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

References

1. Tzitzéica, G. (1907). "Sur une nouvelle classes de surfaces". Comptes rendus de l'Académie des Sciences. 144: 1257–1259. JFM 38.0642.01.
2. Polyanin, Andrei D.; Zaitsev, Valentin F. (2016-04-19). Handbook of Nonlinear Partial Differential Equations (2nd ed.). Chapman & Hall/CRC. pp. 540–542. doi:10.1201/b11412. ISBN 978-0-429-15037-1.

Further reading

• Griffiths, Graham W.; Schiesser, William E. (2012). "Introduction to Traveling Wave Analysis". Traveling Wave Analysis of Partial Differential Equations. Amsterdam: Elsevier/Academic Press. doi:10.1016/b978-0-12-384652-5.00001-7. ISBN 978-0-12-384652-5.
• Enns, Richard H.; McGuire, George C. (1997). Nonlinear physics with Maple for scientists and engineers. Boston: Birkhäuser. ISBN 0-8176-3838-5. OCLC 36130678.
• Shingareva, Inna; Lizárraga-Celaya, Carlos (2011). Solving nonlinear partial differential equations with Maple and Mathematica. Vienna: Springer. ISBN 978-3-7091-0517-7. OCLC 755068833.
• 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