*Application of ξ*145

Let us now demonstrate the results of Tables A2 and A3 by the Lie invariants of the symmetry vector *ξ*145 and construct the similarity solution for the system.

The application of *ξ*145 in the nonrotating system (1)–(3) reduces the PDEs in the following system:

$$\left(hu\right)\_z + \left(hv\right)\_{w} = \begin{array}{c} 0 \end{array} \tag{13}$$

$$
\alpha + \mu u\_z + vu\_w + h^{\gamma - 2}h\_z = \begin{array}{c} 0 \end{array} \tag{14}
$$

$$
\beta + \mu v\_z + \nu v\_w + h^{\gamma - 2} h\_w = \begin{array}{c} 0 \end{array} \tag{15}
$$

where *z* = *x* − *α*2 *t*2 and *w* = *y* − *β*2 *t*2. System (13)–(15) admits the Lie point symmetries:

$$
\partial\_z \text{ , } \partial\_w \text{ , } z\partial\_z + w\partial\_w + \frac{2}{\gamma - 1}h\partial\_h + u\partial\_u + v\partial\_v \tag{16}
$$

Reduction with the symmetry vector *∂z* + *<sup>c</sup>∂w* provides the following system of first-order ODEs:

$$\left|Fh\_{\sigma}\right| = \left(\left|\mathfrak{a}-\beta\right|h^{2}\right)h^{2},\tag{17}$$

$$Fv\_{\mathcal{T}} = \frac{\left(\mathfrak{a} - c\mathcal{B}\right)ch^{\gamma} - al\mathfrak{a}\left(\mathfrak{v} - cu\right)^{2}}{v - cu},\tag{18}$$

$$\left|Fh\_{\sigma}\right| = \frac{\left(\mathfrak{a} - c\mathfrak{z}\mathfrak{k}\right)cu^{\gamma} - \beta h\left(v - cu\right)^{2}}{v - cu}.\tag{19}$$

where *F* = .1 + *c*2/ *hγ* − *h* (*v* − *cu*)<sup>2</sup> and *σ* = *z* + *cw*.

By performing the change of variable *dσ* = *f d<sup>τ</sup>*, function *f* can be removed from the above system. For *h* (*τ*) = 0, the system (17)–(19) admits a solution *u* = *u*0, *v* = *v*0, which is a critical point. The latter special solutions are always unstable when *αc* > *β*.

We proceed with our analysis by considering the rotating system.

#### **4. Symmetries and Optimal System for Rotating Shallow Water**

For the rotating system (*f* = <sup>0</sup>), the Lie symmetries are:

*Y*1 = *∂t* , *Y*2 = *∂x* , *Y*3 = *∂y* , *Y*4 = *y∂x* − *<sup>x</sup>∂y* + *<sup>v</sup>∂u* − *<sup>u</sup>∂v* , *Y*5 = sin (*f t*) *∂x* + cos (*f t*) *∂y* + *f* (cos (*f t*) *∂u* − sin (*f t*) *∂v*) *Y*6 = cos (*f t*) *∂x* − sin (*f t*) *∂y* − *f* (sin (*f t*) *∂u* + cos (*f t*) *∂v*) *Y*7= (*γ* − 1) .*x∂x*+ *<sup>y</sup>∂y*+ *<sup>u</sup>∂u*+ *<sup>v</sup>∂v*/ + 2*h∂h*

The commutators and the adjoint representation are given in Table 2 and Table A4. The Lie symmetries for the rotating system form a smaller dimension Lie algebra than the non-rotating system. That is not the case when *γ* = 2, where the two Lie algebras have the same dimension and are equivalent under point transformation [22]. Therefore, for *γ* > 2, the Coriolis force cannot be eliminated by a point transformation as in the *γ* = 2 case.

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


As for the admitted Lie symmetries admitted by the given system of PDEs with or without the Coriolis terms for *γ* > 2, we remark that the rotating and the nonrotating systems have a common Lie subalgebra of one-parameter point transformations consisting of the symmetry vectors *Y*1, *Y*2, *Y*3, *Y*4, and *Y*7 or for the nonrotating system *X*1, *X*2, *X*3, *X*6, and *X*8.

*Symmetry* **2019**, *11*, 1115

We proceed with the determination of the one-dimensional optimal system and the invariant functions. Specifically, the relative invariants for the adjoint representation are calculated to be *a*1 , *a*7 and *a*8. From Table 2 and Table A4, we can find the one-dimensional optimal system, which is:

$$\begin{aligned} \chi\_{1}, \chi\_{2}, \chi\_{3}, \chi\_{4}, \chi\_{5}, \chi\_{6}, \chi\_{7}, \chi\_{7}, \chi\_{12} &= \chi\_{1} + a \chi\_{2}, \chi\_{13} = \chi\_{1} + a \chi\_{3}, \\\\ \chi\_{14} = \chi\_{1} + a \chi\_{4}, \ \chi\_{15} = \chi\_{1} + a \chi\_{5}, \ \chi\_{16} = \chi\_{1} + a \chi\_{6}, \ \chi\_{17} = \chi\_{1} + a \chi\_{7}, \\\\ \chi\_{23} = \chi\_{2} + a \chi\_{3}, \ \chi\_{45} = \chi\_{4} + a \chi\_{5}, \ \chi\_{46} = \chi\_{4} + a \chi\_{6}, \ \chi\_{56} = \chi\_{5} + a \chi\_{6}, \\\\ \chi\_{47} = \chi\_{4} + a \chi\_{6}, \ \chi\_{123} = \chi\_{1} + a \chi\_{2} + \beta \chi\_{3}, \ \chi\_{147} = \chi\_{1} + a \chi\_{4} + \beta \chi\_{7}. \end{aligned}$$

The Lie invariants, which correspond to all the above one-dimensional Lie algebras, are presented in Tables A5 and A6.

