Email this article Solutions of Some Wave Mechanics Models
- Authors: Kaptsov O.V.1, Kaptsov D.O.1
-
Affiliations:
- Institute of Computational Modelling SB RAS
- Issue: Vol 87, No 2 (2023)
- Pages: 176-185
- Section: Articles
- URL: https://journals.rcsi.science/0032-8235/article/view/138850
- DOI: https://doi.org/10.31857/S003282352302008X
- EDN: https://elibrary.ru/TZNDZR
- ID: 138850
Cite item
Open Access
Access granted
Subscription Access
Abstract
We consider one-dimensional second order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found.
About the authors
O. V. Kaptsov
Institute of Computational Modelling SB RAS
Author for correspondence.
Email: kaptsov@icm.krasn.ru
Russia, Krasnoyarsk
D. O. Kaptsov
Institute of Computational Modelling SB RAS
Author for correspondence.
Email: hot.dok@gmail.com
Russia, Krasnoyarsk
References
- Brekhovskikh L.M. Waves in Layered Media. Moscow: Nauka, 1973. (in Russian)
- Kulikovskii A.G., Sveshnikova Elena I. Nonlinear Waves in Elastic Media. Boca Raton. CRC Press, 1995.
- Ovsyannikov L.V. Lectures on Basic Gas Dynamics., Moscow; Izhevsk: Inst. Comput. Sci., 2003. (in Russian)
- Rabotnov Yu.N. Elements of Hereditary Mechanics of Solids. Moscow: Nauka, 1977. (in Russian)
- Ovsyannikov L.V. Group analysis of differential equations. Moscow: Nauka, 1978. (in Russian)
- Ibragimov N. Transformation Groups in Mathematical Physics. Dordrecht: Reidel, 1985.
- Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevskii L.P. Soliton Theory: Inverse Scattering Method. Moscow: Nauka, 1980.
- Ablowitz M.J., Segur H. Solitons and the Inverse Scattering Transform. Philadelphia: Soc. Industr.&Appl. Math., 1981.
- Sidorov A.F., Shapeev V.P., Yanenko N.N. Method of Differential Relations and Its Applications in Gas Dynamics. Novosibirsk: Nauka, 1984.
- Euler L. Integral Calculus. Vol. 3. Moscow: GIFML, 1958.
- Darboux J.G. Lectures on the General Theory of Surfaces and Geometrical Applications of the Analysis of Infinitesimals. Vol. 2. Izhevsk: Inst. Comput. Sci., 2013.
- Kaptsov O.V. Methods for Integrating Partial Differential Equations. Moscow: Fizmatlit, 2009. (in Russian)
- Cherny G.G. Gas Dynamics. Moscow: Nauka, 1988. (in Russian)
- Novatsky V.K. Wave Problems of the Theory of Plasticity. Moscow, Mir, 1978.
- Medwin H., Clay C. Fundamentals of Acoustical Oceanography. Acad. Press, 1997.
- Ames W.F., Lohner R.J., Adams E. Group properties of // Int. J. Nonlin. Mech., 1981, vol. 16, pp. 439–447.
- Bluman G.W., Kumei S. On invariance properties of the wave equation // J. Math. Phys., 1987, vol. 28, pp. 307–318.
- Bluman G.W., Cheviakov A.F. Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation// J. Math. Anal. Appl., 2007, vol. 333, pp. 93–111.
- Pelinovsky E., Kaptsov O. Traveling waves in shallow seas of variable depths // Symmetry, 2022, vol. 14 (7), pp. 1448.
- Aksenov A.V. Symmetries and relations between solutions of the class Euler–Poisson–Darboux equations // Dokl. Math., 2001, vol. 64, no. 3, pp. 421–424.
- Kamke E. Handbook on First Order Partial Differential Equations. Moscow: Nauka, 1966.
- Galaktionov V., Svirshchevskii S. Exact Solutions and Invariant Subspaces of Nonlinear PDEs in Mechanics and Physics. Chapman&Hall/CRC Appl. Math.&Nonlin. Sci., 2006.
