Lie Symmetries and Similarity Solutions for a Shallow-Water Model with Bed Elevation in Lagrange Variables

Abstract

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, $Q\left(t,x\right)=A_g{u}^m,m\ge 1$. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for $m=3$, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form.

1. Introduction

The Saint Venant equations, a type of shallow-water equations, are a set of the hyperbolic nonlinear differential equations that model the flows of incompressible fluids in the limits where the horizontal length scales are much larger than those of the vertical depths [1,2,3]. Shallow-water equations have been used for the modelling and description of various physical phenomena. Atmospheric dynamics and the stability of the jet streams were investigated in [4], while applications of the shallow-water equation for the description of planetary atmospheres can be found in [5,6,7,8,9]. From a computational and analytical perspective, in [10,11], the authors developed high-order numerical schemes to resolve complex wave interactions and shocks. On the other hand, the qualitative and dynamical properties of these equations were investigated in [12,13,14]. Nonlinear phenomena and bifurcation analysis were presented [15,16,17]. However, the introduction of multifluids or the consideration of other physical effects, such as the phase transitions, Coriolis force, matter interaction, and viscosity, has led to modifications of the shallow-water equations, which describe bottom topography [18], dispersive phenomena [19,20], two-phase flows [21,22,23], Hall effect [24,25,26], and many others.

In this study, we consider the introduction of the Exner term [27] in the shallow-water equations, which describe the dynamics of water flow and sediment transport in systems with bed evolution of the bottom surface. In the Saint Venant–Exner (SVE) model, the bed evolution is described by a new dynamical variable [27,28,29,30,31,32].

The one dimensional SVE system in the Euler variables is expressed as follows: [33]

$$h_{,t}+{\left(uh\right)}_{,x}=0,$$

(1)

$${\left(uh\right)}_{,t}+{\left(h{u}^2+{\displaystyle \frac{1}{2}}g{h}^2\right)}_{,x}+gh{B}_{,x}=0,$$

(2)

$$B_{,t}+\xi Q_{,x}=0,$$

(3)

where $h\left(t,x\right)$ is the height above the bottom of the surface, $u\left(t,x\right)$ is the velocity of the fluid in the x-direction, $B\left(t,x\right)$ describes the bed level, and $Q\left(t,x\right)=Q\left(u\left(t,x\right),h\left(t,x\right)\right)$ defines the volumetric bedload sediment flux. As far as the coupling parameters are concerned, g is the gravitational constant and $\xi ={\displaystyle \frac{1}{1-\epsilon }}$, in which $\epsilon$ is the porosity of the bed surface.

In this study, we consider that the bedload sediment flux depends on the velocity, that is, $Q\left(t,x\right)=A_g{u}^m$, where $A_g$ and m are constants with $m\ge 1$ [34]. The parameter m is constrained by real-world data to lie within the range $1\le m\le 4$ [33], while the case $m=3$ was examined in detail in [33]. Thus, for this bedload sediment flux, Equation (3) reads as

$$B_{,t}+\beta u^{m-1}{u}_{,x}=0,$$

(4)

where parameter $\beta$ is defined as $\beta =m{A}_g\xi$.

Within the framework of the Lagrange variables ${\varphi }_{,t}=u\left(s,t\right),{\varphi }_{,s}={\displaystyle \frac{1}{h\left(s,t\right)}},$ the SVE system is simplified as follows:

$$\begin{array}{ccc}\hfill {\varphi }_{,tt}-g{\displaystyle \frac{{\varphi }_{,ss}}{{\varphi }_{,s}^3}}+{\displaystyle \frac{g}{{\varphi }_{,s}}}{B}_{,s}& =& 0,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill B_{,t}+\beta {\left({\varphi }_{,t}\right)}^{m-1}{\displaystyle \frac{{\varphi }_{,ts}}{{\varphi }_{,s}}}& =& 0.\hfill \end{array}$$

(6)

Differentiating Equation (5) with respect to the time parameter t, and differentiating Equation (6) with respect to the independent parameter s, we end up with the third-order nonlinear differential equation

$${\left({\varphi }_{,s}{\varphi }_{,tt}-g{\displaystyle \frac{{\varphi }_{,ss}}{{\varphi }_{,s}^2}}\right)}_{,t}-g\beta {\left({\left({\varphi }_{,t}\right)}^{m-1}{\displaystyle \frac{{\varphi }_{,ts}}{{\varphi }_{,s}}}\right)}_{,s}=0.$$

(7)

In this work, we apply Lie symmetry analysis for the determination of the symmetry vectors and similarity solutions for the nonlinear Equation (7). Lie symmetry analysis has been a powerful method for studying certain nonlinear differential equations. It has been widely applied to shallow-water systems such as in [35,36,37,38,39], where Lie symmetries were applied to reduce the partial differential equations into ordinary differential equations and construct similarity solutions. Moreover, in [40,41,42,43], conservation laws were determined by the application of the symmetry methods. Multiphase and compressible flows were investigated by symmetry analysis in [44,45,46,47,48,49]. However, according to our knowledge, there is no study that investigates the SVE with a bed elevation.

