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Tzitzeica equation

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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. The Tzitzeica equation has also been used in nonlinear physics, being an integrable 1+1 dimensional Lorentz invariant system.

u x y = exp ⁡ ( u ) − exp ⁡ ( − 2 u ) . {\displaystyle u_{xy}=\exp(u)-\exp(-2u).}

On substituting

w ( x , y ) = exp ⁡ ( u ( x , y ) ) {\displaystyle 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 {\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 u ( x , y ) = ln ⁡ ( w ( x , y ) ) {\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
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