Lin–Tsien equation

The Lin–Tsien equation (named after C. C. Lin and H. S. Tsien) is an integrable partial differential equation

${\displaystyle 2u_{tx}+u_{x}u_{xx}-u_{yy}=0.}$

Integrability of this equation follows from its being, modulo an appropriate linear change of dependent and independent variables, a potential form of the dispersionless KP equation. Namely, if ${\displaystyle u}$ satisfies the Lin–Tsien equation, then ${\displaystyle v=u_{x}}$ satisfies, modulo the said change of variables, the dispersionless KP equation. The Lin-Tsien equation admits a (3+1)-dimensional integrable generalization, see. ^[1]