Emergence of fractal structures from breather interactions in the $(2+1)$-dimensional Konopelchenko--Dubrovsky equation
Fractal structures generated through nonlinear breather interactions are investigated for the $(2+1)$-dimensional Konopelchenko--Dubrovsky (KD) equation by means of the Hirota bilinear method. The bilinear form of the system is first derived, after which breather interaction solutions are constructed analytically through suitable auxiliary functions. It is shown that the interaction of breather waves in the coupled nonlinear environment gives rise to highly intricate multiscale patterns exhibiting self-similar behaviour under successive magnification. To characterize the geometric complexity of the obtained structures, a three-dimensional voxel-based box-counting method is employed. The computed dimensions are found to be non-integer, confirming the fractal nature of the generated patterns. In addition, relative error analysis, standard error estimation, bootstrap standard deviation and convergence analysis are performed to examine the robustness and reproducibility of the estimated dimensions. The present work suggests that nonlinear breather interactions in coupled dispersive systems may provide a natural mechanism for the emergence of fractal geometries and complex multiscale structures. The combined analytical and quantitative framework developed here may provide further insight into nonlinear energy localization and scale-dependent structures arising in fluid dynamics, plasma physics and nonlinear wave propagation.