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

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, Qt,x=Agum,m≥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.