Let us demonstrate the application of the Lie invariants by the following, from which we can see that the Lie invariants reduce the nonlinear field equations into a system of integrable first-order ODEs, which can be solved with quadratures.

#### *4.1. Application of χ*12

We consider the travel-wave similarity solution in the *x*-plane provided by the symmetry vector *χ*12 and the vector field *Y*3. The resulting equations are described by the following system of first order ODEs:

$$v\_z \quad = \ f \frac{u}{u-u} \tag{20}$$

$$\begin{array}{rcl}\vec{F}u\_z & = & f\left(\mathfrak{a} - \mathfrak{u}\right)vh \end{array} \tag{21}$$

$$\begin{array}{rcl} \bar{F}h\_{\bar{z}} & = & fvh^2 \end{array} \tag{22}$$

where *F*¯ = *hγ* − (*a* − *u*) 2 *h* and *z* = *t* − *αx*. Because we performed reduction with a subalgebra admitted by the nonrotating system, by setting *f* = 0 in (20)–(22), we ge<sup>t</sup> the similarity solution for the nonrotating system, where in this case, it is found to be *h* (*z*) = *h*0, *u* (*z*) = *u*0 and *v* (*z*) = *v*0.

We perform the substitution *dz* = *F*¯*fvd<sup>τ</sup>*, and the latter system is simplified as follows:

$$\frac{v}{F}v\_{\tau} = \begin{array}{c} \frac{u}{a-u} \end{array} \tag{23}$$

$$\left(\mu\_{\tau}\right) = \left(\left(\kappa - \mu\right)h\right) \tag{24}$$

$$h\_{\text{Tr}} = \, \, h^2 \tag{25}$$

from which we ge<sup>t</sup> the solution:

$$h\left(\tau\right) = \left(h\_0 - \tau\right)^{-1}, \ u\left(\tau\right) = \alpha + \nu\_0 - \frac{u\_0}{h\_0}\tau\tag{26}$$

and:

$$\left(\upsilon\left(t\right)^{2} = 2\int \frac{\left(a + u\_{0} - \frac{u\_{0}}{h\_{0}}\tau\right)}{\frac{u\_{0}}{h\_{0}}\left(h\_{0} - \tau\right)} \left(\left(h\_{0} - \tau\right)^{-\gamma} + \left(\frac{u\_{0}}{h\_{0}}\right)^{2}\tau - \frac{\left(u\_{0}\right)^{2}}{h\_{0}}\right)d\tau. \tag{27}$$

#### *4.2. Application of χ*23

Consider now the reduction with the symmetry vector fields *χ*23. The resulting system of1+1 differential equations admits five Lie point symmetries, and they are:

$$\begin{aligned} \left(\partial\_t, \,\partial\_w, \left(\sin\left(ft\right) + a\cos\left(ft\right)\right)\right)\partial\_w &+ f\left(\sin\left(ft\right)\partial\_u + \cos\left(ft\right)\partial\_v\right) \\ \left(a\sin\left(ft\right) - \cos\left(ft\right)\right)\partial\_w &- f\left(\cos\left(ft\right)\partial\_u - \sin\left(ft\right)\partial\_v\right) \\ \left(\left(a\sin\left(ft\right) - \cos\left(ft\right)\right)\partial\_w &- f\left(\cos\left(ft\right)\partial\_u - \sin\left(ft\right)\partial\_v\right) \\ \end{aligned} \right) \left(\gamma - 1\right)\left(\partial\_w + u\partial\_u + v\partial\_v\right) + 2h\partial\_h\right) \end{aligned}$$

where *w* = *y* − *αx*. For simplicity of our calculations, let us assume *γ* = 3.

Reduction with the scaling symmetry provides the following system of first order ODEs:

$$H\_t \quad = \ \ 2H \left( a \mathcal{U} - V \right) \; , \tag{28}$$

$$dL\_t = -aH^2 + \mu \left( aLl - V \right) + fV,\tag{29}$$

$$dV\_t = -H^2 - v\left(a\mathcal{U} - V\right) - f\mathcal{U},\tag{30}$$

where *h* = *wH*, *u* = *wU*, and *v* = *wU*. The latter system is integrable and can be solved with quadratures.

Reducing with respect to the symmetry vector(*α* sin (*f t*) − cos (*f t*)) *∂w* − *f* (cos (*f t*) *∂u* − sin (*f t*) *∂v*), we find the reduced system:

$$\frac{H\_t}{H} = -\frac{a\cos\left(ft\right) + \sin\left(ft\right)}{\cos\left(ft\right) - a\sin\left(ft\right)}\tag{31}$$

*Ut* = −*α f* sin (*f t*) *V* − cos (*f t*) *U* cos (*f t*) − *α* sin (*f t*) , (32)

$$\begin{array}{rcl} V\_t &=& -f \frac{\sin\left(ft\right)V - \cos\left(ft\right)U}{\cos\left(ft\right) - a\sin\left(ft\right)}, \end{array} \tag{33}$$

where now:

$$\begin{array}{rcl}h &=& H(t) \\ \end{array},\tag{34}$$

$$\left(\mu\right) = \frac{\cos\left(ft\right)}{\cos\left(ft\right) - a\sin\left(ft\right)}fw + \mathcal{U}\left(t\right),\tag{35}$$

$$w = -\frac{\sin\left(ft\right)}{\cos\left(ft\right) - a\sin\left(ft\right)}fw + V\left(t\right). \tag{36}$$

System (31)–(33) is integrable, and the solution is expressed in terms of quadratures.
