Soliton fractals in the Korteweg-de Vries equation

We have studied the process of creation of solitons and generation of fractal structures in the Korteweg-de Vries (KdV) equation when the relation between the nonlinearity and dispersion is abruptly changed. We observed that when this relation is changed nonadiabatically the solitary waves present in the system lose their stability and split up into ones that are stable for the set of parameters. When this process is successively repeated the trajectories of the solitary waves create a fractal treelike structure where each branch bifurcates into others. This structure is formed until the iteration where two solitary waves overlap just before the breakup. By means of a method based on the inverse scattering transformation, we have obtained analytical results that predict and control the number, amplitude, and velocity of the solitary waves that arise in the system after every change in the relation between the dispersion and the nonlinearity. This complete analytical information allows us to define a recursive L system which coincides with the treelike structure, governed by KdV, until the stage when the solitons start to overlap and is used to calculate the Hausdorff dimension and the multifractal properties of the set formed by the segments defined by each of the two "brothers" solitons before every breakup.