New Approach on the General Shape Equation of Axisymmetric Vesicles

TL;DR

This paper transforms the complex general shape equation for axisymmetric vesicles into a system of two differential equations, including a linear one, enabling easier analysis and broader solution constraints.

Contribution

It introduces a novel method to convert the nonlinear shape equation into a system of differential equations, revealing all known solutions and new constraints.

Findings

The shape equation is reducible to a system of two differential equations.

The system includes a linear differential equation.

New constraint conditions for solutions are identified.

Abstract

The general Helfrich shape equation determined by minimizing the curvature free energy describes the equilibrium shapes of the axisymmetric lipid bilayer vesicles in different conditions. It is a non-linear differential equation with variable coefficients. In this letter, by analyzing the unique property of the solution, we change this shape equation into a system of the two differential equations. One of them is a linear differential equation. This equation system contains all of the known rigorous solutions of the general shape equation. And the more general constraint conditions are found for the solution of the general shape equation.