Generalized Nonlinear Equation and Solutions for Fluid Contour Deformations

TL;DR

This paper extends the nonlinear equation for fluid surface deformations to arbitrary geometries, introduces an infinite order differential equation for solitary waves, and recovers the Korteweg-de Vries equation in the shallow water limit.

Contribution

It generalizes the nonlinear fluid surface equation to any geometry and derives a new infinite order differential equation for solitary waves.

Findings

Derivation of a generalized nonlinear equation for fluid surfaces.

Introduction of an infinite order differential equation for solitary waves.

Recovery of the Korteweg-de Vries equation in the shallow water limit.

Abstract

We generalize the nonlinear one-dimensional equation for a fluid layer surface to any geometry and we introduce a new infinite order differential equation for its traveling solitary waves solutions. This equation can be written as a finite-difference expression, with a general solution that is a power series expansion with coefficients satisfying a nonlinear recursion relation. In the limit of long and shallow water, we recover the Korteweg-de Vries equation together with its single-soliton solution.