KdV equation as a minimising L^2-energy equation, by Irina Markina
Main content
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.