Recently, Lie symmetry analysis for the SVE system, without the bed elevation, in the Eulerian variables was presented in [50]. However, as discussed in detail in [40,41,42,43], the use of Lagrangian variables has opened some new directions in the Lie symmetry analysis. The existence of the bed elevation introduces new dynamical variables in the SVE system, which, as we shall discuss in the following, lead to symmetry breaking [41,42,43]. The application of the Lie symmetry analysis for the study of the SVE system provides constraints for the bedload sediment flux, such that the system possesses invariant transformations. The latter can be applied to reduce the system of equations and, when it is possible, to find similarity solutions. The structure of the paper is as follows.

Section 2 includes the Lie symmetry analysis for the SVE nonlinear equation in Lagrangian variables (7). Specifically, we provide, for the first time, the complete symmetry classification of Equation (7). We find that, in Lagrangian variables, the SVE model under our study possesses four or five Lie point symmetries, depending on the form of the bedload sediment flux. Furthermore, the one-dimensional optimal system is presented. In Section 3, we apply the Lie symmetry vectors to construct similarity transformations. In particular, for the elements of the one-dimensional optimal system, we calculate the Lie invariants. These invariants are used to reduce the third-order partial differential equation to an ordinary differential equation. We then apply Lie symmetry analysis again to determine the symmetries of the resulting ordinary differential equation. The new invariants are used to further reduce the equation, and, when feasible, to derive an analytic solution. Finally, in Section 4, we present our conclusions.

2. Lie Symmetry Analysis

Consider the differential equation

$$\mathbf{H}\left(t,s,\varphi ,{\varphi }_{,t},{\varphi }_{,s},{\varphi }_{,tt},{\varphi }_{,ts},{\varphi }_{,ss},...\right)=0,$$

(8)

in which t and s are the independent variables and $\varphi =\varphi \left(t,s\right)$ is the dependent variable.

Then, under the action of the one-parameter infinitesimal point transformation

$$\begin{array}{ccc}\hfill \overline{t}& =& t+\epsilon {\xi }^1\left(t,s,\varphi \right),\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \overline{s}& =& s+\epsilon {\xi }^2\left(t,s,\varphi \right),\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \overline{\varphi }& =& \varphi +\epsilon \eta \left(t,s,\varphi \right),\hfill \end{array}$$

(11)

with infinitesimal generator

$$X={\xi }^1\left(t,s,\varphi \right){\mathsf{\partial }}_t+{\xi }^2\left(t,s,\varphi \right){\mathsf{\partial }}_s+\eta \left(t,s,\varphi \right){\mathsf{\partial }}_{\varphi },$$

(12)

Equation (8) remains invariant, if and only if [51,52,53,54]

$$\underset{\epsilon \to 0}{lim}{\displaystyle \frac{\overline{\mathbf{H}}\left(\overline{t},\overline{s},\overline{\varphi },{\overline{\varphi }}_{,\overline{t}},{\overline{\varphi }}_{,\overline{s}},...;\epsilon \right)-\mathbf{H}\left(t,s,\varphi ,{\varphi }_{,t},{\varphi }_{,s},...;\epsilon \right)}{\epsilon }}=0,$$

(13)

or, equivalently,

$${\mathcal{L}}_{X^{\left[n\right]}}\left(\mathbf{H}\right)=0,$$

(14)

where ${\mathcal{L}}_{X^{\left[n\right]}}$ is the Lie derivative with respect to the prologation/extension $X^{\left[n\right]},$ and n is the highest derivative in the differential Equation (8). For functions, it follows that

$${\mathcal{L}}_{X^{\left[n\right]}}\left(\mathbf{H}\right)=X^{\left[n\right]}\left(\mathbf{H}\right).$$

(15)

The prologation of the vector field is defined as [51,52,53,54]

$$\begin{array}{ccc}\hfill {\eta }_i^{\left[1\right]}& =& D_i\eta -{\varphi }_j{D}_i\left({\xi }^j\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & ...\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\eta }_{i_1{i}_2...i_{n-1}i}^{\left[n\right]}& =& D_i\left({\eta }_{i_1{i}_2...i_{n-1}}^{\left[n-1\right]}\right)-{\varphi }_{i_1{i}_2...i_{n-1}j}{D}_i\left({\xi }^j\right),\hfill \end{array}$$

(18)

where indices i refer to the independent variables $\left\{t,s\right\}$.

