Nonlinear waves in electromigration dispersion in a capillary

TL;DR

This paper derives exact solutions for a nonlinear advection-diffusion equation modeling electromigration dispersion in capillaries, revealing stability, bistability, and asymmetric kink solutions relevant to electroosmotic flow.

Contribution

It introduces an exact reduction to a Darboux equation for traveling waves and analyzes the stability and bifurcation structure of solutions in electromigration dispersion.

Findings

Exact solutions include asymmetric kink solutions.

The Darboux equation exhibits bistability with coexisting solutions.

Stability depends solely on wave speed, not physical parameters.

Abstract

We construct exact solutions to an unusual nonlinear advection--diffusion equation arising in the study of Taylor--Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave anzats. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions ({\it i.e.}, non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.