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]

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

On substituting

$w(x, y) = exp(u(x, y))$

the equation becomes

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

Obtain the traveling solution of the original equation by the reverse transformation $u(x,y)=ln(w(x,y))$.

References

1. ↑ G. Tzitz´eica, “Geometric infinitesimale-sur une nouvelle classes de surfaces,”Comptes Rendus de l’Acad´emie des Sciences, vol. 144, pp. 1257–1259, 1907.
2. ↑ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS

Additional reading

• Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations Academy Press
• Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
• Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
• Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
• Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
• Dongming Wang, Elimination Practice,Imperial College Press 2004
• David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
• George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759