Public lecture

The onset of instabilities in partial differential equations

  • 3 February 2022 - 3 February 2022
  • 1:00 pm - 2:00 pm
  • Virtual IAS

From a mathematical point of view, the description of natural phenomena that evolve through time and space is encapsulated as solutions of nonlinear partial differential equations. Given that the task of finding explicit solutions is very difficult, a more qualitative approach often allows us to predict the behaviour of the phenomena without having to exhibit a solution for the equation. We are particularly interested in stability, a central problem in the field of nonlinear wave propagation, and a fairly broad subject.

Stability and instability are important for our understanding of the world as many natural phenomena such as volcanic eruptions and aircraft turbulence occur as a result of the development of instabilities in a system. Due to the lack of structural properties, characterising stability/instabilities is a theoretically arduous problem that is often tackled by using numerical techniques. For this reason, we consider the so-called integrable equations, which arise as a compatibility condition for a pair of linear equations (Lax pair) and have rich and numerous properties.

In this talk we introduce the topics being developed, which deal with the characterisation of the onset of instabilities for nonlinear integrable equations through the study of their stability spectra, a structure recently introduced in the literature. As an initial discussion, we compare the stability spectrum to the Lax spectrum of the scalar and vector nonlinear Schroedinger equations and give an overview of further directions for the project.

This research seminar will be given by Newton International Fellow Dr Priscila Leal da Silva

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Kieran Teasdale
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