We study a coupled dispersionless system in \((1+1)\) dimensions with a time-dependent coefficient in the nonlinear coupling. The model consists of a real field u(x, t) and a complex field \(\psi (x,t)\) , driven by a prescribed modulation function \(\alpha (t)\) . A \(2\times 2\) 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 \(\alpha (t)\) . 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 \(\alpha (t)\) 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.
