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An analogue of the Moutard transformation for the Goursat equation\(\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } \)

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Abstract

We present a new Bäcklund-type transformation for the nonlinear equation\(\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } \) studied by É. Goursat. Goursat found a linearization transformation and some properties of this equation, which make it similar to the Moutard equation uxy=M(x, y)u. However, this Goursat transformation does not provide proper superposition formulas. We give the necessary extended superposition formulas.

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References

  1. T. Moutard,J. École Polytech. 45, 1–11 (1878).

    Google Scholar 

  2. C. Athorne and J. J. C. Nimmo,Inverse Problems,7, 809–826 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. V. B. Matveev and M. A. Salle,Darboux Transformations and Solitons, Springer, Berlin (1991).

    MATH  Google Scholar 

  4. J. J. C. Nimmo and W. K. Schief,Proc. Roy. Soc. London A,453, 255–279 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. É. Goursat,Bull. Soc. Math. France,25, 36–48 (1897).

    MathSciNet  Google Scholar 

  6. É. Goursat,Bull. Soc. Math. France,28, 1–6 (1900).

    MathSciNet  MATH  Google Scholar 

  7. G. Darboux,Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Vols. 1–4, Gautier-Villars, Paris (1887–1896).

    MATH  Google Scholar 

  8. C. Athorne,Inverse Problems,9, 217–232 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. B. G. Konopelchenko, “Weierstrass representation for surfaces in 4D space and their integrable deformations via DS hierarchy,” Preprint math-dg/9807129 (1998).

  10. L. P. Eisenhart,A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York (1909).

    Google Scholar 

  11. L. Bianchi,Lezioni di geometria differenziale (3rd ed.), Vols. 1–4, Zanichelli, Bologna (1923–1927).

    MATH  Google Scholar 

  12. R. M. Miura, ed.,Bäcklund Transformations (Lect. Notes Math., Vol. 515), Springer, Berlin (1976).

    MATH  Google Scholar 

  13. E. I. Ganzha and S. P. Tsarev,Russ. Math. Surv.,51, 1200–1202, (1996); Preprint solv-int/9606003 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  14. E. I. Ganzha, “On completeness on the Moutard transformations,” Preprint solv-int/9606001 (1996).

  15. E. I. Ganzha, “On completeness of the Ribaucour transformations for triply orthogonal curvilinear coordinate systems in R3,” Preprint solv-int/9606002 (1996).

  16. O. V. Kaptsov and Yu. V. Shanko, “Trilinear representation and Moutard transformation for the Tzitzéica equation,” Preprint solv-int/9704014 (1997).

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Authors and Affiliations

  1. Krasnoyarsk State Pedagogical University, Krasnoyarsk, Russia

    E. I. Ganzha

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  1. E. I. Ganzha
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 50–57, January, 1999.

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Ganzha, E.I. An analogue of the Moutard transformation for the Goursat equation\(\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } \) . Theor Math Phys 122, 39–45 (2000). https://doi.org/10.1007/BF02551168

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