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On the possibility of exact reciprocal transformations for one-soliton solutions to equations of the Lobachevsky class

Abstract

Problems on reciprocal transformation of solutions to equations of Λ2-class (equations related to special coordinate nets on the Lobachevsky plane Λ2) are discussed. A method of construction of solutions to one analytic differential equation of Λ2-class by a given solution of another analytic differential equation of this class is proposed. The reciprocal transformation of one-soliton solutions of the sine-Gordon equation and one-soliton solutions of the modified Korteweg-de Vries equation (MKdV) is obtained. This result confirms the possibility of construction of such transition.

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References

  1. 1.

    A. V. Pogorelov, Differential Geometry [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  2. 2.

    A. G. Popov, Dokl. Akad. Nauk SSSR, 312, No. 5, 1109–1111 (1990).

    Google Scholar 

  3. 3.

    E. G. Poznyak and A. G. Popov, “Geometry of the sine-Gordon equation,” in: Itogi Nauki Tekhn., Probl. Geom., 23, All-Russian Institute for Scientific and Technical Information, Moscow (1991), pp. 99–130.

    Google Scholar 

  4. 4.

    E. G. Poznyak and A. G. Popov, “Lobachevsky geometry and equations of mathematical physics,” Dokl. Ross. Akad. Nauk, 332, No. 4, 418–421 (1993).

    Google Scholar 

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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.

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Ratinsky, M.S. On the possibility of exact reciprocal transformations for one-soliton solutions to equations of the Lobachevsky class. J Math Sci 141, 1071–1074 (2007). https://doi.org/10.1007/s10958-007-0034-4

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Keywords

  • Fundamental Form
  • Asymptotic Form
  • Vries Equation
  • Gauss Equation
  • Asymptotic Direction