Variable-Coefficient Exact Solutions for Nonlinear Differential Equations by a New Bernoulli Equation-Based Subequation Method

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, andmany powerful and efficient methods to find exact solutions have been presented so far. For example, these methods include the known homogeneous balance method [1, 2], the tanh method [3–5], the inverse scattering transform [6], the Backlund transform [7, 8], the Hirotas bilinear method [9, 10], the generalized Riccati subequation method [11, 12], the Jacobi elliptic function expansion [13, 14], the F-expansion method [15], the exp-function expansion method [16, 17], and the (G/G)-expansion method [18, 19]. However, we notice that most of the existing methods are accompanied with constant coefficients, while very few methods are concerned with variable coefficients. 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 KaupKupershmidt equation. Some conclusions are presented at the end of the paper.


Introduction
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 [1,2], the tanh method [3][4][5], the inverse scattering transform [6], the Backlund transform [7,8], the Hirotas bilinear method [9,10], the generalized Riccati subequation method [11,12], the Jacobi elliptic function expansion [13,14], the -expansion method [15], the exp-function expansion method [16,17], and the (  /)-expansion method [18,19].However, we notice that most of the existing methods are accompanied with constant coefficients, while very few methods are concerned with variable coefficients.
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.

Description of the Bernoulli Equation-Based Subequation Method
We consider the following Bernoulli equation: where  ̸ = 0 is a complex number and  = ().The solutions of (1) are denoted by where  is an arbitrary constant.In particular, when  is a real number and  = 1/, we obtain

Mathematical Problems in Engineering
When  = 1/,  =  λ, where λ is a real number and  is the unit of imaginary number, we obtain Suppose that a nonlinear equation, say in two or three independent variables , , , is given by  (,   ,   ,   ,   ,   ,   ,   , . ..) = 0, where  = (, , ) is an unknown function and  is a polynomial in  = (, , ) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.
Step 1. Suppose that and then ( 5) can be turned into the following form: Step 2. Suppose that the solution of ( 7) can be expressed by a polynomial in  as follows: where  = () satisfies (1) and   (, , ),  −1 (, , ),. . .,  0 (, , ) are all unknown functions to be determined later with   (, , ) ̸ = 0.The positive integer  can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (7).
Step 4. Solving the equations system in Step 3, and using the solutions of (1), we can construct exact coefficient function solutions of (7).
Remark 1.As the partial differential equations in Step 3 are usually overdetermined, we may choose some special forms of   ,  −1 , . . .,  0 as done in the following.
where  1 is an arbitrary constant and  1 () and () are two arbitrary functions with respect to the variables  and , respectively.
Case 2. Consider  (, , ) where  1 ,  2 , and  3 are arbitrary constants and  1 () and  2 () are two arbitrary functions with respect to the variables  and , respectively.
where  1 ,  2 , and  3 are arbitrary constants and  1 () is an arbitrary function.
where  1 is an arbitrary constant with  1  > 0 and  1 () is an arbitrary function.
Substituting the results in the four cases mentioned previously into (13), and combining with the solutions of (1) as denoted in (2), we can obtain a rich variety of exact solutions to the asymmetric (2+1)-dimensional NNV system as follows.
Remark 2. In [20][21][22], some exact solutions for the asymmetric (2+1)-dimensional NNV system are established using different methods.We note that the established solutions mentioned previously are different from them essentially as they are new exact solutions with variable functions coefficients and have been reported by other authors in the literature.
Remark 3. The previous established solutions for the Kaup-Kupershmidt equation cannot be obtained by the methods in [23][24][25] and are new exact solutions to our best knowledge.

Conclusions
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.