Group Classification of the System of Equations of Two-Dimensional Shallow Water over Uneven Bottom
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Group Classification of the System of Equations of Two-Dimensional Shallow Water over Uneven Bottom A. V. Aksenov∗,∗∗,∗∗∗,1 and K. P. Druzhkov∗,∗∗,∗∗∗∗,2 ∗
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory 1, Moscow, 119234 Russia ∗∗ Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia ∗∗∗ National Research Nuclear University MEPhI 31 Kashirskoe Shosse, Moscow, 115409 Russia ∗∗∗∗ Moscow Institute of Physics and Technology (National Research University) Dolgoprudny, Moscow Region, 141700 Russia, E-mail: 1 [email protected], 2 [email protected] Received February 2, 2020; Revised February 15, 2020; Accepted March 15, 2020
Abstract. A system of equations of two-dimensional shallow water over an uneven bottom is considered. An overdetermined system of equations for finding the corresponding symmetries is obtained. The compatibility of this overdetermined system of equations is investigated. A general form of the solution of the overdetermined system is found. The kernel of the symmetry operators is found. The cases of kernel extensions of symmetry operators are presented. The results of group classification indicate that the system of equations of two-dimensional shallow water over an uneven bottom cannot be linearized by point transformation, in contrast to the system of equations of one-dimensional shallow water in the cases of horizontal and inclined bottom profiles. DOI 10.1134/S1061920820030012
1. INTRODUCTION The system of equations of one-dimensional shallow water above an uneven bottom was considered in [1]. The group classification problem was solved, and all hydrodynamic conservation laws were found there. It was also shown that the system of equations of one-dimensional shallow water is linearizable by a point transformation of variables only in cases of horizontal and inclined bottom profiles. In dimensionless variables, the system of two-dimensional shallow-water equations over an uneven bottom has the following form [2]: ut + uux + vuy + ηx = 0 , vt + uvx + vvy + ηy = 0 , ηt + (η + h)u x + (η + h)v y = 0 .
(1.1)
Here u = u(x, y, t), v = v(x, y, t) are the components of a depth-averaged horizontal velocity, η = η(x, y, t) is a deviation of the free surface, h = h(x, y), z = −h is a bottom profile, and η + h 0. Remark 1. System (1.1) is similar to that of equations of one-dimensional gas dynamics [3, 4]. In this paper, the problem of group classification is solved for the system of equations (1.1). This is a further development of [5, 6]. 2. SYSTEM OF DETERMINING EQUATIONS We seek symmetry operators of system (1.1) in the form X = ξ 1 (x, y, t, u, v, η)∂x + ξ 2 (x, y, t, u, v, η)∂y + ξ 3 (x, y, t, u, v, η)∂t + η 1 (x, y, t, u, v, η)∂u + η 2 (x, y, t, u, v, η)∂v + η 3 (x, y, t, u, v, η)∂η . 277
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AKSENOV AND DRUZHKOV
Applying the criterion of invariance [7], we obtain an overdetermined linear homogeneous system of determining equations ξu1 = 0,
ξv1 =
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