*Optimal System*

Consider the *n*-dimensional Lie algebra *Gn* with elements *X*1, *X*2, ... *Xn*. Then, we shall say that the two vector fields [2]:

$$Z = \sum\_{i=1}^{n} a\_i X\_i \; , \; W = \sum\_{i=1}^{n} b\_i X\_i \; , \; a\_i \; b\_i \; \text{are constraints.} \tag{9}$$

are equivalent iff there:

$$\mathbf{W} = \lim\_{j \to i} \mathbf{W} \left( \exp \left( \varepsilon\_i X\_i \right) \right) \mathbf{Z} \tag{10}$$

or:

$$\mathcal{W} = \mathfrak{c}\mathcal{Z} \text{ , } \mathfrak{c} = \text{const.} \tag{11}$$

where the operator [2]:

$$\operatorname{Ad}\left(\exp\left(\varepsilon X\_{i}\right)\right)X\_{\rangle} = X\_{\rangle} - \varepsilon \left[X\_{i\prime}X\_{\rangle}\right] + \frac{1}{2}\varepsilon^{2} \left[X\_{i\prime}\left[X\_{i\prime}X\_{j}\right]\right] + \dots \tag{12}$$

is called the adjoint representation.

Therefore, in order to perform a complete classification for the similarity solutions of a given differential equation, we should determine all the one-dimensional independent symmetry vectors of the Lie algebra *Gn*.

We continue our analysis by calculating the Lie point symmetries for the system (1)–(3) for the case where the system is rotating (*f* = 0) and nonrotating (*f* = <sup>0</sup>).

#### **3. Symmetries and the Optimal System for Nonrotating Shallow Water**

We start our analysis by applying the symmetry condition (6) for the Coriolis free system (1)–(3) with *f* = 0. We found that the system of PDEs admits eight Lie point symmetries, as are presented in the following [11]:

$$\begin{aligned} X\_1 &=& \partial\_t \, \, \, X\_2 = \partial\_x \, \, X\_3 = \partial\_y \, \\ X\_4 &=& t\partial\_x + \partial\_u \, \, X\_5 = t\partial\_y + \partial\_v \, \\ X\_6 &=& y\partial\_x - x\partial\_y + v\partial\_u - u\partial\_v \, \\ X\_7 &=& t\partial\_t + x\partial\_x + y\partial\_y \, \\ X\_8 &=& (\gamma - 1) \left( x\partial\_x + y\partial\_y + u\partial\_u + v\partial\_v \right) + 2h\partial\_{lt} \end{aligned}$$

The commutators of the Lie symmetries and the adjoint representation are presented in Table 1 and Table A1, respectively.


**Table 1.** Commutators of the admitted Lie point symmetries for the nonrotating 2D shallow water.

We continue by determining the one-dimensional optimal system. Let us consider the generic symmetry vector:

$$Z^8 = a\_1 X\_1 + a\_2 X\_2 + a\_3 X\_3 + a\_4 X\_4 + a\_5 X\_5 + a\_6 X\_6 + a \gamma X \gamma + a \varsigma X \varsigma$$

From Table A1, we see that by applying the following adjoint representations:

$$Z^{8} = \operatorname{Ad}\left(\exp\left(\varepsilon\_{5}X\_{5}\right)\right) \operatorname{Ad}\left(\exp\left(\varepsilon\_{4}X\_{4}\right)\right) \operatorname{Ad}\left(\exp\left(\varepsilon\_{3}X\_{3}\right)\right) \operatorname{Ad}\left(\exp\left(\varepsilon\_{2}X\_{2}\right)\right) \operatorname{Ad}\left(\exp\left(\varepsilon\_{1}X\_{1}\right)\right) Z^{8}$$

parameters *ε*1, *ε*2, *ε*3, *ε*4, and *ε*5 can be determined such that:

