Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 45, pp. 1-38.

Title: Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation

Authors: Franz Achleitner  (Vienna Univ. of Technology, Austria)
         Carlota M. Cuesta (Univ. of the Basque Country, Bilbao, Spain)
         Xuban Diez-Izagirre (Univ. of the Basque Country, Donostia-San Sebastian, Spain)

Abstract:
We study travelling wave solutions of a generalised Korteweg-de Vries-Burgers 
equation with a non-local diffusion term and a concave-convex flux. 
This model equation arises in the analysis of a shallow water flow by 
performing formal asymptotic expansions associated to the triple-deck 
regularisation (which is an extension of classical boundary layer theory). 
The resulting non-local operator is a fractional type derivative with order 
between $1$ and $2$. Travelling wave solutions are typically analysed in 
relation to shock formation in the full shallow water problem. 
We show rigorously the existence of travelling waves that, formally, in the 
limit of vanishing diffusion and dispersion would give rise to non-classical 
shocks, that is, shocks that violate the Lax entropy condition. 
The proof is based on arguments that are typical in dynamical systems. 
The nature of the non-local operator makes this possible, since the 
resulting travelling wave equation can be seen as a delayed 
integro-differential equation. Thus, linearization around critical points, 
continuity with respect to parameters and a shooting argument, are the main 
steps that we have proved and adapted for solving this problem.


Submitted December 4, 2024. Published April 30, 2025.
Math Subject Classifications: 47J35, 26A33, 35C07.
Key Words: Non-local evolution equation;  fractional derivative; travelling waves; non-classical shocks.