2.1. Symmetries of the SVE Equation

The Lie symmetry condition (14) for the nonlinear Equation (7) states that the coefficients of the infinitesimal generator X are as follows:

$${\xi }^1={\xi }^1\left(t,s\right),{\xi }^2={\xi }^2\left(t,s\right),\eta ={\eta }^1\left(t,s\right)\varphi +{\eta }^2\left(t,s\right),$$

where functions ${\xi }^1\left(t,s\right),{\xi }^2\left(t,s\right),{\eta }^1\left(t,s\right)$, and ${\eta }^2\left(t,s\right)$ satisfy the linear system of partial differential equations:

$$\begin{array}{ccc}\hfill {\eta }_{,s}^1\varphi +{\eta }_{,s}^2& =& 0,{\eta }_{,t}^1\varphi +{\eta }_{,t}^2=0,{\eta }_{,ss}^1\varphi +{\eta }_{,ss}^2=0,\hfill \\ \hfill {\eta }_{,tt}^1\varphi +{\eta }_{,tt}^2& =& 0,{\eta }_{,ttt}^1\varphi +{\eta }_{,ttt}^2=0,{\eta }_{,ts}^1\varphi +{\eta }_{,ts}^2=0,\hfill \\ \hfill {\eta }_{,tss}^1\varphi +{\eta }_{,tss}^2& =& 0,{\eta }_{,ss}^1-{\xi }_{,tss}^1=0,3{\eta }_{,tt}^1-{\xi }_{,ttt}^1=0,\hfill \\ \hfill {\xi }_{,tt}^1-2{\eta }_{,t}^1& =& 0,2{\eta }_{,t}^1-{\xi }_{,tss}^2=0,{\eta }_{,s}^1-{\eta }_{,ts}^1=0,\hfill \\ \hfill {\eta }_{,t}^1-{\xi }_{,ts}^2& =& 0,{\eta }_{,t}^1+{\xi }_{,ts}^2=0,{\eta }^1-{\xi }_{,t}^1=0,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill m\left({\xi }_{,ts}^1-{\eta }_{,s}^1\right)+{\eta }_{,s}^1& =& 0,{\xi }_{,ts}^2-4{\eta }_{,t}^1+3{\xi }_{,tt}^1=0,\hfill \\ \hfill {\xi }_{,ts}^1-2{\eta }_{,s}^1+{\xi }_{,ss}^2& =& 0,3{\eta }^1+{\xi }_{,s}^2+2{\xi }_{,t}^1=0,\hfill \\ \hfill m\left({\xi }_{,t}^1-{\eta }_1\right)-{\xi }_{,t}^1+{\xi }_{,s}^2& =& 0,\left(2-m\right){\eta }_{,s}^1+\left(1+m\right){\xi }_{,ts}^1-{\displaystyle \frac{1}{2}}{\xi }_{,ss}^2=0,\hfill \end{array}$$

$${\eta }_{,t}^1=0,{\xi }_{,s}^1=0,{\xi }_{,t}^2=0.$$

The number of independent solutions for the latter system depends on the value of parameter m.

Proposition 1 

(Theorem). The SVE equation (7) in the Lagrange variables admits the four Lie symmetries

$$X_1={\mathsf{\partial }}_t,X_2={\mathsf{\partial }}_s,X_3={\mathsf{\partial }}_{\varphi },X_4=t{\mathsf{\partial }}_t+s{\mathsf{\partial }}_s+\varphi {\mathsf{\partial }}_{\varphi },$$

for an arbitrary value of the parameter m. The Lie algebras spanned by these vector fields are isomorphic to the Lie algebra $A_{4,5}^{ab}$ in the Patera et al. classification scheme [55,56]. In the case of $m=3$, the SVE system admits the additional symmetry vector field

$$X_5=t{\mathsf{\partial }}_t-2s{\mathsf{\partial }}_s,$$

where now the Lie algebras spanned by these vector fields are isomorphic to the Lie algebra $A_{5,19}^{apq}$ in the Patera et al. classification scheme [55,56]. The commutators of the symmetry vectors are presented in

Table 1. Commutators of the admitted Lie symmetries for the SVE equation.

