Tzitzeica equation

The Tzitzeica equation is a nonlinear partial differential equation devised by Gheorghe Țițeica in 1907 in the study of differential geometry, describing surfaces of constant affine curvature.^[1] The Tzitzeica equation has also been used in nonlinear physics, being an integrable 1+1 dimensional Lorentz invariant system.^[2]

${\displaystyle u_{xy}=\exp(u)-\exp(-2u).}$

On substituting

${\displaystyle w(x,y)=\exp(u(x,y))}$

the equation becomes

${\displaystyle w(x,y)_{y,x}w(x,y)-w(x,y)_{x}w(x,y)_{y}-w(x,y)^{3}+1=0}$.

One obtains the traveling solution of the original equation by the reverse transformation ${\displaystyle u(x,y)=\ln(w(x,y))}$.