Addendum to "Travelling waves for a non-local Korteweg-de Vries-Burgers equation" [J. Differential Equations 257 (2014), no. 3, 720--758]

TL;DR

This paper completes a previous study on traveling wave solutions of a non-local Korteweg-de Vries-Burgers equation by proving a key assumption, thereby solidifying the existence and uniqueness results.

Contribution

It provides a proof for the assumption that exponentially decaying functions are the only bounded solutions of the linearized equation, closing the previous proof gap.

Findings

Confirmed the uniqueness of bounded solutions as exponentially decaying functions.

Validated the existence and uniqueness of traveling wave solutions for the non-local KdV-Burgers equation.

Strengthened the theoretical foundation of the previous work.

Abstract

We add a theorem to [J. Differential Equations 257 (2014), no. 3, 720--758] by F. Achleitner, C.M. Cuesta and S. Hittmeir. In that paper we studied travelling wave solutions of a Korteweg-de Vries-Burgers type equation with a non-local diffusion term. In particular, the proof of existence and uniqueness of these waves relies on the assumption that the exponentially decaying functions are the only bounded solutions of the linearised equation. In this addendum we prove this assumption and thus close the existence and uniqueness proof of travelling wave solutions.