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Anton A. Tolchennikov

Anton A. Tolchennikov

Candidate of Physical and Mathematical Sciences (PhD), Assistant Professor

Department of Differential Geometry and Applications
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
Main Building, 1 Leninskiye Gory, Moscow 119991, GSP-1, Russia

E-mail: tolchennikovaa@gmail.com

Publications Full list of publications

1. Life and career

Anton A. Tolchennikov was born in 1984 in Kaluga. He graduated from the Faculty of Mechanics and Mathematics in 2006 and entered the postgraduate programme of the Department of Differential Geometry and Applications.

In 2010, under the supervision of Professor Andrey I. Shafarevich, he defended his candidate dissertation “Spectral properties of the Laplace operator on decorated graphs and on surfaces with delta-potentials”.

Since 2010 he has run seminars on mathematical analysis and on the equations of mathematical physics at the Faculty of Physics and Technology of MIPT. In 2012 he began working at the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences. He joined the Department of Differential Geometry and Applications as assistant professor in 2014.

2. Degrees and positions

Born1984, Kaluga.
EducationFaculty of Mechanics and Mathematics, MSU, graduated 2006; postgraduate programme at the Department of Differential Geometry and Applications.
Candidate of Physical and Mathematical Sciences (PhD)2010; advisor Professor A. I. Shafarevich. Thesis: “Spectral properties of the Laplace operator on decorated graphs and on surfaces with delta-potentials”.
PositionsInstitute for Problems in Mechanics RAS (since 2012); Assistant Professor, Department of Differential Geometry and Applications, MSU (since 2014).
Research fieldsSpectral properties of operators on decorated graphs; asymptotic theory of partial differential equations; wave propagation on metric graphs; shallow-water and tsunami asymptotics; equations of mathematical physics.

3. Research

The main scientific interests of A. A. Tolchennikov are spectral properties of operators on decorated graphs and the asymptotic theory of partial differential equations.

His candidate work, carried out under Professor A. I. Shafarevich, was devoted to spectral properties of the Laplace–Beltrami operator defined on so-called decorated graphs, that is, cell complexes of a special form obtained by gluing points of closed manifolds with the help of segments. In particular, an estimate of the dimension of the kernel of such an operator was obtained in terms of the number of cycles of the graph formed only by the segments of the decorated graph.

Jointly with V. L. Chernyshev he studied the Schroedinger equation on metric graphs with an initial condition in the form of a localized wave packet. Reaching a vertex of the graph, part of the wave packet is reflected and part passes through, so that the number of carriers (points in whose neighbourhood the solution is localized) grows with time. For an arbitrary finite graph the leading term of the asymptotics was found, and in some examples the dependence of the next term on the ordering of the edges was described; for some decorated graphs the number of carriers was also found.

Under the supervision of Professor S. Yu. Dobrokhotov (Institute for Problems in Mechanics RAS) he worked on asymptotic formulae for the solution of the shallow-water system on a sphere with a localized initial condition, a problem that allows one to model sources of long waves in the ocean and tsunami generation. Related work concerns the asymptotics of linear water waves generated by localized sources, the two-dimensional Dirac equation for graphene, the Maxwell system with temporal dispersion, and semiclassical asymptotics near caustics.

4. Teaching

Since 2010 A. A. Tolchennikov has run seminars on mathematical analysis and on the equations of mathematical physics at MIPT (Faculty of Physics and Technology), and since 2014 he has taught at the Department of Differential Geometry and Applications of MSU.

5. Publications

Full list of publications — 137 records (department page and ISTINA profile)

