Advertisement

Journal of Mathematical Sciences

, Volume 236, Issue 6, pp 594–602 | Cite as

Special Class of Liouville-Type Hyperbolic Equations

  • A. V. ZhiberEmail author
  • A. M. Yur’eva
Article
  • 8 Downloads

Abstract

In this paper, we obtain differential substitutions for a special class of Liouville-type equations that simplify these equations. We present necessary and sufficient conditions of the existence of y-integrals of the second order. We examine the principal case of necessary and sufficient conditions of the existence of y-integrals.

Keywords

x-integral y-integral Liouville-type equation differential substitution 

AMS Subject Classification

35Q51 37K60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. V. Kaptsov, Methods of Integration of Partial Differential Equations [in Russian], Fizmatlit, Moscow (2009).Google Scholar
  2. 2.
    O. V. Kaptsov, “On the Goursat classification problem,” Programmirovanie, 2, 68–71 (2012).MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. E. Laine “Sur l’application de la methode de Darboux aux equations s = f(x, y, z, p, q),” Compt. Rend. V., 182, 1126–1127 (1926).zbMATHGoogle Scholar
  4. 4.
    A. V. Zhiber and V. V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type,” Usp. Mat. Nauk, 56, No. 1 (337), 63–106 (2001).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics with Computer Center of Ufa Scientific Center of Russian Academy of SciencesUfaRussia
  2. 2.Bashkir State UniversityUfaRussia

Personalised recommendations