Darboux transformations and integrable discretisation

Within the Lax-Darboux scheme, Darboux transformations provide discretisations of integrable partial differential equations (PDEs) as integrable differential-difference equations (DΔEs). In the present thesis, we apply this method to the seven non-commutative derivative nonlinear Schrödinger equations (DNLS) identified by Olver and Sokolov. The considered Lax representations, already appearing in the literature, arise also from solving a classification problem.

Focusing on polynomial Darboux matrices of degree n ∈ N, we construct a model for reduction group-invariant Darboux transformations, namely the rank-1 Darboux matrices M_↑(n) and M_↓(n), that generates evolutionary systems. We study the constant, linear, and quadratic cases, whose related discretisations, derived through reduction procedures, consist of non-commutative integrable systems with non-commutative constants.

We demonstrate that the constant Darboux matrices induce a scaling transformation, the linear Darboux matrices are associated with Volterra-type equations, and reductions of the quadratic Darboux matrices yield two-component systems, including the relativistic Toda, the Merola-Ragnisco-Tu, and the Ablowitz-Ladik equations.

Examining the relationship between linear and quadratic Darboux transformations, we provide the necessary conditions for a Darboux matrix to be factorisable with a specific linear Darboux matrix as a factor.

Finally, since the DNLS equations are known to be connected by non-local gauge transformations, we extend this correspondence to Darboux transformations and the associated systems of equations. This thesis concludes with four appendices, devoted, respectively, to Lax representations of the non-commutative DNLS equations, the Darboux system associated with the polynomial matrix M(n), the properties of the resulting DΔEs, and the Lax pairs of two novel equations.