Hugoniot-Maslov chains for the system of shallow-water equations taking into account energy exchange.

*(English. Russian original)*Zbl 1331.76029
Math. Notes 78, No. 5, 740-743 (2005); translation from Mat. Zametki 78, No. 5, 796-799 (2005).

From the text: Singular solutions of the system of shallow-water equations with structure of the “type of the square root” of a quadratic form were introduced in [V. P. Maslov, Usp. Mat. Nauk 35, No. 2, 252–253 (1980), see Sessions of the Petrovskiĭ Seminar on differential equations and problems of mathematical physics, http://mi.mathnet.ru/eng/umn/v35/i2/p251]. To such solutions one assigns a Hugoniot-Maslov chain, which is an infinite chain of ordinary differential equations essentially determining the solution dynamics. These chains have many interesting properties (see the literature cited in the bibliography). The aim of this note is to describe the Hugoniot-Maslov chain for a more complicated system of equations describing the dynamics of two-dimensional atmospheric vortices with the atmosphere-ocean energy exchange taken into account [F. V. Dolzhanskiĭ, V. A. Krymov and D. Yu. Manin, Sov. Phys., Usp. 33, No. 7, 495–520 (1990); translation from Usp. Fiz. Nauk 160, No. 7, 1–47 (1990), doi:10.3367/UFNr.0160.199007a.0001]. The dimensionless form of this system is studied here.

##### MSC:

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

35L67 | Shocks and singularities for hyperbolic equations |

35Q35 | PDEs in connection with fluid mechanics |

86A10 | Meteorology and atmospheric physics |

##### Keywords:

shallow-water equations; square-root singular solution; two-dimensional vortex dynamics; atmospheric vertices; Hugoniot-Maslov chain
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\textit{S. Yu. Dobrokhotov} et al., Math. Notes 78, No. 5, 740--743 (2005; Zbl 1331.76029); translation from Mat. Zametki 78, No. 5, 796--799 (2005)

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##### References:

[1] | V. P. Maslov, Uspekhi Mat.Nauk [Russian Math. Surveys], 35 (1980), no. 2, 252–253. |

[2] | S. Yu. Dobrokhotov, Russ. J. Math. Phys., 6 (1999), no. 2, 137–173, no. 3, 282–313. |

[3] | S. Yu. Dobrokhotov, K. V. Pankrashkin, and E. S. Semenov, Russ. J. Math. Phys., 8 (2001), no. 1, 25–54. |

[4] | E. S. Semenov, Mat. Zametki [Math. Notes], 71 (2002), no. 6, 902–913. |

[5] | S. Yu. Dobrokhotov and B. Tirozzi, Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 384 (2002), no. 6, 741–746. |

[6] | S. Dobrokhotov, E. Semenov, and B. Tirozzi, Contemporary Mathematics: Fundamental Directions, 2 (2003), 5–44. |

[7] | S. Yu. Dobrokhotov, E. S. Semenov, and B. Tirozzi, Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 139 (2004), no. 1, 500–512. |

[8] | F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, Uspekhi Fiz. Nauk, 160 (1990), no. 7, 1–47. |

[9] | J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1979. Russian translation: Mir, Moscow, 1984. · Zbl 0429.76001 |

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