Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 091, 20 pages      arXiv:1111.2027      https://doi.org/10.3842/SIGMA.2014.091

A Reciprocal Transformation for the Constant Astigmatism Equation

Adam Hlaváč and Michal Marvan
Mathematical Institute in Opava, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava, Czech Republic

Received May 07, 2014, in final form August 14, 2014; Published online August 25, 2014

Abstract
We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter $\lambda = \pm1$. It is also a nonlocal symmetry.

Key words: constant astigmatism equation; exact solution; constant astigmatism surface; orthogonal equiareal pattern; reciprocal transformation; sine-Gordon equation.

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References

  1. Bäcklund A.V., Om ytor med konstant negativ krökning, Lunds Univ. Årsskrift 19 (1883), 1-48.
  2. Baran H., Marvan M., On integrability of Weingarten surfaces: a forgotten class, J. Phys. A: Math. Theor. 42 (2009), 404007, 16 pages, arXiv:1002.0989.
  3. Baran H., Marvan M., Classification of integrable Weingarten surfaces possessing an $\mathfrak{sl}(2)$-valued zero curvature representation, Nonlinearity 23 (2010), 2577-2597, arXiv:1002.0992.
  4. Bianchi L., Ricerche sulle superficie elicoidali e sulle superficie a curvatura costante, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1879), 285-341.
  5. Bianchi L., Lezioni di Geometria Differenziale, Vol. I, E. Spoerri, Pisa, 1902.
  6. Bianchi L., Lezioni di Geometria Differenziale, Vol. II, E. Spoerri, Pisa, 1903.
  7. Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor'kova N.G., Krasil'shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential equations of mathematical physics, Translations of Mathematical Monographs, Vol. 182, Amer. Math. Soc., Providence, RI, 1999.
  8. Ferapontov E.V., Reciprocal transformations and their invariants, Differ. Equ. 25 (1989), 898-905.
  9. Ferapontov E.V., Autotransformations with respect to the solution, and hydrodynamic symmetries, Differ. Equ. 27 (1991), 885-895.
  10. Ferapontov E.V., Rogers C., Schief W.K., Reciprocal transformations of two-component hyperbolic systems and their invariants, J. Math. Anal. Appl. 228 (1998), 365-376.
  11. Ganchev G., Mihova V., On the invariant theory of Weingarten surfaces in Euclidean space, J. Phys. A: Math. Theor. 43 (2010), 405210, 27 pages, arXiv:0802.2191.
  12. Goursat E., Le Probléme de Bäcklund, Gauthier-Villars, Paris, 1925.
  13. Hlaváč A., Marvan M., Another integrable case in two-dimensional plasticity, J. Phys. A: Math. Theor. 46 (2013), 045203, 15 pages.
  14. Hlaváč A., Marvan M., On Lipschitz solutions of the constant astigmatism equation, J. Geom. Phys., to appear.
  15. Hoenselaers C.A., Miccichè S., Transcendental solutions of the sine-Gordon equation, in Bäcklund and Darboux Transformations. The Geometry of Solitons (Halifax, NS, 1999), CRM Proc. Lecture Notes, Vol. 29, Amer. Math. Soc., Providence, RI, 2001, 261-271.
  16. Kingston J.G., Rogers C., Reciprocal Bäcklund transformations of conservation laws, Phys. Lett. A 92 (1982), 261-264.
  17. von Lilienthal R., Bemerkung über diejenigen Flächen bei denen die Differenz der Hauptkrümmungsradien constant ist, Acta Math. 11 (1887), 391-394.
  18. Lipschitz R., Zur Theorie der krummen Oberflächen, Acta Math. 10 (1887), 131-136.
  19. Manganaro N., Pavlov M.V., The constant astigmatism equation. New exact solution, J. Phys. A: Math. Theor. 47 (2014), 075203, 8 pages, arXiv:1311.1136.
  20. Marvan M., Some local properties of Bäcklund transformations, Acta Appl. Math. 54 (1998), 1-25.
  21. Pavlov M.V., Zykov S.A., Lagrangian and Hamiltonian structures for the constant astigmatism equation, J. Phys. A: Math. Theor. 46 (2013), 395203, 6 pages, arXiv:1212.6239.
  22. Polyanin A.D., Zaitsev V.F., Handbook of exact solutions for ordinary differential equations, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL, 2003.
  23. Prus R., Sym A., Rectilinear congruences and Bäcklund transformations: roots of the soliton theory, in Nonlinearity & Geometry, Luigi Bianchi Days (Warsaw, 1995), Editors D. Wójcik, J. Cieśliński, Polish Scientific Publishers, Warsaw, 1998, 25-36.
  24. Ribaucour A., Note sur les développées des surfaces, C. R. Math. Acad. Sci. Paris 74 (1872), 1399-1403.
  25. Rogers C., Schief W.K., Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
  26. Rogers C., Schief W.K., Szereszewski A., Loop soliton interaction in an integrable nonlinear telegraphy model: reciprocal and Bäcklund transformations, J. Phys. A: Math. Theor. 43 (2010), 385210, 16 pages.
  27. Rogers C., Shadwick W.F., Bäcklund transformations and their applications, Mathematics in Science and Engineering, Vol. 161, Academic Press, Inc., New York - London, 1982.
  28. Rogers C., Wong P., On reciprocal Bäcklund transformations of inverse scattering schemes, Phys. Scripta 30 (1984), 10-14.
  29. Sadowsky M.A., Equiareal pattern of stress trajectories in plane plastic strain, J. Appl. Mech. 8 (1941), A74-A76.
  30. Sadowsky M.A., Equiareal patterns, Amer. Math. Monthly 50 (1943), 35-40.
  31. Weatherburn C.E., Differential geometry of three dimensions, Cambridge University Press, Cambridge, 1927.

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