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In this thesis we study the Darboux transformations related to particular Laxoperators of NLS type which are invariant under the action of the so-calledreduction group. Specifically, we study the cases of: 1) the nonlinearSchr\"odinger equation (with no reduction), 2) the derivative nonlinearSchr\"odinger equation, where the corresponding Lax operator is invariant underthe action of the $\mathbb{Z}_2$-reduction group and 3) a deformation of thederivative nonlinear Schr\"odinger equation, associated to a Lax operatorinvariant under the action of the dihedral reduction group. These reductiongroups correspond to recent classification results of automorphic Lie algebras.  We derive Darboux matrices for all the above cases and we use them toconstruct novel discrete integrable systems together with their Laxrepresentations. For these systems of difference equations, we discuss theinitial value problem and, moreover, we consider their integrable reductions.Furthermore, the derivation of the Darboux matrices gives rise to manyinteresting objects, such as B\"acklund transformations for the correspondingpartial differential equations as well as symmetries and conservation laws oftheir associated systems of difference equations.  Moreover, we employ these Darboux matrices to construct six-dimensionalYang-Baxter maps for all the afore-mentioned cases. These maps can berestricted to four-dimensional Yang-Baxter maps on invariant leaves, which arecompletely integrable; we also consider their vector generalisations.

Darboux transformations, discrete integrable systems and related Yang-Baxter maps