Does a fractal-like structure emerge from the interaction of breathers in a coupled PDE?

A systematic and direct approach to demonstrate the existence of quasi-fractal structures in the coupled Boussinesq system introducing a novel perspective within the Hirota bilinear technique has been investigated. This coupled system arises from two-layered fluid flow in shallow water waves. In this coupled system, there are two components: one representing the depth-averaged horizontal velocity field and the other representing the height of the water surface above the bottom level. When breathers for each component interact in a confined region, the amplitude and horizontal velocity undergo abrupt changes. Repeated iterations of this process yield fractal fork-like structures, formed until the interaction where two breather waves overlap just before breakup. Upon zooming into this graph, we observe a structure that is qualitatively identical to the original one, which indicates the scale-invariant property of fractal structures. We also calculate the box dimensions of the quasi-fractal structures, confirming that these fractals have non-integer dimensions. This study pioneers the systematic investigation of quasi-fractal structures in coupled nonlinear systems. Further exploration reveals various interacting phenomena, including other line structures with fractal excitations for specific parameter values. This study enhances the understanding of the Hirota bilinear method by demonstrating, for the first time, the emergence of quasi-fractal structures through this approach.