  1. On the kernel of Laplace–Beltrami operators with zero-radius potential and on a decorated graph. Matematicheskii Sbornik, vol. 199, no. 7 (2008).
  2. Kernel and trace formula for the exponential of the Laplace–Beltrami operator on a decorated graph. Russian Journal of Mathematical Physics, vol. 15, no. 1 (2008), pp. 128–139.
  3. A. A. Tolchennikov, V. L. Chernyshev, A. I. Shafarevich. Asymptotic properties of classical dynamical systems in quantum problems on singular spaces. Russian Journal of Mathematical Physics (2010).
  4. The number of points moving on a metric tree. Russian Journal of Mathematical Physics, vol. 24, no. 3 (2017), pp. 290–298.
  5. S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov. Asymptotics of linear water waves generated by a localized source near the focal points on the leading edge. Russian Journal of Mathematical Physics, vol. 24, no. 4 (2017), pp. 544–552.
  6. V. L. Chernyshev, A. A. Tolchennikov. The second term in the asymptotics for the number of points moving along a metric graph. Regular and Chaotic Dynamics, vol. 22, no. 8 (2017), pp. 940–951.
  7. P. S. Petrov, S. A. Sergeev, A. A. Tolchennikov. Modelling of pulse signals in 3D propagation problems of deep-water acoustics based on the modified Maslov canonical operator. Russian Journal of Mathematical Physics, vol. 25, no. 1 (2018), pp. 102–112.
  8. Simple exact solutions and asymptotics of localized solutions of the two-dimensional massless Dirac equation for graphene. Theoretical and Mathematical Physics, 196:1 (2018), pp. 64–69.
  9. V. L. Chernyshev, A. A. Tolchennikov. Asymptotic estimate for the counting problems corresponding to the dynamical system on some decorated graphs. Ergodic Theory and Dynamical Systems, vol. 38 (2018), pp. 1697–1708.
  10. V. Chernyshev, A. Tolchennikov. Polynomial approximation for the number of all possible endpoints of a random walk on a metric graph. Electronic Notes in Discrete Mathematics, vol. 70 (2018), pp. 31–35.
  11. S. Ya. Sekerzh-Zenkovich, A. A. Tolchennikov. Application of an asymptotic solution of the problem of linear wave propagation on water to the approximation of tsunami mareograms of 2011 obtained at two DART stations. Russian Journal of Mathematical Physics, 26:1 (2019), pp. 70–74.
  12. On the behaviour of the solution of the Klein–Gordon equation with a localized initial condition. Theoretical and Mathematical Physics, 199:2 (2019), pp. 761–770.
  13. S. Yu. Dobrokhotov, A. A. Tolchennikov. Solution of the two-dimensional Dirac equation with a linear potential and a localized initial condition. Russian Journal of Mathematical Physics, vol. 26, no. 2 (2019), pp. 139–151.
  14. Asymptotics of the trace of the resolvent of the Laplace operator on a metric graph with arbitrary boundary conditions. Russian Journal of Mathematical Physics, vol. 26, no. 2 (2019), pp. 208–218.
  15. P. S. Petrov, S. A. Sergeev, A. A. Tolchennikov. On the use of asymptotic formulae based on the modified Maslov canonical operator in modelling the propagation of pulsed acoustic signals in three-dimensional shallow-sea waveguides. Acoustic Journal, vol. 65, no. 6 (2019), pp. 799–807.
  16. S. Yu. Dobrokhotov, M. V. Klimenko, I. A. Nosikov, A. A. Tolchennikov. Variational method for computing ray trajectories and fronts of tsunami waves generated by a localized source. Computational Mathematics and Mathematical Physics, 60 (2020), pp. 1392–1401.
  17. S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov. Uniform formulae for the asymptotic solution of a linear pseudodifferential equation describing water waves generated by a localized source. Russian Journal of Mathematical Physics, 27 (2020), pp. 185–191.
  18. V. L. Chernyshev, D. S. Minenkov, A. A. Tolchennikov. The number of endpoints of a random walk on a semi-infinite metric path graph. Theoretical and Mathematical Physics, 207 (2021), pp. 487–493.
  19. On the effect of intersection of characteristics in a two-dimensional massless Dirac equation with linear potential and localized initial condition. Russian Journal of Mathematical Physics, 28 (2021), pp. 265–269.
