<p>This study examines the Painlevé integrability and complex dynamics of a dispersive coupled Burgers system, focussing on its fractal and chaotic structures. The Painlevé test is performed to determine the system’s integrability. Analytical solutions are derived using the Riccati method, while logarithmic, sine, cosine and Jacobi elliptic functions are employed to investigate the emergence of fractal structures. The chaotic nature of the system is further analysed through Poincaré section and Lyapunov exponent computations. The results provide valuable insights into the nonlinear wave interactions, pattern formation and turbulence modelling. This system has potential applications in fluid dynamics, particularly in describing shock waves and turbulence in compressible flows. Moreover, understanding the integrability, fractality and chaos in the present system can lead to advancement in predictive modelling across various scientific and engineering disciplines.</p>