Abstract
Abstract Exact traveling wave solutions (TWS)s of the one-dimensional (1D) shallow water equations are derived and studied in the case of a viscous fluid. These TWSs, which satisfy special cases of Abel’s equation, are shown to take the form of kinks, which are not classic Taylor shocks, and to admit bifurcations and steepening. Stability issues are also addressed, asymptotic/limiting case expressions are presented, the possibility of hysteresis is explored, and it is established that bistability only occurs for left-running waves. Last, it is shown that the free surface height is capable of behaving similarly to the strain exhibited by a class of nonlinear viscoelastic media.
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