[, ]	${\mathbf{X}}_1$	${\mathbf{X}}_2$	${\mathbf{X}}_3$	${\mathbf{X}}_4$	${\mathbf{X}}_5$
${\mathbf{X}}_1$	0	0	0	$X_1$	$X_1$
${\mathbf{X}}_2$	0	0	0	$X_2$	$-2X_2$
${\mathbf{X}}_3$	0	0	0	$X_3$	0
${\mathbf{X}}_4$	$-X_1$	$-X_2$	$-X_3$	0	0
${\mathbf{X}}_5$	$X_1$	$-2X_2$	0	0	0

.

It is easy to see that Equation (7) admits fewer symmetries than the corresponding model without the bed-elevation term, as presented in [40,41,42,43]. Thus, the inclusion of this term introduces a symmetry-breaking effect.

2.2. One-Dimensional Optimal System

A novel application of Lie symmetries lies in deriving similarity transformations that simplify differential equations by reducing the number of independent variables or by lowering the order of the equation. Using this family of transformations, it is possible to rewrite the given equation as another known differential equation or into a simpler form from which an analytic or analytic solution can be extracted.

The solutions derived through Lie invariants are commonly referred to as similarity solutions. To systematically obtain all possible similarity solutions, it is essential to determine the one-dimensional optimal system of subalgebras.

For the n-dimensional Lie algebra $G_n$ with elements $X_1,X_2,\dots ,X_n,$ the vector fields [51,52,53,54]

$$Z=\sum _{i=1}^n{a}_i{X}_i,W=\sum _{i=1}^n{b}_i{X}_i,a_i,b_i\mathrm{are}\mathrm{constants}.$$

(19)

are equivalent, that is,

$$W=Ad\left(exp\left({\epsilon }_i{X}_i\right)\right)\mathbf{Z}$$

(20)

or

$$W=cZ,$$

(21)

where c is a constant. Operator $Ad\left(exp\left({\epsilon }_i{X}_i\right)\right)$ is defined as

$$Ad\left(exp\left({\epsilon }_i{X}_i\right)\right)X_j=X_j-{\epsilon }_i\left[X_i,X_j\right]+{\displaystyle \frac{1}{2}}{\epsilon }_i^2\left[X_i,\left[X_i,X_j\right]\right]+...$$

and it is called the adjoint representation.

It is essential to compute the adjoint representation of the admitted Lie symmetries in order to identify all possible independent similarity transformations. The resulting set of one-dimensional subalgebras that are not connected through the adjoint action constitutes the so-called one-dimensional optimal system for the given differential equation.

In

Table 2. Adjoint representation for the admitted Lie symmetries for the SVE equation.

Ad(exp(ε${\mathit{X}}_{\mathit{A}}$))${\mathit{X}}_{\mathit{B}}$	${\mathbf{X}}_1$	${\mathbf{X}}_2$	${\mathbf{X}}_3$	${\mathbf{X}}_4$	${\mathbf{X}}_5$
${\mathbf{X}}_1$	$X_1$	$X_2$	$X_3$	$X_4-\epsilon X_1$	$X_5-\epsilon X_1$
${\mathbf{X}}_2$	$X_1$	$X_2$	$X_3$	$X_4-\epsilon X_2$	$X_5+2\epsilon X_2$
${\mathbf{X}}_3$	$X_1$	$X_2$	$X_3$	$X_4-\epsilon X_3$	$X_5$
${\mathbf{X}}_4$	$e^{\epsilon }{X}_1$	$e^{\epsilon }{X}_2$	$e^{\epsilon }{X}_3$	$X_4$	$X_5$
${\mathbf{X}}_5$	$e^{\epsilon }{X}_1$	$e^{-2\epsilon }{X}_2$	$X_3$	$X_4$	$X_5$

, we present the adjoint representations for the Lie algebras $A_{4,5}^{ab}$ and $A_{5,19}^{apq},$ which correspond to an arbitrary value of the parameter m and to the case with $m=3,$ respectively.

With the use of Table 2 we conclude that the one-dimensional system is consisted by the following one-dimensional Lie algebras

$$\begin{array}{ccc}& & \left\{X_1\right\},\left\{X_2\right\},\left\{X_3\right\},\hfill \\ & & \left\{X_1+{\alpha }_2{X}_2+{\alpha }_3{X}_3\right\},\hfill \\ & & \left\{X_4\right\},\left\{X_5\right\},\left\{X_4+\alpha X_5\right\}.\hfill \end{array}$$

3. Similarity Transformations

In this section, we apply the Lie symmetry vectors to determine similarity transformations for the SVE Equation (7). In particular, we reduce the partial differential equation to an ordinary differential equation. We then investigate the group properties of the reduced equation. Additional symmetries are considered for further reduction, and, when feasible, we provide the closed-form expression for the similarity solution.

