SYMMETRIES AND INVARIANT SOLUTIONS OF GENERALIZED MODIFIED LIN – REISSNER – TSIEN EQUATIONS
https://doi.org/10.26583/vestnik.2024.6.4
EDN: ORVJUU
Abstract
The article provides a group analysis of nonlinear second-order partial differential equations that model the propagation of shear waves in a nonlinear elastic cylindrical shell interacting with an external elastic medium. The equations contain cubic nonlinearity and generalize the well-known models of Lin – Reissner – Tsian and Khokhlov – Zabolotskaya. Their classical symmetries are found using a universal algorithm of commutative algebra, which consists of constructing a Gröbner basis of a system of defining equations to find the explicit form of the generating function of the symmetry group. To construct solutions that are invariant under a group of shifts in the space of independent variables, the hodograph method was used, which made it possible to move from a nonlinear partial differential equation to a system of linear equations with variable coefficients. For the self-similar regime, invariant under extensions, a nonlinear equation is obtained, the linear part of which is exactly solved in terms of Bessel functions and trigonometric functions. The conditions necessary for the physical realizability of exact solutions are established.
Supporting agencies: The study was supported by the Russian Science Foundation grant No. 24-29-00071, https://rscf.ru/project/24-29-00071/.
About the Authors
A. I. Zemlyanukhin
Yuri Gagarin State Technical University of Saratov
Russian Federation
Russian Federation
A. V. Bochkarev
Yuri Gagarin State Technical University of Saratov
Russian Federation
Russian Federation
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For citations:
Zemlyanukhin A.I., Bochkarev A.V. SYMMETRIES AND INVARIANT SOLUTIONS OF GENERALIZED MODIFIED LIN – REISSNER – TSIEN EQUATIONS. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2024;13(6):403-410. (In Russ.) https://doi.org/10.26583/vestnik.2024.6.4. EDN: ORVJUU
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