**1. Introduction**

A powerful mathematical treatment for the determination of exact solutions for nonlinear differential equations is the Lie symmetry analysis [1–3]. Specifically, Lie point symmetries help us in the simplification of differential equations by means of similarity transformations, which reduce the differential equation. The reduction process is based on the existence of functions that are invariant under a specific group of point transformations. When someone uses these invariants as new dependent and independent variables, the differential equation is reduced. The reduction process differs between ordinary differential equations (ODEs) and partial differential equations (PDEs). For ODEs, Lie point symmetries are applied to reduce the order of ODE by one; while on PDEs, Lie point symmetries are applied to reduce by one the number of independent variables, while the order of the PDEs remains the same. The solutions that are found with the application of those invariant functions are called similarity solutions. Some applications on the determination of similarity solutions for nonlinear differential equations can be found in [4–9] and the references therein.

A common characteristic in the reduction process is that the Lie point symmetries are not preserved during the reduction; hence, we can say that the symmetries can be lost. That is not an accurate statement, because symmetries are not "destroyed" or "created" under point transformations, but the "nature" of the symmetry changes. In addition, Lie symmetries can be used to construct new similarity solutions for a given differential equation by applying the adjoint representation of the Lie group [10].

It is possible that a given differential equation admits more than one similarity solution when the given differential equation admits a "large" number of Lie point symmetries. Hence, in order for someone to classify a differential equation according to the admitted similarity solutions, all the inequivalent Lie subalgebras of the admitted Lie symmetries should be determined.

The first group classification problem was carried out by Ovsiannikov [11], who demonstrated the construction of the one-dimensional optimal system for the Lie algebra. Since then, the classification of the one-dimensional optimal system has become a main tool for the study of nonlinear differential equations [12–15].

In this work, we focus on the classification of the one-dimensional optimal system for the two-dimensional rotating ideal gas system described by the following system of PDEs [16–18]:

$$(h\_t + (h\nu)\_x + (h\nu)\_y) = \quad 0,\tag{1}$$

$$u\_t + uu\_x + vu\_y + h^{\gamma - 2}h\_x - fv \quad = \quad 0,\tag{2}$$

$$v\_t + \imath v\_x + \imath v\_y + h^{\gamma - 2} h\_y + f\mu \quad = \quad 0. \tag{3}$$

where *u* and *v* are the velocity components in the *x* and *y* directions, respectively, *h* is the density of the ideal gas, *f* is the Coriolis parameter, and *γ* is the polytropic parameter of the fluid. Usually, *γ* is assumed to be *γ* = 2 where Equations (1)–(3) reduce to the shallow water system. However, in this work, we consider that *γ* > 2. In this work, polytropic index *γ* is defined as *Cp Cv*= *γ* − 1.

 Shallow water equations describe the flow of a fluid under a pressure surface. There are various physical phenomena that are described by the shallow water system with emphasis on atmospheric and oceanic phenomena [19–21]. Hence, the existence of the Coriolis force becomes critical in the description of the physical phenomena.

In the case of *γ* = 2, the complete symmetry analysis of the system (1)–(3) is presented in [22]. It was found that for *γ* = 2, the given system of PDEs is invariant under a nine-dimensional Lie algebra. The same Lie algebra, but in a different representation, is also admitted by the nonrotating system, i.e., *f* = 0. One of the main results of [22] is that the transformation that relates the two representations of the admitted Lie algebras for the rotating and nonrotating system transforms the rotating system (1)–(3) into the nonrotating one. For other applications of Lie symmetries on shallow water equations, we refer the reader to [23–28].

For the case of an ideal gas [17], i.e., parameter *γ* > 1 from our analysis, it follows that this property is lost. The nonrotating system and the rotating one are invariant under a different number of Lie symmetries and consequently under different Lie algebras. For each of the Lie algebras, we have the one-dimensional optimal system and all the Lie invariants. The results are presented in tables. We demonstrate the application of the Lie invariants by determining some similarity solutions for the system (1)–(3) for *γ* > 2. The paper is structured as follows.

