Cusps in interfacial problems

TL;DR

This paper reviews various interfacial problems involving free surface motion, highlighting the formation of cusp and swallowtail singularities, and proposes a universal classification based on their scaling behaviors.

Contribution

It provides a unified review and classification of cusp and swallowtail singularities in interfacial free surface problems, connecting fluid flow and wave motion examples.

Findings

Identifies two main types of singularities: cusp and swallowtail.

Establishes a universal scaling form for these singularities.

Proposes a tentative classification scheme for interfacial singularities.

Abstract

A wide range of equations related to free surface motion in two dimensions exhibit the formation of cusp singularities either in time, or as function of a parameter. We review a number of specific examples, relating in particular to fluid flow and to wave motion, and show that they exhibit one of two types of singularity: cusp or swallowtail. This results in a universal scaling form of the singularity, and permits a tentative classification.