On the uniqueness of flow in a recent tsunami model

We give an elementary proof of uniqueness for the integral curve starting from the vertical axis in the phase-plane analysis of the recent model [A. Constantin, R.S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res. 40 (2008), 175--211]. Our technique can be applied easily in circumstances where the reparametrization device from [A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states, Arch. Rational Mech. Anal. doi: 10.1007/s00205-010-0347-1] might lead to some serious difficulties.