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On Simple Solutions of Some Equations of Mathematical Physics

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Abstract

All solutions to the Burgers, Hopf, Helmholtz, Klein–Gordon, sine-Gordon, Schrödinger, and Monge–Ampere equations having analytical complexity one (simple solutions) are described. It turns out that all simple solutions of the Burgers and Hopf equation are represented by elementary functions. An example of a family of solutions of complexity two to the Burgers equation is presented. Simple solutions to the Helmholtz (or Klein–Gordon) equation are expressed in terms of Bessel functions and elementary functions. For the Laplace and wave equations, an explicit description is given for the simple solutions that are expressed in terms of Jacobi elliptic functions. Open problems of the theory of analytic complexity (the analytical spectrum of an equation) are discussed.

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Funding

The research was financially supported by the RFBR grant \(18-51-41011\) Uzb t.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    V. K. Beloshapka

Authors
  1. V. K. Beloshapka
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Correspondence to V. K. Beloshapka.

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Beloshapka, V.K. On Simple Solutions of Some Equations of Mathematical Physics. Russ. J. Math. Phys. 27, 309–325 (2020). https://doi.org/10.1134/S1061920820030036

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  • DOI: https://doi.org/10.1134/S1061920820030036

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