Hirota-Satsuma dynamics as a non-relativistic limit of KdV equations |
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Affiliation: | Laboratoire de Physique Théorique et Hautes Energies, Sorbonne Université and CNRS UMR 7589, F-75005 Paris, France |
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Abstract: | 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 BMS3/GCA2 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. |
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Keywords: | KdV equation Hirota-Satsuma dynamics Inonu-Wigner contraction BMS symmetry Lie-Poisson equations |
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