Specifically, for a given generator X of the invariant infinitesimal transformation, we determine the canonical variables. To achieve this, we solve the differential equation

$$X\left(F\left(t,s,\varphi \right)\right)=0,$$

(22)

The latter equation leads to the construction of the characteristic functions $W^{\left[0\right]}=W^{\left[0\right]}\left(t,s,\varphi \right)$, $W^{\left[1\right]}=W^{\left[1\right]}\left(t,s,\varphi \right)$, etc., such that

$$X\left(F\left(W^{\left[0\right]}\left(t,s,\varphi \right),W^{\left[1\right]}\left(t,s,\varphi \right)\right)\right)=0.$$

(23)

The characteristic functions, also known as invariant functions, are used to reduce the order of the differential equation in the case of ordinary differential equations, or to reduce the number of dependent variables. This approach is applied in the following lines to solve Equation (7).

In

Table 3. One dimensional optimal system and Lie invariants.

Symmetry	Invariants
${\mathbf{X}}_1$	$s,\varphi$
${\mathbf{X}}_2$	$t,\varphi$
${\mathbf{X}}_3$	$t,s$
${\mathbf{X}}_4$	$s{t}^{-1},\varphi t^{-1}$
${\mathbf{X}}_5$	$s{t}^2,\varphi$
${\mathbf{X}}_1+{\mathbf{f}\mathbf{f}}_2{\mathbf{X}}_2+{\mathbf{f}\mathbf{f}}_3{\mathbf{X}}_3$	$s-a_{2}t,\varphi -a_{3}t$
${\mathbf{X}}_4+\mathbf{f}\mathbf{f}{\mathbf{X}}_5$	${st}^{{\displaystyle \frac{2\alpha -1}{\alpha +1}}},{\varphi t}^{-{\displaystyle \frac{1}{\alpha +1}}}$

, we present invariant functions which follow from the solution of the differential Equation (22) for all the elements of the one-dimensional optimal system.

3.1. Reduction with $\left\{X_1\right\}$

The symmetry vector $X_1$ provides the invariant variables $t,\varphi =\varphi \left(t\right)$. Thus, it describes stationary solutions, which are trivial solution for Equation (7).

3.2. Reduction with $\left\{X_2\right\}$

The symmetry vector $X_2$ provides the invariant variables $s,\varphi =\varphi \left(s\right)$. Thus, it describes static solutions, which are trivial solution for Equation (7).

3.3. Reduction with $\left\{X_3\right\}$

The symmetry vector $X_3$ does not lead to a reduction in independent variables.

3.4. Reduction with $\left\{X_4\right\}$

From the vector field $X_4$, we calculate the invariants $\zeta ={\displaystyle \frac{s}{t}}$ and $\varphi =t\mathsf{\Phi }\left(\zeta \right)$, where $\mathsf{\Phi }\left(\zeta \right)$ now satisfies the differential equation

$$\begin{array}{ccc}\hfill 0& =& -{\zeta }^2{\left({\mathsf{\Phi }}_{,\zeta }\right)}^3\left(\zeta {\left({\mathsf{\Phi }}_{,\zeta }{\mathsf{\Phi }}_{,\zeta \zeta }\right)}_{,\zeta }+3{\mathsf{\Phi }}_{,\zeta }{\mathsf{\Phi }}_{,\zeta \zeta }\right)+g{\mathsf{\Phi }}_{,\zeta }{\left(\zeta {\mathsf{\Phi }}_{,\zeta \zeta }\right)}_{,\zeta }-2g\zeta {\mathsf{\Phi }}_{,\zeta \zeta }^2\hfill \\ & & +g\beta \zeta \left({\mathsf{\Phi }}_{,\zeta }{\left(\mathsf{\Phi }-\zeta {\mathsf{\Phi }}_{,\zeta }\right)}^{m-2}\right)\left(m-1\right){\zeta }^2{\left({\mathsf{\Phi }}_{,\zeta \zeta }\right)}^2\hfill \\ & & -{\left(\mathsf{\Phi }-\zeta {\mathsf{\Phi }}_{,\zeta }\right)}^{m-1}{\mathsf{\Phi }}_{,\zeta }\left(\left(\zeta {\mathsf{\Phi }}_{,\zeta \zeta \zeta }+2{\left({\mathsf{\Phi }}_{,\zeta \zeta }\right)}^2\right){\mathsf{\Phi }}_{,\zeta }-\zeta {\left({\mathsf{\Phi }}_{,\zeta \zeta }\right)}^2\right).\hfill \end{array}$$

(24)

For $m=3$, the latter equation admits the additional symmetry vector $\zeta {\mathsf{\partial }}_{\zeta }+\frac{1}{3}\mathsf{\Phi }\mathsf{\partial }\mathsf{\Phi }$, which is nothing other than the reduced symmetry vector $X_5$. From this vector field, we derive the scaling solution $\mathsf{\Phi }={\mathsf{\Phi }}_0{\zeta }^{\frac{1}{3}}$, with $\left(4\beta g-1\right){\mathsf{\Phi }}_0^3+27g=0.$ The solution possesses a singularity at $t=0$, while the solution decays with the time. This solution can describe a shock.

