The Lin–Tsien equation (named after C. C. Lin and H. S. Tsien) is an integrable partial differential equation
2 u t x + u x u x x − u y y = 0. {\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 u {\displaystyle u} satisfies the Lin–Tsien equation, then v = u x {\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.
References
D. Zwillinger. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997. Ames, William F. (1965). Nonlinear partial differential equations in engineering, Vol. 18. Academic Press. p. 173. ISBN 9780080955247. Rogers, C.; Ames, William F. (1989). Nonlinear boundary value problems in science and engineering. Academic Press. p. 373. ISBN 9780080958705.