On the uniqueness of flow in a recent tsunami model

TL;DR

This paper provides a simple proof of the uniqueness of flow trajectories in a recent tsunami model, improving understanding of water wave propagation with vorticity over variable depths.

Contribution

It introduces an elementary proof technique for flow uniqueness in a tsunami model, applicable where previous reparametrization methods face difficulties.

Findings

Proves uniqueness of integral curves in the phase-plane analysis.

The technique simplifies analysis compared to previous methods.

Applicable to models with complex vorticity and variable depth.

Abstract

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.