CLASSES OF NONLINEAR PDEs RELATED TO METRICS OF CONSTANT CURVATURE

Nome: 
Keti Tenenblat
Instituição: 
Universidade de Brasilia
Data do Evento: 
sexta-feira, 30 de Agosto de 2024 - 14:00
Local do evento
Sala 253
Descrição: 

In this talk, I will survey some aspects relating classes of PDEs with metrics on a 2-
dimensional manifold with non zero constant Gaussian curvature. The notion of a differential

equation (or system of equations) describing pseudo-spherical surfaces (curvature -1) or sphe-
rical surfaces (curvature 1) will be introduced. Such equations have remarkable properties. Each

equation is the integrability condition of a linear problem explicitly given. The linear problem
may provide solutions for the equation by using Bäcklund type transformations or by applying
the inverse scattering method. Moreover, the geometric properties of the surfaces may provide
infinitely many conservation laws. Very well known equations such as the sine-Gordon, Korteveg
de Vries, Non Linear Schrödinger, Camassa-Holm, short-pulse equation, elliptic sine-Gordon, etc.
are examples of large classes of equations related to metrics with non zero constant curvature.
Classical and more recent results characterizing and classifying certain types of equations will be
mentioned. Examples and illustrations will be included. Some higher dimensions generalizations
will be mentioned.