This work investigates the emergence of finite-scale self-similar waveforms in a \((2+1)\) -dimensional Broer–Kaup (BK) system, a nonlinear dispersive model relevant to fluids, plasmas and optical systems. Exact analytical solutions are obtained via a Riccati transformation. By incorporating oscillatory structures based on trigonometric, logarithmic and Jacobi elliptic functions into a selected solution, a class of waveforms exhibiting multiscale recursive patterns is constructed. Geometrical visualisation through successive zoom-in operations demonstrates that both surface and contour representations preserve their global structure across multiple magnification levels, confirming finite-scale self-similarity. To quantify this geometric complexity, a three-dimensional voxel-based box-counting method is applied for the surfaces, while a grid-based box-counting method is employed for the contour plots. The resulting non-integer dimensions exhibit a gradual decrease with increasing resolution, consistent with physical smoothing at fine scales. Complementary statistical diagnostics, including relative error analysis, standard error estimation and bootstrap resampling, are incorporated to ensure the robustness and reproducibility of the fractal dimension estimates. The combined analytical construction, visualisation and quantitative analysis provide a comprehensive framework for characterising multiscale geometry in higher-dimensional nonlinear systems. The findings not only enrich the mathematical understanding of dispersive wave phenomena but also offer physical insight into scale-dependent complexity and energy localisation with potential relevance to hydrodynamic turbulence, nonlinear ion-acoustic waves in plasmas and optical wave propagation in structured media.
