On meromorphic factoring and entire solutions of certain nonlinear partial differential equations

Abstract

In the first section of this article, we employ the parametrization of rational plane algebraic curves to characterize global solutions of certain nonlinear first-order partial differential equations of complex variables associated with rational curves. In particular, we utilize recent results that tell us that the parametrizing meromorphic functions share a common right factor. We present our main results by primarily examining examples associated with singular irreducible cubic curves. In the second section, we examine regular and entire solutions of certain quadratic second-order partial differential equations.

Keywords:

• entire solutions
• factorization of meromorphic funtions
• plane algebraic curves
• nonlinear partial differential equations
• Fermat type equations

AMS Subject Classifications:

• 30F10
• 32A15
• 35F20
• 35G20

Acknowledgments

The authors would like to thank the referee and to thank Prof. Dimtry Khavinson and Prof. Bao Qin Li for helpful suggestions.