Expansion shock waves in regularized shallow-water theory.
We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularized shallow-water equations that include the Benjamin-Bona-Mahony and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.
Department of Mathematical Sciences , Loughborough University , Loughborough LE11 3TU, UK.
Gennady A
GA
El
Department of Applied Mathematics , University of Colorado , Boulder, CO 80309, USA.
Mark A
0000-0001-5883-6562
MA
Hoefer
Department of Mathematics , North Carolina State University , Raleigh, NC 27695, USA.
Michael
M
Shearer
Benjamin–Bona–Mahony equation; Boussinesq equations; Lax entropy condition; non-local dispersion