Analysis and PDE seminar: Spring 2024

Date and time: Thursday, January 25, at 14.15

Place: Aud. Sigma

Speaker: Irina Markina, Professor,  Department of Mathematics, UiB

Title: KdV equation as a minimising L^2-energy equation.

Abstract: This talk is oriented to two parts of the Analysis and PDE group: the group with an interest in differential geometry and the group that is interested in non-linear partial differential equations. The motion of a rigid body in 3-D space is successfully described as a motion in the group of Euclidean transformations (rotations and translations) by making use of the Euler angles. V. Arnold proposed to describe fluid motion by replacing the finite-dimensional group of Euclidean transformations with the infinite-dimensional group of diffeomorphic transformations of a suitable space. We will consider the simplest infinite dimensional group, which is the group of diffeomorphism Diff(S) of the unit circle S, which corresponds to the description of periodic solutions of one variable. I will define the group (slightly different from Diff(S)), its Lie algebra, the metric on it (or the energy) and finally show that the equation describing the geodesics (the curves minimizing the energy) is the famous KdV equation in the fluid mechanics.

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Date and time: Thursday, February 1, at 14.15

Place: Aud. Sigma

Speaker: René Langøen, PhD. student,  Department of Mathematics, UiB

Title: Stokes graphs of a quadratic differential related to a Rabi model

Abstract: To study the behaviour of solutions to a second-order linear differential equation 𝑦″+𝑄(𝑧,𝑡)𝑦=0 one can associate the quadratic differential 𝑄(𝑧)𝑑𝑧² on the punctured Riemann sphere and consider its Stokes graph. We consider an ODE related to a Rabi problem describing a light-atom interaction in physics. The associated quadratic differential is meromorphic with two finite poles. The integrability condition for this type of ODE under isomonodromic deformations is related to a non-linear second-order differential equation, known as Painlevé V. In my talk, I will explain a classification of the Stokes graphs according to the nature of the zeros of the meromorphic quadratic differential originated in the Rabi model. This is a joint work with I. Markina (University of Bergen) and A. Solynin (Texas Tech, USA).

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