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TMF, 1997, Volume 112, Number 1, Pages 47–66 (Mi tmf1026)  

This article is cited in 4 scientific papers (total in 4 papers)

Reduction of Hugoniot–Maslov chains for trajectories of solitary vortices of the “shallow water” equations to the Hill equation

S. Yu. Dobrokhotov

A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: According to V. Maslov's idea, many of 2D quasilinear hyperbolic systems of PDE posses only 3 types of singularities in generic positions with properties of “structure” self-similarity and stability. They are shock waves, “narrow” solitons and “square root” point singularities (solitary vortices). Their propogations are described by infinite chains of ODE that generalize the well known Hugoniot conditions for shock waves. After some resonable closing of the chain for solitary vortices of the “shallow water” equations we obtain the nonlinear system of 16 ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. It means that in some approximation the trajectory of solitary vortex can be decribed by the Hill equation. This result can be used also for prediction of a future trajectory of the centre of solitary vortices via its observable part.

DOI: https://doi.org/10.4213/tmf1026

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English version:
Theoretical and Mathematical Physics, 1997, 112:1, 827–843

Bibliographic databases:

Received: 03.02.1997

Citation: S. Yu. Dobrokhotov, “Reduction of Hugoniot–Maslov chains for trajectories of solitary vortices of the “shallow water” equations to the Hill equation”, TMF, 112:1 (1997), 47–66; Theoret. and Math. Phys., 112:1 (1997), 827–843

Citation in format AMSBIB
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\by S.~Yu.~Dobrokhotov
\paper Reduction of Hugoniot--Maslov chains for trajectories of solitary vortices of the ``shallow water'' equations to the Hill equation
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\yr 1997
\vol 112
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\pages 47--66
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\transl
\jour Theoret. and Math. Phys.
\yr 1997
\vol 112
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\pages 827--843
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Dobrokhotov, SY, “Hugoniot-Maslov chains for solitary vortices of the shallow water equations, I. - Derivation of the chains for the case of variable Coriolis forces and reduction to the Hill equation”, Russian Journal of Mathematical Physics, 6:2 (1999), 137  mathscinet  zmath  isi
    2. S. Yu. Dobrokhotov, “Integrability of truncated Hugoniot–Maslov chains for trajectories of mesoscale vortices on shallow water”, Theoret. and Math. Phys., 125:3 (2000), 1724–1741  mathnet  crossref  crossref  mathscinet  zmath
    3. S. Yu. Dobrokhotov, E. S. Semenov, B. Tirozzi, “Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory”, Journal of Mathematical Sciences, 124:5 (2004), 5209–5249  mathnet  crossref  mathscinet  zmath
    4. Dobrokhotov S., Tirozzi B., “A perturbative theory of the evolution of the center of typhoons”, Zeta Functions, Topology and Quantum Physics, Developments in Mathematics, 14, 2005, 31–50  crossref  mathscinet  zmath  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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