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Asymptotics of the Riemann–Hilbert Problem for the Somov Model of Magnetic Reconnection of Long Shock Waves

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Abstract

We consider the Riemann–Hilbert problem in a domain of complicated shape (the exterior of a system of cuts), with the condition of growth of the solution at infinity. Such a problem arises in the Somov model of the effect of magnetic reconnection in the physics of plasma, and its solution has the physical meaning of a magnetic field. The asymptotics of the solution is obtained for the case of infinite extension of four cuts from the given system, which have the meaning of shock waves, so that the original domain splits into four disconnected components in the limit. It is shown that if the coefficient in the condition of growth of the magnetic field at infinity consistently decreases in this case, then this field basically coincides in the limit with the field arising in the Petschek model of the effect of magnetic reconnection.

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Notes

  1. For the boundary elements (“simple ends” in the Carathéodory terminology); see [20], [21]. In the case where this does not lead to misunderstandings, we will simply call them boundary points.

  2. The boundary elements of other kinds are represented by a continuum consisting of more than one point.

  3. Not equal to infinity or not identically zero at the end points of the domain.

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Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation as a part of the program of Moscow Center for Fundamental and Applied Mathematics (grant no. 075-15-2019-1621).

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Authors and Affiliations

  1. Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, 119333, Russia

    S. I. Bezrodnykh & V. I. Vlasov

  2. Sternberg Astonomical Insitute of Lomonosov Moscow State University, Moscow, 119992, Russia

    S. I. Bezrodnykh

  3. Moscow Center for Fundamental and Applied Mathematics of Lomonosov Moscow State University, Moscow, 119991, Russia

    V. I. Vlasov

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  1. S. I. Bezrodnykh
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  2. V. I. Vlasov
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Correspondence to S. I. Bezrodnykh.

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Bezrodnykh, S.I., Vlasov, V.I. Asymptotics of the Riemann–Hilbert Problem for the Somov Model of Magnetic Reconnection of Long Shock Waves. Math Notes 110, 853–871 (2021). https://doi.org/10.1134/S0001434621110225

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