The Lie symmetry analysis is applied for the study of a modified one-dimensional Saint–Venant system in which the density depends on the average temperature of the fluid. The geometry of the bottom we assume that is a plane, while the viscosity term is considered to be nonzero, as the gravitational force is included. The modified shallow water system is consisted by three hyperbolic first-order partial differential equations. The admitted Lie symmetries form a four-dimensional Lie algebra, the A 3,3 ⊕ A 1. However, for the viscosity free model, the admitted Lie symmetries are six and form the A 5,19 ⊕ A 1 Lie algebra. For each Lie algebra we determine the one-dimensional optimal system and we present all the possible independent reductions provided by the similarity transformations. New exact and analytic solutions are calculated for the modified Saint–Venant system.

中文翻译：

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密度依赖于平均温度的修正浅水方程的相似变换

Lie 对称性分析用于研究改进的一维圣维南系统，其中密度取决于流体的平均温度。我们假设底部的几何形状是一个平面，而粘度项被认为是非零的，因为包括重力。修正后的浅水系统由三个双曲一阶偏微分方程组成。承认的李对称形成一个四维李代数，A 3,3⊕A 1个. 然而，对于无粘性模型，承认的李对称性是六个，形成A 5,19⊕A 1个谎言代数。对于每个李代数，我们确定一维最优系统，并且我们展示了相似变换提供的所有可能的独立约简。为修改后的圣维南系统计算新的精确解和解析解。