Tzitzeica equation explained

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,} 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

u(x,y) =ln (w(x,y)) .

References

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

Further reading

• Book: Griffiths, Graham W.. Schiesser. William E.. Introduction to Traveling Wave Analysis. 2012. Traveling Wave Analysis of Partial Differential Equations. Elsevier/Academic Press. Amsterdam. 10.1016/b978-0-12-384652-5.00001-7. 978-0-12-384652-5.
• Book: Enns. Richard H.. Nonlinear physics with Maple for scientists and engineers. 1997. Birkhäuser. McGuire. George C.. 0-8176-3838-5. Boston. 36130678.
• Book: Shingareva. Inna. Solving nonlinear partial differential equations with Maple and Mathematica. 2011. Springer. Lizárraga-Celaya. Carlos. 978-3-7091-0517-7. Vienna. 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
• George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998