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Symmetry of the barochronous motions of a gas. (Russian, English) Zbl 1035.35006
Sib. Mat. Zh. 44, No. 5, 1098-1109 (2003); translation in Sib. Math. J. 44, No. 5, 857-866 (2003).
The author studies the nonlinear system of differential equations \[ u_t^i + u^iu_j^i = 0,\quad u_i^i = -\rho_t/\rho, \quad u_j^i = \partial u^i/\partial x^j, \quad \rho = \rho(t),\quad i = 1,2,3, \] which manifests constancy of algebraic invariants of the Jacobi matrix for smooth vector fields in the three-dimensional space \(\mathbb R^3\). The system under consideration arises in the context of the theory of gas dynamics equations and describes the barochronous motions of a gas. For such a system, the author looks for an admissible local Lie group and proves that the above-mentioned system admits an eighteen-dimensional Lie algebra. The structure of the Lie algebra obtained is presented in detail that enables us to construct a general solution to the original system. The author also exposes some new problems which appear in the framework of the problem of the barochronous motions of a gas.
MSC:
35A30 Geometric theory, characteristics, transformations in context of PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
22E70 Applications of Lie groups to the sciences; explicit representations
58J70 Invariance and symmetry properties for PDEs on manifolds
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