltage discharges, RF and Microwave guiding and lightning deflection.

Dana Contreras Julio (University of Auckland,  New Zealand) 

Title: Close Enough for Continuation: A Numerical Method for Computing Pseudo-Orbits on Invariant Manifolds as Seeds for Boundary Value Problems in Maps

Abstract: In the study of discrete dynamical systems, identifying special orbits, specifically connecting orbits between saddle fixed or periodic points, is essential for understanding complex dynamics. Such special orbits are closely associated with stable or unstable manifolds and can be found numerically and then continued in parameters as solutions of suitably formulated boundary value problems (BVPs). However, obtaining the initial "seed" solution required for Newton's method to converge to the BVP remains a significant numerical challenge.

Pseudo-orbits are guaranteed to have a true finite orbit of the diffeomorphism nearby, and hence, they serve as robust and natural seeds for BVPs. In this talk, we present a numerical method for computing one-dimensional invariant manifolds such that a pseudo-orbit is readily available for every computed point. We illustrate the efficiency of this method using two examples of robust non-hyperbolic dynamics, where we identify pseudo-orbits to systematically find many orbits in a chosen section, as well as numerous homoclinic and heteroclinic connections. 

Alannah Neff (University of Edinburgh,  UK) 

Title: Solute transport in the cranial subarachnoid space

Abstract: Cerebrospinal fluid (CSF) is a clear, Newtonian fluid that fills the subarachnoid space (SAS)  surrounding the brain and spinal cord and plays a central role in maintaining brain health. It is responsible for the delivery of nutrients and the clearance of metabolic waste and neurotoxic proteins. Disruptions to CSF transport are thought to contribute to the accumulation of proteins such as amyloid-β, which is implicated in neurological diseases including Alzheimer’s disease. CSF motion within the SAS is driven by brain pulsations associated with the cardiac and respiratory cycles, leading to predominantly oscillatory flow. Although this motion is largely periodic, it can 
generate secondary transport mechanisms that produce a net movement of solutes over long  timescales. Understanding how these pulsations give rise to effective solute transport remains an open question. In this talk, I present simplified mathematical models of oscillatory CSF flow in the cranial SAS 
aimed at understanding the link between brain pulsations and solute transport. Using lubrication  theory to reduce the governing fluid equations, and exploiting the separation of timescales between fast oscillatory motion and slow net transport, we derive an effective long-time transport equation 
for solutes. We then investigate how both temporally and spatially varying oscillations influence solute transport within the SAS.