Accelerating solutions of the Korteweg-de Vries equation

The Korteweg-de Vries equation is a fundamental nonlinear equation that describes solitons with constant velocity. On the contrary, here we show that this equation also presents accelerated wavepacket solutions. This behavior is achieved by putting the KdV equation in terms of the Painlevé I equation. The accelerated waveform solutions are explored numerically showing their accelerated behavior explicitly.