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On the structure and classification of solutions to certain nonlinear differential equations

Abhijit Banerjee, Sujoy Majumder, Shantanu Panja, Junfeng Xu

Abstract

This paper is devoted to the study of meromorphic solutions of nonlinear differential equations, specifically the equation \[ (f^n)^{(k)}(g^n)^{(k)} = α^2, \] where $k$ and $n$ are positive integers with $n>2k$, and $α$ is a common small function of $f$ and $g$. Our main results provide a detailed characterization of the solutions, improving upon earlier works by Fang-Qiu [5], Fang [4], Zhang-Xu [19], and Li-Yi [9]. Notably, we identify and correct significant errors in the proof of Lemma 2.11 [13], which represents the most recent contribution in this area and provide a resolved and rigorous treatment of the problem. Equations of this type arise naturally in various areas of mathematics and applied sciences such as in the study of complex dynamical systems, integrable systems and value distribution theory in complex analysis. Moreover, understanding the meromorphic solutions helps to realize the growth behavior of solutions, stability analysis and modeling of phenomena in physics and engineering. By characterizing these solutions, one can develop methods to solve broader classes of nonlinear differential equations and explore their qualitative properties, which are essential for both theoretical studies and practical applications.

On the structure and classification of solutions to certain nonlinear differential equations

Abstract

This paper is devoted to the study of meromorphic solutions of nonlinear differential equations, specifically the equation where and are positive integers with , and is a common small function of and . Our main results provide a detailed characterization of the solutions, improving upon earlier works by Fang-Qiu [5], Fang [4], Zhang-Xu [19], and Li-Yi [9]. Notably, we identify and correct significant errors in the proof of Lemma 2.11 [13], which represents the most recent contribution in this area and provide a resolved and rigorous treatment of the problem. Equations of this type arise naturally in various areas of mathematics and applied sciences such as in the study of complex dynamical systems, integrable systems and value distribution theory in complex analysis. Moreover, understanding the meromorphic solutions helps to realize the growth behavior of solutions, stability analysis and modeling of phenomena in physics and engineering. By characterizing these solutions, one can develop methods to solve broader classes of nonlinear differential equations and explore their qualitative properties, which are essential for both theoretical studies and practical applications.
Paper Structure (17 sections, 10 theorems, 134 equations, 1 table)

This paper contains 17 sections, 10 theorems, 134 equations, 1 table.

Table of Contents

  1. Preliminaries and Fundamental Results
  2. Logarithmic Derivative Estimates
  3. Small Functions and Growth Comparison
  4. Exponent of Convergence and Exceptional Values
  5. Normal Families and Spherical Convergence
  6. Zalcman’s Lemma
  7. Value Sharing and Counting Functions
  8. Solutions of the Nonlinear Differential Equation $\boldsymbol{(f^n)^{(k)}(g^n)^{(k)}=\alpha^2}$
  9. Motivation and Analytical Framework
  10. Historical Development and Known Solvable Cases
  11. Two Principal Directions of Extension
  12. Polynomial Right-Hand Side and Reduced Growth Conditions
  13. Higher-Order Derivatives and Exact Classification
  14. Common Poles
  15. Normal Family Approach and the Open Problem
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $f$ and $g$ be two non-constant meromorphic functions having common poles, and let $\alpha$ be a non-zero small function of both $f$ and $g$ such that $0$ and $\infty$ are Borel exceptional values of $\alpha$. Assume further that $\rho(\alpha)<\rho(f)$. Let $k$ and $n$ be positive integers satis Then the following conclusions hold.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 6 more