Conservation laws and symmetries of difference equations

• Main Author: Rasin, Olexandr G.
• Published: University of Surrey 2007
• Subjects: 515.625
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.442672

This thesis deals with conservation laws and symmetries of difference equations. The main new results in the field of conservation laws are: • We have improved the effectiveness of Hydon's direct method for constructing conservation laws; • A classification of all three-point conservation laws for a large class of integrable difference equations that has been described by Nijhoff, Quispel and Capel is presented. We show that every nonlinear equation from this class has at least two nontrivial conservation laws. • We deal with conservation laws for all integrable difference equations that belong to the famous Adler-Bobenko-Suris classification. All inequivalent three-point conservation laws are found, as are three five-point conservation laws for each equation. • We describe a method of generating conservation laws from known ones; this method can be used to generate higher-order conservation laws from those that are listed here. • An example of conservation laws for a Toda type system is presented. The connection between these conservation laws and symmetries is shown. • Conservation laws for non autonomous quad-graph equations are found. • We include a Maple program for deriving three-point conservation laws for quad~ graph equations. The main new results in the field of symmetries are: • Symmetries of all integrable difference equations that belong to the Adler-BobenkoSuris classification are described. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. • We demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations . • A program for deriving five-point symmetries for quad-graph equations is presented.