In Section 2, we briefly discuss the theory of Lie symmetries for differential equations and the adjoint representation. The nonrotating system (1)–(3) is studied in Section 3. Specifically, we determine the Lie points symmetries, which form an eight-dimensional Lie algebra. The commutators and the adjoint representation are presented. We make use of these results, and we perform, a classification of the one-dimensional optimal system. We found that in total, there are twenty-three one-dimensional independent Lie symmetries and possible reductions, and the corresponding invariants are determined and presented in tables. In Section 4, we perform the same analysis for the rotating system. There, we find that the admitted Lie symmetries form a seven-dimensional Lie algebra, while there are twenty independent one-dimensional Lie algebras. We demonstrate the results by reducing the system of PDEs (1)–(3) into an integrable system of three first-order ODEs, the solution of which is given by quadratures. In Section 5, we discuss our results and draw our conclusions. Finally, in Appendix A, we present the tables, which include the results of our analysis.

#### **2. Lie Symmetry Analysis**

Let *H<sup>A</sup>* .*xi*, <sup>Φ</sup>*A*, Φ*Ai* , .../ = 0 be a system of partial differential equations (PDEs) where Φ*<sup>A</sup>* denotes the dependent variables and *xi* are the independent variables. At this point, it is important to mention that we make use of the Einstein summation convention. By definition, under the action of the infinitesimal one-parameter point transformation (1PPT):

$$\bar{\mathbf{x}}^i = \mathbf{x}^i \left( \mathbf{x}^j, \Phi^B; \varepsilon \right), \ \Phi^A = \Phi^A \left( \mathbf{x}^j, \Phi^B; \varepsilon \right), \tag{4}$$

which connects two different points *P* .*xj*, Φ*<sup>B</sup>*/ → *Q* .*x*¯*j*, Φ¯ *B*,*<sup>ε</sup>*/, the differential equation *H<sup>A</sup>* = 0 remains invariant if and only if *H* ¯ *A* = *<sup>H</sup>A*, that is [2]:

$$\lim\_{\varepsilon \to 0} \frac{\bar{H}^A\left(\bar{y}^i, \bar{u}^A, ...; \varepsilon\right) - H^A\left(y^i, u^A, ...\right)}{\varepsilon} = 0. \tag{5}$$

The latter condition means that the Φ*<sup>A</sup>* (*P*) and Φ*<sup>A</sup>* (*Q*) are connected through the transformation. The lhs of Expression (5) defines the Lie derivative of *H<sup>A</sup>* along the vector field *X* of the one-parameterpointtransformation(4),inwhich*X* isdefinedas:

$$X = \frac{\partial \mathfrak{X}^i}{\partial \mathfrak{e}} \partial\_i + \frac{\partial \Phi}{\partial \mathfrak{e}} \partial\_A.$$

Thus, Condition (5) is equivalent to the following expression: [2]

$$
\mathcal{L}\_X \left( H^A \right) = 0,\tag{6}
$$

where L denotes the Lie derivative with respect to the vector field *<sup>X</sup>*[*n*], which is the *n*th-extension of generator *X* of the transformation (4) in the jet space *xi*, <sup>Φ</sup>*A*, Φ*<sup>A</sup>*,*i* , Φ*<sup>A</sup>*,*ij*, ... <sup>g</sup>iven by the expression [2]:

$$X^{[n]} = X + \eta^{[1]} \partial\_{\Phi\_i^A} + \dots + \eta^{[n]} \partial\_{\Phi\_{i\_i i\_j \dots i\_n}^A} \tag{7}$$

in which:

$$\eta^{[n]} = D\_i \eta^{[n-1]} - u\_{i\_1 i\_2 \dots i\_{n-1}} D\_i \left(\frac{\partial \mathfrak{E}^j}{\partial \varepsilon}\right) \; ; \; i \succeq 1 \; \; ; \; \eta^{[0]} = \left(\frac{\partial \Phi^A}{\partial \varepsilon}\right) \; \tag{8}$$

Condition (6) provides a system of PDEs whose solution determines the components of the *X*, consequently the infinitesimal transformation. The vector fields *X*, which satisfy condition (6), are called Lie symmetries for the differential equation *H<sup>A</sup>* = 0. The Lie symmetries for a given differential equation form a Lie algebra.

Lie symmetries can be used in different ways [2] in order to study a differential equation. However, their direct application is on the determination of the so-called similarity solutions. The steps that we follow to determine a similarity solution are based on the determination and application of the Lie invariant functions.

Let *X* be a Lie symmetry for a given differential equation *H<sup>A</sup>* = 0, then the differential equation *X* (*F*) = 0, where *F* is a function, provides the Lie invariants where by replacing in the differential equation *H<sup>A</sup>* = 0, we reduce the number of the independent variables (in the case of PDEs) or the order of the differential equation (in the case of ordinary differential equations (ODEs)).
