We present a multi-parameter family of rational solutions to the complex Korteweg-de Vries(KdV) equations. This family of solutions includes particular cases with high-amplitude peaks at the centre, as well as a multitude of cases in which high-order rogue waves are partially split into lower-order fundamental components. We present an empirically-found symmetry which introduces a parameter controlling the splitting of the rogue wave components into multi-peak solutions, and allows for nonsingular solutions at higher order in certain cases.
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