Abstract
We study the potential double wave equation and the system of spatial double wave equations. In the class of solutions of multiple wave type, these equations are reduced to an ODE and the system of ODEs respectively. We find some exact solutions and obtain formulas for the contact lines of the corresponding double waves with a simple wave, show that in a neighborhood of an arbitrary point in the plane of self-similar variables there exists a special flow of potential double wave type, and construct a spatial double wave type flow around a specified smooth body.
Similar content being viewed by others
References
Sidorov A. F. and Yanenko N. N., “On the question about unsteady plane polytropic gas flows with rectangular characteristics,” Dokl. Akad. Nauk SSSR, vol. 123, no. 5, 832–834 (1958).
Ermolin E. V., Rubina L. I., and Sidorov A. F., “On the problem of two pistons,” J. Appl. Math. Mech., vol. 32, no. 5, 967–976 (1968).
Rubina L. I. and Ulyanov O. N., “A geometric method for solving nonlinear partial differential equations,” Trudy Inst. Mat. i Mekh. UrO RAN, vol. 16, no. 2, 130–145 (2010).
Rubina L. I. and Ulyanov O. N., “Towards the differences in behavior of solutions of linear and nonlinear heat-conduction equations,” Vestnik Yuzhno-Uralsk. Gos. Univ. Ser. Mat., Mech., Fiz., vol. 5, no. 2, 52–59 (2013).
Rubina L. I. and Ulyanov O. N., “On solving certain nonlinear acoustics problems,” Acoust. Phys., vol. 61, no. 5, 527–533 (2015).
Rubina L. I. and Ulyanov O. N., “On some method for solving a nonlinear heat equation,” Sib. Math. J., vol. 53, no. 5, 872–881 (2012).
Sidorov A. F., “Two-dimensional processes of the unbounded unshocked compression of a gas,” J. Appl. Math. Mech., vol. 61, no. 5, 787–796 (1997).
Sidorov A. F., “New regimes of the unbounded unshocked compression of a gas,” Dokl. Akad. Nauk, vol. 364, no. 2, 199–202 (1999).
Sidorov A. F., “Some three-dimensional gas flows adjacent to regions of rest,” Prikl. Mat. Mekh., vol. 32, no. 3, 369–380 (1968).
Rubina L. I. and Ulyanov O. N., “One method for solving systems of nonlinear partial differential equations,” Proc. Steklov Inst. Math., vol. 288, no. suppl. 1, 180–188 (2015).
Fushchich V. I., Zhdanov R. Z., and Revenko I. V., “General solutions of the nonlinear wave equation and of the eikonal equation,” Ukrainian Math. J., vol. 43, no. 11, 1364–1379 (1991).
Collins C. B., “Complex potential equations I. A technique for solution,” Math. Proc. Cambridge Philos. Soc., vol. 80, no. 2, 165–187 (1976).
Cartan È., Exterior Differential Systems and Its Geometric Applications [Russian translation], Moscow Univ., Moscow (1962).
Sidorov A. R., Shapeev V. P., and Yanenko N. N., The Method of Differential Connections and Its Applications to Gas Dynamics [Russian], Nauka, Novosibirsk (1984).
Ibragimov N. H., Meleshko S. V., and Rudenko O. V., “Group analysis of evolutionary integro-differential equations describing nonlinear waves: The general model,” J. Phys. A: Math. Theor., 2011. No. 44.315201.
Courant R., Partial Differential Equations, Interscience, New York (1962).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 4, pp. 859–873.
Rights and permissions
About this article
Cite this article
Rubina, L.I., Ulyanov, O.N. On Double Wave Type Flows. Sib Math J 60, 673–684 (2019). https://doi.org/10.1134/S0037446619040128
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446619040128