Effects of non–commutativity on soliton behavior in baroclinic flows with time–dependent coefficients

This paper investigates a non-commutative extension of the [Formula: see text] and [Formula: see text][Formula: see text] system with time-dependent coefficients. In this framework, [Formula: see text] describes the evolution of a baroclinic wave packet, while [Formula: see text] captures the corresponding adjustment of the background flow, providing a model for marginally unstable baroclinic dynamics in fluid systems. The non-commutative [Formula: see text] system is formulated within a Lax-integrable structure, ensuring a consistent mathematical foundation for generating analytical solutions. Explicit soliton solutions are constructed using an iterative Darboux transformation expressed through quasideterminants. These formulas clarify how non-commutative algebra modifies the profiles, phase evolution and interaction behaviour of the resulting waves. Numerical simulations further illustrate a variety of nonlinear phenomena, including localized bright and dark solitons, parabolic waveforms, periodic soliton trains and coherent S-type structures arising under different choices of the time-dependent coefficients. The results demonstrate that non-commutativity, together with temporal modulation of the system parameters, plays a decisive role in shaping soliton dynamics. This study therefore provides new insight into how non-abelian effects influence nonlinear wave propagation in the time-dependent [Formula: see text] model relevant to fluid dynamics.