▁field ▁theory ▁and ▁partial ▁differential ▁equations , ▁a ▁Tod a ▁field ▁theory ▁( named ▁after ▁Mor ik azu ▁Tod a ) ▁is ▁derived ▁from ▁the ▁following ▁L agr ang ian : ▁ ▁Here ▁x ▁and ▁t ▁are ▁spac etime ▁coordinates , ▁( , ) ▁is ▁the ▁K illing ▁form ▁of ▁a ▁real ▁r - dimensional ▁Cart an ▁algebra ▁ ▁of ▁a ▁K ac – Mo ody ▁algebra ▁over ▁, ▁α i ▁is ▁the ▁it h ▁simple ▁root ▁in ▁some ▁root ▁basis , ▁ni ▁is ▁the ▁C ox eter ▁number , ▁m ▁is ▁the ▁mass ▁( or ▁bare ▁mass ▁in ▁the ▁quantum ▁field ▁theory ▁version ) ▁and ▁ β ▁is ▁the ▁coupling ▁constant . ▁ ▁Then ▁a ▁Tod a ▁field ▁theory ▁is ▁the ▁study ▁of ▁a ▁function ▁ φ ▁mapping ▁ 2 - dimensional ▁M ink owski ▁space ▁satisfying ▁the ▁corresponding ▁E uler – L ag range ▁equations . ▁ ▁If ▁the ▁K ac – Mo ody ▁algebra ▁is ▁finite , ▁it ' s ▁called ▁a ▁Tod a ▁field ▁theory . ▁If ▁it ▁is ▁aff ine , ▁it ▁is ▁called ▁an ▁aff ine ▁Tod a ▁field ▁theory ▁( after ▁the ▁component ▁of ▁ φ ▁which ▁dec ou ples ▁is ▁removed ) ▁and ▁if ▁it ▁is ▁hyper bol ic , ▁it ▁is ▁called ▁a ▁hyper bol ic ▁Tod a ▁field ▁theory . ▁ ▁Tod a ▁field ▁theories ▁are ▁integra ble ▁models ▁and ▁their ▁solutions ▁describe ▁solit ons . ▁ ▁Ex amples ▁ ▁Li ou ville ▁field ▁theory ▁is ▁associated ▁to ▁the ▁A 1 ▁Cart an