Variable depth KDV equations and generalizations to more nonlinear regimes

TL;DR

This paper extends the KdV equation to variable bottom topographies in highly nonlinear water wave regimes, providing new models, their justification, and analysis of wave breaking phenomena.

Contribution

It introduces a new class of variable depth KdV equations, generalizing existing models and establishing their consistency and justification in highly nonlinear water wave regimes.

Findings

Derived new variable depth KdV models.

Proved model consistency and full justification.

Analyzed wave breaking in generalized equations.

Abstract

We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known, that for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified by A. Constantin, D. Lannes "The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations" when the bottom is flat. We generalize here this result with a new class of equations taking into account variable bottom topographies. Of course, the many variable depth KdV equations existing in the literature are recovered as particular cases. Various regimes for the topography regimes are investigated and we prove consistency of these models, as well as a full justification for some of them. We also study the problem of wave breaking for our new variable depth and highly nonlinear generalizations of the KDV equations.