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

Abstract

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.

1. 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 [3–5], 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 Kaup-Kupershmidt equation. Some conclusions are presented at the end of the paper.

2. Description of the Bernoulli Equation-Based Subequation Method

We consider the following Bernoulli equation: 𝐺^′+𝜆𝐺=𝐺², (1) where 𝜆≠0 is a complex number and 𝐺=𝐺(𝜉). The solutions of (1) are denoted by 𝐺(𝜉)=𝜆1+𝜆𝑑𝑒^𝜆𝜉, (2) where 𝑑 is an arbitrary constant. In particular, when 𝜆 is a real number and 𝑑=1/𝜆, we obtain 𝐺(𝜉)=𝜆21−tanh𝜆𝜉2. (3)

When 𝑑=1/𝜆, 𝜆=𝑖𝜆, where 𝜆 is a real number and 𝑖 is the unit of imaginary number, we obtain 𝐺(𝜉)=𝜆2−𝜆𝑖2tan𝜆𝜉2. (4)

Suppose that a nonlinear equation, say in two or three independent variables 𝑥, 𝑦, 𝑡, is given by 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑦,…=0, (5) where 𝑢=𝑢(𝑥,𝑦,𝑡) is an unknown function and 𝑃 is a polynomial in 𝑢=𝑢(𝑥,𝑦,𝑡) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.

Step 1. Suppose that 𝑢𝑥,𝑦,𝑡=𝑢(𝜉),𝜉=𝜉𝑥,𝑦,𝑡, (6) and then (5) can be turned into the following form: 𝑃𝑢,𝑢^′,𝑢^′′,…=0. (7)

Step 2. Suppose that the solution of (7) can be expressed by a polynomial in 𝐺 as follows: 𝑢(𝜉)=𝑎_𝑚𝑥,𝑦,𝑡𝐺^𝑚+𝑎_𝑚−1𝑥,𝑦,𝑡𝐺^𝑚−1+⋯+𝑎₀𝑥,𝑦,𝑡, (8) where 𝐺=𝐺(𝜉) satisfies (1) and 𝑎_𝑚(𝑥,𝑦,𝑡),𝑎_𝑚−1(𝑥,𝑦,𝑡),…,𝑎₀(𝑥,𝑦,𝑡) are all unknown functions to be determined later with 𝑎_𝑚(𝑥,𝑦,𝑡)≠0. The positive integer 𝑚 can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (7).

Step 3. Substituting (8) into (7) and using (1), collecting all terms with the same order of 𝐺 together, the left-hand side of (7) is converted to another polynomial in 𝐺. Equating each coefficient of this polynomial to zero yields a set of partial differential equations for 𝑎_𝑚(𝑥,𝑦,𝑡),𝑎_𝑚−1(𝑥,𝑦,𝑡),…,𝜉(𝑥,𝑦,𝑡),𝜆.

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 𝑎_𝑚,𝑎_𝑚−1,…,𝑎₀ as done in the following.

3. Application of the Bernoulli Equation-Based Subequation Method to Some NLDEs

3.1. Asymmetric (2+1)-Dimensional NNV System

We consider the (2+1)-dimensional asymmetric NNV system [20–22]: 𝑢_𝑡−𝑢_𝑥𝑥𝑥+𝛼(𝑢𝑣)_𝑥𝑢=0,_𝑥+𝛽𝑣_𝑦=0, (9) where 𝛼 and 𝛽 are arbitrary nonzero constants.

