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Transcendental meromorphic solutions and the complex Schrödinger equation with delay

Tingbin Cao, Risto Korhonen, Wenlong Liu

TL;DR

The paper addresses transcendental meromorphic solutions of the delay differential-difference equation $f'(z)=a(z)f(z+1)+\dfrac{P(z,f)}{Q(z,f)}$, situating it within Nevanlinna theory to classify subnormal solutions. It proves strict degree constraints $\deg_f(Q)\le1$ and $\deg_f(P)\le3$ under subnormal growth and shows that when $\deg_f(P)-\deg_f(Q)=2$ the equation reduces to two canonical forms, with further Riccati-type reductions obtained in the general small-coefficient setting. Through a sequence of lemmas and intricate pole-zero analyses, the authors establish the existence of generic poles, the admissibility of coefficient functions, and the conversion of higher-degree forms to Riccati equations, thereby simplifying the structure of solutions. The results are supported by explicit examples that demonstrate both the Riccati reductions and Schrödinger-with-delay phenomena, highlighting the practical impact for constructing and understanding such transcendental solutions. Overall, the work provides a coherent framework for reducing complex delay equations to tractable Riccati forms and clarifies the growth constraints governing transcendental meromorphic solutions in this setting.

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.

Transcendental meromorphic solutions and the complex Schrödinger equation with delay

TL;DR

The paper addresses transcendental meromorphic solutions of the delay differential-difference equation , situating it within Nevanlinna theory to classify subnormal solutions. It proves strict degree constraints and under subnormal growth and shows that when the equation reduces to two canonical forms, with further Riccati-type reductions obtained in the general small-coefficient setting. Through a sequence of lemmas and intricate pole-zero analyses, the authors establish the existence of generic poles, the admissibility of coefficient functions, and the conversion of higher-degree forms to Riccati equations, thereby simplifying the structure of solutions. The results are supported by explicit examples that demonstrate both the Riccati reductions and Schrödinger-with-delay phenomena, highlighting the practical impact for constructing and understanding such transcendental solutions. Overall, the work provides a coherent framework for reducing complex delay equations to tractable Riccati forms and clarifies the growth constraints governing transcendental meromorphic solutions in this setting.

Abstract

In this article, we focus on studying the differential-difference equation 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 minus the degree of is 2, we demonstrate that the equation reduces to a Riccati differential equation. Finally, examples are provided to support our results.

Paper Structure

This paper contains 8 sections, 18 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Lemmas
  5. Proof of Theorem \ref{['T1']}
  6. Proof of Theorem \ref{['T3']}
  7. Proof of Theorem \ref{['T2']}
  8. Examples

Key Result

Theorem 1

bib16 where $a(z)$ is rational, $P(z,w)$ is a polynomial in $w$ having rational coefficients in $z$, and $Q(z, w)$ is a polynomial in $w(z)$ with roots that are nonzero rational functions of $z$ and not roots of $P(z,w)$. If the hyper-order of $w(z)$ is less than one, then

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Corollary 6
  • Remark 2
  • Definition 1
  • Definition 2
  • ...and 25 more