Transcendental meromorphic solutions and the complex Schr\"{o}dinger equation with delay

TL;DR

This paper investigates transcendental meromorphic solutions of a differential-difference equation related to the complex Schrödinger equation with delay, classifying solutions and establishing conditions for their existence.

Contribution

It characterizes all forms of the equation admitting transcendental meromorphic solutions with subnormal growth, including reduction to Riccati equations when degree differences are two.

Findings

All possible forms of the equation with solutions of subnormal growth are derived.

Necessary conditions for the existence of transcendental meromorphic solutions are established.

When degree difference is 2, the equation reduces to a Riccati differential equation.

Abstract

In this article, we focus on studying the differential-difference equation \[ f'(z) = a(z)f(z+1) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \] where the two nonzero polynomials \( P(z, f(z)) \) and \( Q(z, f(z)) \) in \( f(z) \), with small meromorphic coefficients, are coprime, and \( a(z) \) is a nonzero small meromorphic function of \( f(z) \). This equation includes the complex Schrodinger equation with delay as a special case. If \( f(z) \) is a transcendental meromorphic solution of the equation with subnormal growth, then we derive all possible forms of the equation. Additionally, under these assumptions, we classify these specific forms based on the degrees of \( P(z, f(z)) \) and \( Q(z, f(z)) \) to establish necessary conditions for the existence of transcendental meromorphic solutions. In particular, when the degree of \( P \) minus the degree of \( Q \) is 2, we demonstrate that the equation reduces to a Riccati differential equation. Finally, examples are provided to support our results.