$$Z^{\prime 8} = a\_6^{\prime} X\_6 + a\_7^{\prime} X\_7 + a\_8^{\prime} X\_8$$

Parameters *a*6, *a*7, and *a*8 are the relative invariants of the full adjoint action. Indeed, in order to determine the relative invariants, we solve the following system of partial differential equations [1]:

$$\Delta\left(\phi\left(a\_i\right)\right) = C\_{ij}^k a^i \frac{\partial}{\partial a\_j}$$

where *Ckij* are the structure constants of the admitted Lie algebra as presented in Table 1. Consequently, in order to derive all the possible one-dimensional Lie symmetries, we should study various cases were none of the invariants are zero, one of the invariants is zero, two of the invariants are zero, or all the invariants are zero.

Hence, for the first three cases, infer the following one-dimensional independent Lie algebras:

$$\begin{array}{c} \mathcal{X}\_6 \text{ , } \mathcal{X}\_7 \text{ , } \mathcal{X}\_8 \text{ , } \mathcal{Z}\_{(67)} = \mathcal{X}\_6 + \kappa \mathcal{X}\_7 \text{ , } \mathcal{Z}\_{(68)} = \mathcal{X}\_6 + \kappa \mathcal{X}\_8 \\\\ \mathcal{Z}\_{(78)} = \mathcal{X}7 + \kappa \mathcal{X}\_8 \text{ , } \mathcal{Z}\_{(678)} = \mathcal{X}\_6 + \kappa \mathcal{X}\_7 + \beta \mathcal{X}\_8. \end{array}$$

We apply the same procedure for the rest of the possible linear combinations of the symmetry vectors, and we find the one-dimensional-dependent Lie algebras:

*X*1, *X*2 , *X*3 , *X*4 , *X*5 , *ξ*(12) = *X*1 + *<sup>α</sup>X*2 , *ξ*(13) = *X*1 + *<sup>α</sup>X*3 , *ξ*(23) = *X*2 + *<sup>α</sup>X*3 , *ξ*(14) = *X*1 + *<sup>α</sup>X*4 , *ξ*(15) = *X*1 + *<sup>α</sup>X*5 , *ξ*(16) = *X*1 + *<sup>α</sup>X*6, *ξ*(34) = *X*3 + *<sup>α</sup>X*4 , *ξ*(25) = *X*2 + *<sup>α</sup>X*5 *ξ*(45) = *X*4 + *<sup>α</sup>X*5 , *ξ*(123) = *X*1 + *<sup>α</sup>X*2 + *βX*3 *ξ*(145) = *X*1 + *<sup>α</sup>X*4 + *βX*5 , *ξ*(125) = *X*1 + *<sup>α</sup>X*2 + *βX*5 , *ξ*(134) = *X*1 + *<sup>α</sup>X*3 + *βX*4,

in which *α* and *β* are constants.

Therefore, by applying one of the above Lie symmetry vectors, we find all the possible reductions from a system of 1 + 2 PDEs to a system of 1 + 1 PDEs. The reduced system will not admit all the remaining Lie symmetries. The Lie symmetries that survive under a reduction process are given as described in the following example.

Let a PDE admit the Lie point symmetries Γ1, Γ2, which are such that [<sup>Γ</sup>1, <sup>Γ</sup>2] = *<sup>C</sup>*112*<sup>X</sup>*1, with *C*112 = 0. Reduction with the symmetry vector Γ1 leads to a reduced differential equation, which admits Γ2 as the Lie symmetry. On the other hand, reduction of the mother equation with respect to the Lie symmetry Γ2 leads to a different reduced differential equation, which does not admit as a Lie point symmetry the vector field Γ1. In case the two Lie symmetries form an Abelian Lie algebra, i.e., *C*112= 0, then under any reduction process, symmetries are preserved by any reduction.

We found that the optimal system admits twenty-three one-dimensional Lie symmetries and possible independent reductions. All the possible twenty-three Lie invariants are presented in Tables A2 and A3.

An application of the Lie invariants is presented below.

 ,
