Nonlinear Waves Seminar - Cloned
Nonlinear Waves
The Nonlinear Waves seminar is motivated by wave phenomena in various systems. However, more broadly, we are interested in computational and applied mathematics. All are welcome to attend. All meetings are in-person, unless specified otherwise.
If you would like to give a talk, email Justin Cole (jcole13@uccs.edu).
Fridays, 4:00pm-4:50pm OSB B215
Fall 2023 Abstracts Date: 10/13 and 10/20 Abstract: Inspired by recent experiments, the Korteweg-de Vries equation with nonzero Dirichlet boundary conditions is considered. Two types of boundary data are examined: step up (which generates a rarefaction wave) and step down (which creates a dispersive shock wave). Soliton dynamics are analytically studied via inverse scattering transform and a soliton perturbation theory. Depending on the initial position and amplitude, an incident soliton will either transmit through or become trapped inside a step-induced wave. A formula for the transmitted soliton and its phase shift is derived. The soliton perturbation theory is able to describe the dynamics of a soliton that becomes trapped inside a rarefaction wave.
Date: 11/10 Abstract: Intracranial saccular aneurysms (ISAs) tend to form at the apex of arterial bifurcations and often assume a nominally spherical shape. In certain cases, the aneurysm growth can become unstable and lead to rupture. While the mechanisms of instability are not well understood, hemodynamics, or the blood flow in the vasculature, almost certainly play an important role. In this talk, a mathematical model of an ISA is presented that describes the shape deformations of an initially spherical membrane interacting with a viscous fluid in the interior. The governing equations are derived from the equations of a thin shell supplemented with a constitutive model that is representative of aneurysmal tissue. Among the key findings are that two families of free vibration modes exist and, for certain values of the membrane properties, one family of nonspherical, axisymmetric modes is unstable to small perturbations. In addition, the presence of a vortical interior flow of sufficient strength can excite resonance of the membrane – an unstable phenomenon that might cause eventual rupture. |
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Past Semesters