3.5. Reduction with $\left\{X_5\right\}(m=3)$

For the case $m=3$ and the symmetry vector $X_5$, we calculate the invariant functions $\sigma =s{t}^2$, $\varphi =\varphi \left(\sigma \right)$. Thus, from Equation (7), we end up with the reduced equation

$$\begin{array}{ccc}\hfill 0& =& 4\sigma {\left({\varphi }_{,\sigma }\right)}^3{\varphi }_{,\sigma \sigma }\left({\varphi }_{,\sigma \sigma }\sigma +3{\varphi }_{,\sigma }\right)+2{\left({\varphi }_{,\sigma \sigma }\right)}^{2}g\hfill \\ & & +\beta \sigma \left(\left({\varphi }_{,\sigma \sigma }\sigma +5{\varphi }_{,\sigma }\right){\varphi }_{,\sigma \sigma }\sigma +\left(2{\varphi }_{,\sigma }+{\sigma }^2{\varphi }_{,\sigma \sigma \sigma }\right){\varphi }_{,\sigma }\right).\hfill \end{array}$$

(25)

The latter equation admits the two symmetry vectors ${\mathsf{\partial }}_{\varphi },3\sigma {\mathsf{\partial }}_{\sigma }+\varphi {\mathsf{\partial }}_{\sigma }$, which are reduced symmetries. They are the vector fields $X^3$ and $X^4$, which form the $A_{2,1}$ Lie algebra.

The existence of these two symmetry vectors allows us to reduce further Equation (25) into a first-order differential equation. Specifically, from the vector field ${\mathsf{\partial }}_{\varphi }$, we calculate the invariants $\mu ={\varphi }_{,\sigma }$, then Equation (25) becomes

$$\begin{array}{ccc}\hfill 0& =& 4\sigma {\left(\mu \right)}^3{\mu }_{,\sigma }\left({\mu }_{,\sigma }\sigma +3\mu \right)+2{\left({\mu }_{,\sigma }\right)}^{2}g\hfill \\ & & +\beta \sigma \left(\left({\mu }_{,\sigma }\sigma +5\mu \right){\mu }_{,\sigma }\sigma +\left(2\mu +{\sigma }^2{\mu }_{,\sigma \sigma }\right)\mu \right).\hfill \end{array}$$

(26)

which possesses as symmetry vector the $3\sigma {\mathsf{\partial }}_{\sigma }-2\mu {\mathsf{\partial }}_{\sigma }$. A closed-form solution of Equation (26) is the $\mu \left(\sigma \right)={\mu }_0{\sigma }^{-{\displaystyle \frac{2}{3}}}$, with ${\mu }_0^3\left(4\beta g-1\right)+g=0$. This is a singular solution at $t=0$ or $\sigma =0$, and it can describe shocks.

From the latter vector field, we derive $\mathsf{\Sigma }={\sigma }^{\frac{2}{3}}\mu ,M={\sigma }^{\frac{5}{3}}{\mu }_{,\sigma }$, that is, Equation (26) is reduced to the first-order differential equation

$$\begin{array}{ccc}\hfill 0& =& M_{,\mathsf{\Sigma }}-{\displaystyle \frac{2}{9}}{\displaystyle \frac{\left(g+\left(4\beta g-1\right){\mathsf{\Sigma }}^3\right)\mathsf{\Sigma }{M}^3}{\left(4{\mathsf{\Sigma }}^3\left(\beta g-1\right)+g\right)}}-{\displaystyle \frac{1}{3}}{\displaystyle \frac{\left(8{\mathsf{\Sigma }}^3\left(1+2\beta g\right)+g\right)M^2}{\left(4{\mathsf{\Sigma }}^3\left(\beta g-1\right)+g\right)}}\hfill \\ & & -{\displaystyle \frac{2\left(2{\mathsf{\Sigma }}^3\left(\beta g-1\right)-g\right)M}{\mathsf{\Sigma }\left(4{\mathsf{\Sigma }}^3\left(\beta g-1\right)+g\right)}},\hfill \end{array}$$

(27)

which is an Abel-type equation.

