Soliton management in a variable-coefficient coupled dispersionless system via Darboux transformation

We study a coupled dispersionless system in [Formula: see text] dimensions with a time-dependent coefficient in the nonlinear coupling. The model consists of a real field u(x, t) and a complex field [Formula: see text], driven by a prescribed modulation function [Formula: see text]. A [Formula: see text] Lax pair of Zakharov-Shabat type is constructed, showing that the system remains integrable with the temporal dependence confined to diagonal terms of the Lax matrices. Using a Darboux transformation adapted to this Lax pair, we derive Wronskian-type formulas for multi-soliton solutions and obtain explicit families of bright and dark traveling waves for several representative choices of [Formula: see text]. In addition, a time reparametrization transforms the model to an autonomous form whose traveling-wave reduction yields a planar Hamiltonian system with equilibria and phase portraits identical to the constant-coefficient coupled dispersionless equations. The modulation [Formula: see text] therefore leaves the intrinsic traveling-wave dynamics unchanged but reshapes how these waves are embedded in the (x, t)-plane, enabling management of soliton amplitudes, velocities and curved or oscillatory trajectories.