Assume that 𝑢(𝑥,𝑦,𝑡)=𝑈(𝜉), where 𝜉=𝜉(𝑥,𝑦,𝑡), and then (9) can be turned into 𝜉^′_𝑡𝑈^′−𝜉³_𝑥𝑈^′′′+3𝜉_𝑥𝜉_𝑥𝑥𝑈^′′+𝜉_𝑥𝑥𝑥𝑈^′+𝛼𝜉_𝑥(𝑈𝑉)^′𝜉=0,_′𝑥𝑈^′+𝛽𝜉_𝑦𝑉^′=0. (10)(11) Suppose that the solutions of (10)-(11) can be expressed by a polynomial in 𝐺 as follows: 𝑈(𝜉)=^𝑚_𝑖=0𝑎_𝑖𝑦,𝑡𝐺^𝑖,𝑉(𝜉)=^𝑛_𝑖=0𝑏_𝑖𝑦,𝑡𝐺^𝑖, (12) where 𝑎_𝑖(𝑦,𝑡), 𝑏_𝑖(𝑦,𝑡) are underdetermined functions and 𝐺=𝐺(𝜉) satisfies (1). Balancing the order of 𝑈^′′′ and (𝑈𝑉)^′ in (10) and the order of 𝑈^′ and 𝑉^′ in (11), we can obtain 𝑚+3=𝑚+𝑛+1, 𝑚+1=𝑛+1⇒𝑚=2, 𝑛=2. So we have 𝑈(𝜉)=𝑎₂𝑦,𝑡𝐺²+𝑎₁𝑦,𝑡𝐺+𝑎₀,𝑉(𝜉)𝑦,𝑡=𝑏₂𝑦,𝑡𝐺²+𝑏₁𝑦,𝑡𝐺+𝑏₀𝑦,𝑡. (13) Substituting (13) into (10)-(11) and collecting all the terms with the same power of 𝐺 together, equating each coefficient to zero yields a set of underdetermined partial differential equations for 𝑎_𝑖(𝑥,𝑦,𝑡), 𝑏_𝑖(𝑥,𝑦,𝑡), 𝑖=0,1,2, and 𝜉(𝑥,𝑦,𝑡). Solving these equations yields the following.

Case 1. Consider 𝜉𝑥,𝑦,𝑡=𝐶₁𝑥−𝐶₁𝛼𝐹₁(𝑡)𝑑𝑡+𝐶³₁𝜆²𝑡+𝐹₂𝑦,𝑎₂𝑦,𝑡=−6𝛽𝐶₁𝐹^′₂𝑦𝛼,𝑎₁𝑦,𝑡=6𝛽𝜆𝐶₁𝐹^′₂𝑦𝛼,𝑎₀𝑦,𝑡=0,𝑏₂𝑦,𝑡=6𝐶²₁𝛼,𝑏₁𝑦,𝑡=−6𝐶²₁𝜆𝛼,𝑏₀𝑦,𝑡=𝐹₁(𝑡), (14) where 𝐶₁ is an arbitrary constant and 𝐹₁(𝑡) and (𝑦) are two arbitrary functions with respect to the variables 𝑡 and 𝑦, respectively.

Case 2. Consider 𝜉𝑥,𝑦,𝑡=𝐶₁𝑥+−𝛼𝛽𝐶₁𝐶₂𝐹₁(𝑡)+𝐶²₁𝛼+𝐶₁𝐶₂𝛽𝜆²𝛽𝐶₂𝑑𝑡+𝐶₂𝐹₂𝑦𝑑𝑦+𝐶₃,𝑎₂𝑦,𝑡=−6𝛽𝐶₁𝐶₂𝐹₂𝑦𝛼,𝑎₁𝑦,𝑡=6𝛽𝜆𝐶₁𝐶₂𝐹₂𝑦𝛼,𝑎₀𝑦,𝑡=𝐹₂𝑦,𝑏₂𝑦,𝑡=6𝐶²₁𝛼,𝑏₁𝑦,𝑡=−6𝐶²₁𝜆𝛼,𝑏₀𝑦,𝑡=𝐹₁(𝑡), (15) where 𝐶₁, 𝐶₂, and 𝐶₃ are arbitrary constants and 𝐹₁(𝑡) and 𝐹₂(𝑦) are two arbitrary functions with respect to the variables 𝑡 and 𝑦, respectively.