3.6. Reduction with $\left\{X_1+\alpha X_2\right\},{\alpha }_3=0$

From the vector field $X_1+\alpha X_2$, we calculate the invariants $\kappa =s-\alpha t$, $\varphi$; thus, Equation (7) becomes

$$\begin{array}{ccc}\hfill 0& =& -\alpha {\varphi }_{,\kappa }\left({\alpha }^2{\left({\varphi }_{,\kappa }{\varphi }_{,\kappa \kappa }\right)}^2+{\varphi }_{,\kappa \kappa \kappa }\left({\alpha }^2{\left({\varphi }_{,\kappa }\right)}^3-g\right)\right)\hfill \\ & & +{\left(-1\right)}^{m-1}{\alpha }^m\beta g{\left({\varphi }_{,\kappa }\right)}^m\left[\left(m-2\right)\left({\varphi }_{,\kappa \kappa }+{\varphi }_{,\kappa }{\varphi }_{,\kappa \kappa \kappa }\right)+{\varphi }_{,\kappa }{\varphi }_{,\kappa \kappa \kappa }\right].\hfill \end{array}$$

(28)

The latter equation admits the symmetry vectors ${\mathsf{\partial }}_{\kappa },{\mathsf{\partial }}_{\varphi }$ and $\kappa {\mathsf{\partial }}_{\kappa }+\varphi {\mathsf{\partial }}_{\varphi }$ for any value of $m\ne 0,$ which are the vector fields $X_1-\alpha X_2,X_3$ and $X_4$, respectively. From $X_4$, we calculate the similarity solution $\varphi \left(\kappa \right)={\varphi }_0\kappa$. However, if we apply the rest of the symmetry vectors, we can reduce the equation further.

From ${\mathsf{\partial }}_{\varphi }$, it follows that $\mu ={\varphi }_{,\kappa }$, such that Equation (28) becomes

$$\begin{array}{ccc}\hfill 0& =& -\alpha \mu \left({\alpha }^2{\left(\mu {\mu }_{,\kappa }\right)}^2+{\mu }_{,\kappa \kappa }\left({\alpha }^2{\left(\mu \right)}^3-g\right)\right)\hfill \\ & & +{\left(-1\right)}^{m-1}{\alpha }^m\beta g\left(\left(m-2\right){\left(\mu \right)}^m\left({\mu }_{,\kappa }+\mu {\mu }_{,\kappa \kappa }\right)\right).\hfill \end{array}$$

(29)

However, the application of the Lie symmetry analysis for Equation (29) indicates that the latter equation is maximally symmetric, that is, it admits eight Lie point symmetries which form the $SL\left(3,R\right)$ Lie algebra. That means that there exists a point transformation where Equation (29) can be written in the equivalent expression of the free particle.

From the autonomous symmetry ${\mathsf{\partial }}_{\kappa }$, we calculate the second similarity transformation $\mathsf{\Xi }=\mu ,M={\mu }_{,\kappa }$, then Equation (29) is expressed as follows:

$$0={\displaystyle \frac{M_{,\mathsf{\Xi }}}{M}}-{\displaystyle \frac{{\alpha }^2{\mathsf{\Xi }}^3+\left({\left(-1\right)}^{m-1}\beta {\alpha }^m\left(2-m\right){\mathsf{\Xi }}^{m-1}+2g\right)}{\mathsf{\Xi }\left(g+{\left(-1\right)}^{m-1}{\alpha }^m{\mathsf{\Xi }}^{m-1}-\alpha {\mathsf{\Xi }}^3\right)}},$$

(30)

with closed-form solution

$$M\left(\mathsf{\Xi }\right)={\displaystyle \frac{M_0\mathsf{\Xi }}{{\mathsf{\Xi }}^{m-1}-\frac{e^{-i\left(m-1\right)\pi }}{\beta g}{\alpha }^{-\left(m-1\right)}\left(\alpha {\mathsf{\Xi }}^2-\frac{g}{\mathsf{\Xi }}\right)}}.$$

(31)

Which is a nonsingular solution.

3.7. Reduction with $\left\{X_1+\alpha X_3\right\},a_2=0$

The application of the symmetry vector $X_1+\alpha X_3$ leads to the trivial similarity solution $\varphi \left(t,s\right)=\alpha s+\mathsf{\Phi }\left(t\right)$, where $\mathsf{\Phi }\left(t\right)$ is an arbitrary function.

3.8. Reduction with $\left\{X_2+\alpha X_3\right\}$

The application of the symmetry vector $X_2+\alpha X_3$ leads to the trivial similarity solution $\varphi \left(t,s\right)=\alpha t+\mathsf{\Phi }\left(s\right)$, where $\mathsf{\Phi }\left(s\right)$ is a solution of the linear equation ${\mathsf{\Phi }}_{,sss}=0$, that is,

$$\mathsf{\Phi }\left(s\right)={\displaystyle \frac{{\mathsf{\Phi }}_2}{2}}{s}^2+{\mathsf{\Phi }}_{1}s+{\mathsf{\Phi }}_0.$$

(32)

This is a nonsingular solution.

