Thomas Ivey, an applied mathematics professor from the College of Charleston in South Carolina, currently serving as a visiting discrete math professor in Boulder, shared some highlights from his most recent research project this Friday afternoon at this week’s AMS colloquium.
Dr. Ivey presented on, “the Stability of Finite-Gap Vortex Filaments,” aimed at finding solutions to the vortex filament equation (VFE), which is used to describe the “self-induced motion of vortex lines in inviscid, incompressible fluid.” The first step to accomplishing this was normalizing all physical quantities involved to reduce the number of variables. Next, solutions to the non-linear Schrödinger (NLS) equation producing both plane wave potentials and rigid traveling wave potentials were found to satisfy certain conditions of the VFE. However, these particular partial differential equation (PDE) solution sets could only describe the appropriate vortex filament under particular circumstances, so more solutions sets were necessary. These other solutions came in the form of Torus knot solutions, which are “rigid solutions of the VFE” with “elastic rod centerlines,” meaning slight changes in the knot size and structure can be apparent over time.
The next question Ivey needed to answer was whether or not these PDE solutions were linearly stable, meaning they exhibited a controlled growth over time and did not ‘explode’ to infinity when expanded. If the solution sets were found to be linearly stable, they would produce a “linearized non-linear Schrödinger equation,” or LNLS. This, in turn, resulted in a linearized VFE, or LVFE, which was used to search for the vector field around the time-dependent filament curve in three-dimensional space, aiding in finding the stability of the solution sets. As Ivey discovered, the stability of the solutions produced all comes down to the coefficient in front of the variable for time, and whether that coefficient is completely real or contains an imaginary part. This is because it was found that an imaginary portion of the coefficient is necessary to keep growth of the VFE fairly constant; an all-real coefficient would result in an exponential ‘explosion’ of the equation towards infinity.
Previous literature on this topic studied by Ivey and Calini seemed to suggest that solutions to the VFE were always linearly unstable, an idea that this new research appears to contradict. Through the use of techniques such as Bäcklund transformations, their work counters what was previously researched in this field and insists that linearly stable solutions are in fact quite possible. However, as of today a problem that is keeping Ivey from converting the VFE to a true integrable system (a system that can be integrated and solved for) is the lack of Lax pairs (time-dependent matrices specific to a certain PDE, in this case the VFE). As part of his planned future research, Ivey hopes to discover these missing Lax pairs and be able to construct a truly integrable system to model finite-gap vortex filaments.