Advertisement

Theoretical and Mathematical Physics

, Volume 177, Issue 2, pp 1471–1478 | Cite as

Periodic solutions of the Hopf equation

  • A. B. Shabat
Article

Abstract

We consider the long-time dynamics of approximate solutions of the boundary-value problem for the Hopf equation on a finite segment. Together with the initial conditions, for instance, we impose the zero Dirichlet conditions on both ends of the segment. In this case, all features of solutions associated with the intersections of characteristics are accumulated on a strip bounded by the vertical characteristics emitted from the boundary points.

Keywords

generalized solution conservation law Galerkin method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. A. Oleinik, Am. Math. Soc. Transl. 2, 26, 95–172 (1963).MathSciNetGoogle Scholar
  2. 2.
    B. L. Rozhdestvenskii, Russ. Math. Surveys, 15, No. 5, 53–111 (1960).ADSCrossRefGoogle Scholar
  3. 3.
    E. Ferapontov, Phys. Lett. A, 158, 112–118 (1991).MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    A. B. Shabat, Vladikavkaz Math. J., 14, No. 4, 83–94 (2012).MathSciNetGoogle Scholar
  5. 5.
    E. Hopf, Commun. Pure Appl. Math., 3, 201–230 (1950).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Aliev Karachaev-Cherkess State UniversityKarachaevskRussia
  2. 2.Landau Institute for Theoretical PhysicsRASChernogolovka, Moscow OblastRussia

Personalised recommendations