Skip to main content
SpringerLink
Log in

Distribution of energy of solutions of the wave equation on singular spaces of constant curvature and on a homogeneous tree

  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

In the paper, the Cauchy problem for the wave equation on singular spaces of constant curvature and on an infinite homogeneous tree is studied. Two singular spaces are considered: the first one consists of a three-dimensional Euclidean space to which a ray is glued, and the other is formed by two three-dimensional Euclidean spaces joined by a segment. The solution of the Cauchy problem for the wave equation on these objects is described and the behavior of the energy of a wave as time tends to infinity is studied.

The Cauchy problem for the wave equation on an infinite homogeneous tree is also considered, where the matching conditions for the Laplace operator at the vertices are chosen in the form generalizing the Kirchhoff conditions. The spectrum of such an operator is found, and the solution of the Cauchy problem for the wave equation is described. The behavior of wave energy as time tends to infinity is also studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. V. Sobolev and M. Solomyak, “Schrödinger Operators on Homogeneous Metric Trees: Spectrum in Gaps,” Rev. Math. Phys. 14 (5), 421–468 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Solomyak, “Laplace and Schrödinger Operators on Regular Metric Trees: the Discrete Spectrum Case,” Function Spaces, Differential Operators and Nonlinear Analysis, The Hans Triebel Anniversary Volume, 161–181 (2003).

    Chapter  Google Scholar 

  3. A. I. Shafarevich and A. V. Tsvetkova, “Cauchy Problem for the Wave Equation on a Homogeneous Tree,” Mat. Zametki 100 (6), 939–947 (2016) [in Russian].

    Article  Google Scholar 

  4. Yu. V. Pokornyi, O. M. Penkin, and V. L. Pryadiev, Differential Equations on Geometric Graphs (Fizmatlit, Moscow, 2005).

    MATH  Google Scholar 

  5. A. I. Shafarevich and A. V. Tsvetkova, “Solutions of the Wave Equation on Hybrid Spaces of Constant Curvature,” Russ. J. Math. Phys. 21 (4), 509–520 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bruning and V. Geyler, “Scattering on Compact Manifolds with Infinitely Thin Horns,” J. Math. Phys. 44, 371–405 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo, 101-1, Moscow, 119526, Russia

    A. V. Tsvetkova

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudny, Institutskii per., 9, Moscow Region, 141700, Russia

    A. V. Tsvetkova

Authors
  1. A. V. Tsvetkova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Tsvetkova.

Additional information

The research was supported by the Russian Foundation for Basic Research (grant 16-31-00339)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsvetkova, A.V. Distribution of energy of solutions of the wave equation on singular spaces of constant curvature and on a homogeneous tree. Russ. J. Math. Phys. 23, 536–550 (2016). https://doi.org/10.1134/S1061920816040099

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920816040099

Search

Navigation