Symmetry structure for differential-difference equations

Having infinitely many generalised symmetries is one of the definition of integrability for non-linear differential-difference equations. Therefore, it is important to develop tools by which we can produce these quantities and guarantee the integrability. Two different methods of producing generalised symmetries are studied throughout this thesis, namely recursion operators and master symmetries. These are objects that enable one to obtain the hierarchy of symmetries by recursive action on a known symmetry of a given equation. Our first result contains new Hamiltonian, symplectic and recursion operators for several (1+1)-dimensional differential-difference equations both scalar and multicomponent. In fact in chapter 5 we give the factorization of the new recursion operators into composition of compatible Hamiltonian and symplectic operators. For the list of integrable equations we shall also provide the inverse of recursion operators if it exists. As the second result, we have obtained the master symmetry of differential-difference KP equation. Since for (2+1)-dimensional differential-difference equations recursion operators take more complicated form, " master symmetries are alternative effective tools to produce infinitely many symmetries. The notion of master symmetry is thoroughly discussed in chapter 6 and as a result of this chapter we obtain the master symmetry for the differential-difference KP (DDKP) equation. Furthermore, we also produce time dependent symmetries through sl(2, C)-representation of the DDKP equation.