 CARMA ANALYSIS SEMINAR
 Speaker: Ekaterina Shemyakova, Western University (Canada) and Russian Academy of Sciences (Moscow)
 Title: The Method of Darboux Transformations for Partial Differential operators
 Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 3:00 pm, Tue, 28^{th} Feb 2012
 Abstract:
Integrability theory is the area of mathematics in which methods are developed for the exact solution of partial differential
equations, as well as for the study of their properties. We concentrate on PDEs appearing in Physics and other applications. Darboux transformations constitute one of the important methods used in integrability theory and, as well as being a method for the exact solution of linear PDEs, they are an essential part of the method of Lax pairs, used for the solution of nonlinear PDEs. A large series of Darboux transformations may be constructed using Wronskians built from some number of individual solutions of the original PDE. In this talk we prove a longstanding conjecture that this construction captures all possible Darboux transformations for transformations of order two, while for transformations of order one the construction captures
everything but two Laplace transformations. An introduction into the theory will be provided.
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