Title:

Conservation laws and symmetries of difference equations

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 threepoint 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 AdlerBobenkoSuris classification. All inequivalent threepoint conservation
laws are found, as are three fivepoint conservation laws for each equation.
• We describe a method of generating conservation laws from known ones; this method
can be used to generate higherorder 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 quadgraph equations are found.
• We include a Maple program for deriving threepoint 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 AdlerBobenkoSuris
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 fivepoint symmetries of integrable
equations on the quadgraph are found. These include mastersymmetries, which
allow one to construct infinite hierarchies of local symmetries.
• We demonstrate a connection between the symmetries of quadgraph equations and
those of the corresponding Toda type difference equations .
• A program for deriving fivepoint symmetries for quadgraph equations is presented.
