All meromorphic solutions for two forms of odd order algebraic differential equations and its applications

In this article, we employ the Nevanlinna's value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for two classes of odd order algebraic differential equations with the weak 〈p,q〉 and dominant conditions. Moreover, we 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 some generalized Bretherton equations 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, and using the traveling wave nobody can find other new exact solutions for many nonlinear partial differential equations by any method.