A short note on the appearance of the simplest antilinear ODE in several physical contexts

TL;DR

This paper reviews how various one-dimensional physical problems can be reduced to a simple antilinear ODE, highlighting advantages of this reformulation and encouraging further research into its applications.

Contribution

It introduces a unified reduction of several physical equations to a simple antilinear ODE, offering new perspectives for analysis and solution methods.

Findings

Reduction of multiple physical problems to a single antilinear ODE

Advantages of the reformulation are discussed

Encourages further investigation of the ODE's properties

Abstract

In this short note, we review several one-dimensional problems such as those involving linear Schroedinger equation, variable-coefficient Helmholtz equation, Zakharov-Shabat system and Kubelka-Munk equations. We show that they all can be reduced to solving one simple antilinear ordinary differential equation $u^{\prime}\left(x\right)=f\left(x\right)\overline{u\left(x\right)}$ or its nonhomogeneous version $u^{\prime}\left(x\right)=f\left(x\right)\overline{u\left(x\right)}+g\left(x\right)$, $x\in\left(0,x_{0}\right)\subset\mathbb{R}$. We point out some of the advantages of the proposed reformulation and call for further investigation of the obtained ODE.