3.9. Reduction with $\left\{X_1+\alpha X_2+{\alpha }_3{X}_3\right\}$

The similarity transformation that follows from the generic vector field $X_1+\alpha X_2+{\alpha }_3{X}_3$ is given by the expressions $\kappa =s-\alpha t,\varphi =a_{3}t+\mathsf{\Phi }\left(\kappa \right)$. Equation (7) is reduced to the third-order ordinary differential equation

$$\begin{array}{ccc}\hfill 0& =& -\alpha {\mathsf{\Phi }}_{,\kappa }\left({\alpha }^2{\left({\mathsf{\Phi }}_{,\kappa }{\mathsf{\Phi }}_{,\kappa \kappa }\right)}^2+{\mathsf{\Phi }}_{,\kappa \kappa \kappa }\left({\alpha }^2{\left({\mathsf{\Phi }}_{,\kappa }\right)}^3-g\right)\right)\hfill \\ & & +{\left(-1\right)}^{m-1}{\alpha }^m\beta g\left(\left(m-2\right){\left({\alpha }_3-\alpha {\mathsf{\Phi }}_{,\kappa }\right)}^m{\mathsf{\Phi }}_{,\kappa }\left({\mathsf{\Phi }}_{,\kappa \kappa }+{\mathsf{\Phi }}_{,\kappa }{\mathsf{\Phi }}_{,\kappa \kappa \kappa }\right)-\left(a_3-\alpha {\mathsf{\Phi }}_{,\kappa }\right){\mathsf{\Phi }}_{,\kappa \kappa \kappa }\right).\hfill \end{array}$$

(33)

The latter equations possess the following symmetry vectors: ${\mathsf{\partial }}_{\kappa },{\mathsf{\partial }}_{\mathsf{\Phi }}$ and $\kappa {\mathsf{\partial }}_{\kappa }+{\mathsf{\Phi }}_{\mathsf{\Phi }}$. Thus, we can continue the reduction process in a similar way with the case $a_3=0$.

Indeed, we find that after the change in variables $\mu ={\mathsf{\Phi }}_{,\kappa }$, the second-order differential equation is maximally symmetric, that is, it admits eight Lie point symmetries. In a similar way to before, the analytic solution for the latter equation is given by the following expression:

$${\int }^{\mu }\left({\displaystyle \frac{g}{{\tau }^2}}-{\alpha }^2\kappa +{\displaystyle \frac{\beta g}{\tau }}{\left({\alpha }_3-\alpha \tau \right)}^{m-1}\right)d\tau -\left(\kappa -{\kappa }_0\right)=0.$$

(34)

Which is, again, a nonsingular solution.

4. Conclusions

In this study, we applied Lie symmetry analysis to determine similarity solutions for a shallow-water system with bed elevation. We employed Lagrangian variables, in which the SVE system of hyperbolic equations is expressed as a master third-order nonlinear partial differential equation. For this equation, we could not find a Lagrangian formulation.

We applied the Lie symmetry conditions and found that the master equation admits four Lie symmetries, forming the $A_{4,5}^{ab}$ Lie algebra. Three of these symmetries are scaling symmetries, forming the $3A_1$ Lie algebra, while the fourth vector field is also a scaling symmetry. However, for a specific value of one of the model’s free parameters, an additional fifth scaling Lie symmetry exists, such that the admitted Lie algebra becomes $A_{5,19}^{apq}$, with $A_{4,5}^{ab}$ as a subalgebra. We computed the adjoint representation of the admitted symmetries and determined the one-dimensional optimal system.

Lie symmetries were then used to find the invariant functions of the point transformations. We used these invariants to construct similarity transformations, which allowed us to reduce and simplify the master equation. Through this approach, we determined, for the first time, analytic solutions for the shallow-water system with bed elevation in Lagrangian variables. Lie symmetry analysis, thus, proves to be a powerful and systematic method for constructing solutions.

However, the Lagrangian properties of the bed-elevation model described by the Exner term were analysed in this study. Such an analysis lies beyond the scope of the present work and will be addressed in a future publication. Finally, we should remark the initial value problem was examined in this work. This is important in order to make a comparison of the obtained results with real-world applications.

Finally, it is important to mention that we did not discuss the initial value problem, which is essential for the solutions obtained here to have a physical application. Nevertheless, the solutions obtained here reveal important information regarding the integrability properties of the model.