A new Bernoulli equation-based subequation method is proposed to establish variable-coefficient exact solutions for nonlinear differential equations. For illustrating the validity of this method, we apply it to the asymmetric (2 + 1)-dimensional NNV system and the Kaup-Kupershmidt equation. As a result, some new exact solutions with variable functions coefficients for them are successfully obtained.

Nonlinear differential equations (NLDEs) can be used to describe many nonlinear phenomena such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. Recently, research for seeking exact analytical solutions of NLDEs has been a hot topic, and many powerful and efficient methods to find exact solutions have been presented so far. For example, these methods include the known homogeneous balance method [

In this paper, by introducing a new ansatz, we develop a new Bernoulli equation-based sub equation method for obtaining variable-coefficient exact solutions for NLDEs. First we give the description of the Bernoulli equation-based subequation method. Then we apply the method to solve the asymmetric (2+1)-dimensional NNV system and the Kaup-Kupershmidt equation. Some conclusions are presented at the end of the paper.

We consider the following Bernoulli equation:

When

Suppose that a nonlinear equation, say in two or three independent variables

Suppose that

Suppose that the solution of (

Substituting (

Solving the equations system in Step

As the partial differential equations in Step

We consider the (2+1)-dimensional asymmetric NNV system [

Assume that

Consider

Consider

Consider

Consider

Substituting the results in the four cases mentioned previously into (

Particularly, by a combination between Cases

Consider

Consider

Consider

By a combination with (

One has

One has

In [

We consider the following Kaup-Kupershmidt equation [

The previous established solutions for the Kaup-Kupershmidt equation cannot be obtained by the methods in [

We have proposed a Bernoulli equation-based subequation method for solving nonlinear differential equations and applied it to find exact solutions with variable functions coefficients of the asymmetric (2+1)-dimensional asymmetric NNV system and the Kaup-Kupershmidt equation. As a result, some new exact solutions for them have been successfully found. Finally, we note that the proposed method can be applied to solve other nonlinear evolution equations.

This work is partially supported by Humanity and Social Science Youth Foundation of Ministry of Education of China (11YJCZH070). The authors would like to thank the reviewers very much for their valuable suggestions on this paper.