All meromorphic solutions of some algebraic differential equations and their applications

Wenjun Yuan,Jianming Qi,Yezhou Li

doi:10.1186/1687-1847-2014-105

Abstract

Abstract In this paper, we employ Nevanlinna’s value distribution theory to investigate the existence of meromorphic solutions of some algebraic differential equations. We obtain the representations of all meromorphic solutions of certain algebraic differential equations with constant coefficients and dominant term. Many results are the corollaries of our result, and we will give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of the Kuramoto-Sivashinsky equation by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics. MSC:30D35, 34A05.

Highlights

Introduction and the main resultNonlinear partial differential equations (NLPDEs) are widely used as models to describe many important dynamical systems in various fields of science, in fluid mechanics, solid state physics, plasma physics, and nonlinear optics
Many results are the corollaries of our result, and we will give the complex method to find all traveling wave exact solutions of corresponding partial differential equations
Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics

Summary

Introduction

Introduction and the main resultNonlinear partial differential equations (NLPDEs) are widely used as models to describe many important dynamical systems in various fields of science, in fluid mechanics, solid state physics, plasma physics, and nonlinear optics. Where ν, μ, b, and A are constants, and obtained that all meromorphic solutions w of equation We employ Nevanlinna’s value distribution theory to investigate the existence of meromorphic solutions of some algebraic differential equations.

Results

Conclusion