Hirota-Satsuma Dynamics as a Non-Relativistic Limit of KdV Equations

We consider a system of two coupled KdV equations (one for left-movers, the other for right-movers) and investigate its ultra-relativistic and non-relativistic limits in the sense of BMS$_3$/GCA$_2$ symmetry. We show that there is no local ultra-relativistic limit of the system with positive energy, regardless of the coupling constants in the original relativistic Hamiltonian. By contrast, local non-relativistic limits with positive energy exist, provided there is a non-zero coupling between left- and right-movers. In these limits, the wave equations reduce to Hirota-Satsuma dynamics (of type IV) and become integrable. This is thus a situation where input from high-energy physics contributes to nonlinear science - in this case, uncovering the limiting relation between integrable structures of KdV and Hirota-Satsuma.