Case 3. Consider 𝜉𝑥,𝑦,𝑡=−𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦2𝛽𝐶₂𝜆²𝐹₁𝑦𝑥+𝐶₂𝐹₁𝑦𝑑𝑦+𝐶₃,𝑎₀𝑦,𝑡=𝐹₁𝑦,𝑏₀𝑦,𝑡=𝐶₁,𝑎₂=3𝑦,𝑡𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦𝜆²𝛼,𝑎₁3𝑦,𝑡=−𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦,𝑏𝜆𝛼₂=3𝑦,𝑡𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆⁴𝐶²₂𝐹²₁𝑦,𝑏₁3𝑦,𝑡=−𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆³𝐶²₂𝐹²₁𝑦, (16) where 𝐶₁, 𝐶₂, and 𝐶₃ are arbitrary constants and 𝐹₁(𝑦) is an arbitrary function.

Case 4. Consider 𝜉𝑥,𝑦,𝑡=−𝐶₁𝛼𝑥𝜆+𝐹₁𝑦,𝑎₂𝑦,𝑡=6𝛽𝐶₁𝐹^′₁𝑦𝜆𝐶₁𝛼,𝑎₁𝑦,𝑡=−6𝛽𝐶₁𝐹^′₁𝑦𝐶₁𝛼,𝑎₀𝑏𝑦,𝑡=0,₂𝑦,𝑡=6𝐶₁𝜆²,𝑏₁𝑦,𝑡=−6𝐶₁𝜆,𝑏₀𝑦,𝑡=𝐶₁, (17) where 𝐶₁ is an arbitrary constant with 𝐶₁𝛼>0 and 𝐹₁(𝑦) 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.

Family 1. One has 𝑢₁𝑥,𝑦,𝑡=−6𝛽𝐶₁𝐹^′₂𝑦𝛼×𝜆1+𝜆𝑑𝑒^{𝜆[𝐶1𝑥−∫𝐶1𝛼𝐹1(𝑡)𝑑𝑡+𝐶31𝜆2𝑡+𝐹2(𝑦)]2}+6𝛽𝜆𝐶₁𝐹^′₂𝑦𝛼×𝜆1+𝜆𝑑𝑒^{𝜆[𝐶1𝑥−∫𝐶1𝛼𝐹1(𝑡)𝑑𝑡+𝐶31𝜆2𝑡+𝐹2(𝑦)]},𝑣₁=𝑥,𝑦,𝑡6𝐶²₁𝛼𝜆1+𝜆𝑑𝑒^{𝜆[𝐶1𝑥−∫𝐶1𝛼𝐹1(𝑡)𝑑𝑡+𝐶31𝜆2𝑡+𝐹2(𝑦)]2}−6𝐶²₁𝜆𝛼𝜆1+𝜆𝑑𝑒^{𝜆[𝐶1𝑥−∫𝐶1𝛼𝐹1(𝑡)𝑑𝑡+𝐶31𝜆2𝑡+𝐹2(𝑦)]}+𝐹₁(𝑡). (18)

Family 2. One has 𝑢₂𝑥,𝑦,𝑡=−6𝛽𝐶₁𝐶₂𝐹₂𝑦𝛼𝜆1+𝜆𝑑𝑒^𝜆𝜉2+6𝛽𝜆𝐶₁𝐶₂𝐹₂𝑦𝛼𝜆1+𝜆𝑑𝑒^𝜆𝜉+𝐹₂𝑦,𝑣₂𝑥,𝑦,𝑡=6𝐶²₁𝛼𝜆1+𝜆𝑑𝑒^𝜆𝜉2−6𝐶²₁𝜆𝛼𝜆1+𝜆𝑑𝑒^𝜆𝜉+𝐹₁(𝑡), (19) where 𝜉=𝐶₁𝑥+∫((−𝛼𝛽𝐶₁𝐶₂𝐹₁(𝑡)+𝐶²₁(𝛼+𝐶₁𝐶₂𝛽𝜆²))/𝛽𝐶₂)𝑑𝑡+𝐶₂∫𝐹₂(𝑦)𝑑𝑦+𝐶₃. 

