Exact Solutions to the System of Equations of Shallow Water over an Inclined Bottom: Arrival of a Wave to the Shore

A linear system of equations is considered, which is reduced by a point transformation to a nonlinear system of equations of shallow water over an inclined bottom. A complex-valued family of solutions to a linear system of equations that are invariant with respect to a one-parameter group of transformations is obtained. A two-parameter family of solutions to a nonlinear system of equations describing the arrival and reflection of a wave from the shore is identified. The effects of splash and wave rollback are detected. Examples of such solutions are given.