  20. V. L. Chernyshev, A. A. Tolchennikov. Asymptotics of the number of endpoints of a random walk on a certain class of directed metric graphs. Russian Journal of Mathematical Physics, vol. 28, no. 4 (2021), pp. 434–438.
  21. S. Yu. Dobrokhotov, A. A. Tolchennikov. Keplerian trajectories and an asymptotic solution of the Schroedinger equation with repulsive Coulomb potential and a localized right-hand side. Russian Journal of Mathematical Physics, vol. 29, no. 4 (2022), pp. 442–452.
  22. V. E. Nazaikinskii, A. A. Tolchennikov. Constructive implementation of semiclassical asymptotic formulae in a neighbourhood of a generic caustic cusp. Russian Journal of Mathematical Physics, vol. 29, no. 4 (2022), pp. 545–554.
  23. V. L. Chernyshev, A. A. Tolchennikov. A metric graph for which the number of possible end positions of a random walk grows minimally. Russian Journal of Mathematical Physics, vol. 29, no. 4 (2022), pp. 427–431.
  24. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov. Uniformization and semiclassical asymptotics for a class of equations degenerating on the boundary of a manifold. Journal of Mathematical Sciences, 270 (2023), pp. 507–530.
  25. S. Yu. Dobrokhotov, S. B. Levin, A. A. Tolchennikov. Keplerian trajectories and global Airy-function-type asymptotics for the scattering problem with a repulsive Coulomb potential. Uspekhi Matematicheskikh Nauk, 78:4 (2023), pp. 205–206.
  26. Trace of the resolvent of the Laplace operator on a metric graph. Russian Journal of Mathematical Physics, vol. 30, no. 4 (2023), pp. 703–711.
  27. I. A. Nosikov, A. A. Tolchennikov, M. V. Klimenko. Boundary value problem of calculating ray characteristics of ocean waves reflected from a coastline. Computational Mathematics and Mathematical Physics, vol. 64, no. 3 (2024), pp. 497–508.
  28. I. A. Bogaevskii, S. Yu. Dobrokhotov, A. A. Tolchennikov. Arnold's Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side. Theoretical and Mathematical Physics, 218:1 (2024), pp. 19–40.
  29. S. Yu. Dobrokhotov, A. A. Tolchennikov. Uniform formulae for the asymptotic solution near the leading front for Maxwell's equations with temporal dispersion and localized initial data. Mathematical Notes, 116 (2024), pp. 458–464.
  30. S. Yu. Dobrokhotov, A. I. Klevin, V. E. Nazaikinskii, A. A. Tolchennikov. Asymptotics of solutions to systems of (pseudo)differential equations with localized right-hand sides. Journal of Physics: Conference Series, vol. 2817 (2024).
  31. M. V. Vakhitov, D. S. Minenkov, A. A. Tolchennikov, V. L. Chernyshev. Asymptotics of the counting function of lengths corresponding to classes of equivalent routes for an infinite binary metric tree. Discrete Mathematics, 37:4 (2025), pp. 3–15.
  32. S. Yu. Dobrokhotov, V. E. Nazaikinskii, I. A. Nosikov, A. A. Tolchennikov. Asymptotics of long waves generated by time-harmonic spatially localized sources in basins with gently sloping shores. Computational Mathematics and Mathematical Physics, vol. 65, no. 5 (2025), pp. 625–640.
  33. S. Yu. Dobrokhotov, I. A. Nosikov, A. A. Tolchennikov. The Maupertuis–Jacobi principle and Fermat's variational principle in the problem of short-wave asymptotics of the solution of the Helmholtz equation with a localized source. Computational Mathematics and Mathematical Physics, vol. 65, no. 4 (2025), pp. 446–459.
  34. S. Yu. Dobrokhotov, S. B. Levin, A. A. Tolchennikov. Global uniform Airy-function-type asymptotics for the scattering problem with a repulsive Coulomb potential and Keplerian trajectories. Matematicheskii Sbornik, vol. 216, no. 8 (2025).
  35. Asymptotic solution of the Maxwell system with a time-harmonic and spatially localized source of the electromagnetic field. Mathematical Notes, 118:6 (2025), pp. 963–966.
  36. I. A. Bogaevskii, S. Yu. Dobrokhotov, A. A. Tolchennikov. Effect of multiplicity switching of terms in the Cauchy problem for the Dirac equation in graphene with a constant electric field and a localized initial condition. Theoretical and Mathematical Physics, 226:2 (2026), pp. 256–276.