On smooth and peaked traveling waves in a local model for shallow water waves

TL;DR

This paper introduces a new local model for shallow water waves derived from conformal transformations, characterizing smooth and peaked traveling waves and analyzing their spectral stability.

Contribution

It presents an exact existence solution for traveling waves in a new shallow water wave model and analytically proves their spectral stability.

Findings

Exact solutions for smooth and peaked traveling waves

Spectral stability of smooth waves established

Model based on conformal transformations of Euler's equations

Abstract

We introduce a new model equation for Stokes gravity waves based on conformal transformations of Euler's equations. The local version of the model equation is relevant for dynamics of shallow water waves. It allows us to characterize the traveling periodic waves both in the case of smooth and peaked waves and to solve the existence problem exactly, albeit not in elementary functions. Spectral stability of smooth waves with respect to co-periodic perturbations is proven analytically based on the exact count of eigenvalues in a constrained spectral problem.