Darboux transformation and quasideterminant solutions of a noncommutative semi-discrete coupled dispersionless integrable system

We present a noncommutative generalization of the semi-discrete coupled dispersionless integrable system. The motivation stems from the role of noncommutative geometry in string theory and D-brane dynamics, and the need to understand soliton dynamics in matrix-valued field theories beyond the standard abelian setting. A matrix Lax pair for the noncommutative semi-discrete coupled dispersionless system is proposed, and the corresponding equations of motion are obtained as the compatibility conditions of this Lax pair. A Darboux transformation is constructed both for the Lax pair and for the nonlinear field equations, and its iteration yields multisoliton solutions written in a compact quasideterminant form. We then investigate noncommutative semi-discrete solutions of the matrix fields and discuss their qualitative behaviour on the space–time lattice. Furthermore, an equivalence is established between the noncommutative semi-discrete coupled dispersionless system and a noncommutative semi-discrete sine-Gordon equation. Finally, by applying an appropriate continuum limit, we recover multisoliton solutions of the corresponding noncommutative continuous coupled dispersionless system.