Abstract
The wave equation on an infinite homogeneous tree is studied. For the Laplace operator, the Kirchhoff conditions are taken as the matching conditions at the vertices. A solution of the Cauchy problem is obtained and the behavior of the wave energy as time tends to infinity is described. It is shown that part of the energy does not go to infinity, but remains on the edges of the trees. The part of the energy remaining on the edges depends on the branching number.
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References
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, in Math. Surveys Monogr. (Amer. Math. Soc., Providence, RI, 2013), Vol. 186.
A. V. Sobolev and M. Solomyak, “Shrödinger operators on homogeneous metric trees: spectrum in gaps,” Rev. Math. Phys. 14 (5), 421–468 (2002).
M. Solomyak, “Laplace and Shrödinger operators on regular metric trees: the discrete spectrum case,” in Function Spaces, Differential Operators and Nonlinear Analysis (Birkhäuser, Basel, 2003), pp. 161–181.
J. Brüning and V. A. Geyler, “Scattering on compact manifolds with infinitely thin horns,” J. Math. Phys. 44 (2), 371–405 (2003).
Yu. V. Pokornyi, O. M. Penkin, V. L. Pryadiev, A. V. Borovskikh, K. P. Lazarev, and S. A. Shabrov, Differential Equations on Geometric Graphs (Fizmatlit, Moscow, 2004) [in Russian].
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).
V. L. Chernyshev and A. I. Shafarevich, “Semiclassical asymptotics and statistical properties of Gaussian packets for the nonstationary Schroedinger equation on a geometric graph,” Russ. J. Math. Phys. 15 (1), 25–34 (2008).
A. A. Tolchennikov, V. L. Chernyshev, and A. I. Shafarevich, “Asymptotic properties and classical dynamical systems in quantum problems on singular spaces,” Nelin. Dinam. 6 (3), 623–638 (2010).
V. L. Chernyshev and A. I. Shafarevich, “Statistics of Gaussian packets on metric and decorated graphs,” Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (20130145) (2014).
O. V. Korovina and V. L. Pryadiev, “Structure of mixed problem solution for wave equation on compact geometrical graph in nonzero initial velocity case,” Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform. 9 (3), 37–46 (2009).
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Original Russian Text © A. V. Tsvetkova, A. I. Shafarevich, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 6, pp. 923–931.
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Tsvetkova, A.V., Shafarevich, A.I. The Cauchy problem for the wave equation on homogeneous trees. Math Notes 100, 862–869 (2016). https://doi.org/10.1134/S0001434616110262
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DOI: https://doi.org/10.1134/S0001434616110262