Family 3. One has 𝑢₃=3𝑥,𝑦,𝑡𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦𝜆²𝛼×𝜆1+𝜆𝑑𝑒^𝜆𝜉2−3𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦×𝜆𝜆𝛼1+𝜆𝑑𝑒^𝜆𝜉+𝐹₁𝑦,𝑣₃=3𝑥,𝑦,𝑡𝛼𝐹₁(𝑦)+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆⁴𝐶²₂𝐹²₁𝑦×𝜆1+𝜆𝑑𝑒^𝜆𝜉2−3𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆³𝐶²₂𝐹²₁𝑦×𝜆1+𝜆𝑑𝑒^𝜆𝜉+𝐶₁, (20)(21) where 𝜉=−((𝛼𝐹₁(𝑦)+(𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂)𝐹²₁(𝑦))/2𝛽𝐶₂𝜆²𝐹₁(𝑦))𝑥+𝐶₂∫𝐹₁(𝑦)𝑑𝑦+𝐶₃.

Family 4. Consider 𝑢₄𝑥,𝑦,𝑡=6𝛽𝐶₁𝐹^′₁𝑦𝜆𝐶₁𝛼𝜆1+𝜆𝑑𝑒^𝜆𝜉2−6𝛽𝐶₁𝐹^′₁𝑦𝐶₁𝛼𝜆1+𝜆𝑑𝑒^𝜆𝜉,𝑣₄𝑥,𝑦,𝑡=6𝐶₁𝜆²𝜆1+𝜆𝑑𝑒^𝜆𝜉2−6𝐶₁𝜆𝜆1+𝜆𝑑𝑒^𝜆𝜉+𝐶₁, (22) where 𝜉=−(𝐶₁𝛼𝑥/𝜆)+𝐹₁(𝑦).

Particularly, by a combination between Cases 1–4 and (3) we can obtain some special hyperbolic function solutions as follows: 𝑢₅𝑥,𝑦,𝑡=−6𝛽𝐶₁𝐹^′₂𝑦𝛼𝜆21−tanh𝜆𝜉2²+6𝛽𝜆𝐶₁𝐹^′₂𝑦𝛼𝜆21−tanh𝜆𝜉2,𝑣₅𝑥,𝑦,𝑡=6𝐶²₁𝛼𝜆21−tanh𝜆𝜉2²−6𝐶²₁𝜆𝛼𝜆21−tanh𝜆𝜉2+𝐹₁(𝑡), (23) where 𝜉=𝐶₁𝑥−∫𝐶₁𝛼𝐹₁(𝑡)𝑑𝑡+𝐶³₁𝜆²𝑡+𝐹₂(𝑦).

Consider 𝑢₆𝑥,𝑦,𝑡=−6𝛽𝐶₁𝐶₂𝐹₂𝑦𝛼𝜆21−tanh𝜆𝜉2²+6𝛽𝜆𝐶₁𝐶₂𝐹₂𝑦𝛼𝜆21−tanh𝜆𝜉2+𝐹₂𝑦,𝑣₆𝑥,𝑦,𝑡=6𝐶²₁𝛼𝜆21−tanh𝜆𝜉2²−6𝐶²₁𝜆𝛼𝜆21−tanh𝜆𝜉2+𝐹₁(𝑡), (24) where 𝜉=𝐶₁𝑥+∫((−𝛼𝛽𝐶₁𝐶₂𝐹₁(𝑡)+𝐶²₁(𝛼+𝐶₁𝐶₂𝛽𝜆²))/𝛽𝐶₂)𝑑𝑡+𝐶₂∫𝐹₂(𝑦)𝑑𝑦+𝐶₃. 

Consider 𝑢₇=3𝑥,𝑦,𝑡𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦𝜆²𝛼×𝜆21−tanh𝜆𝜉2²−3𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦×𝜆𝜆𝛼21−tanh𝜆𝜉2+𝐹₁𝑦,𝑣₇=3𝑥,𝑦,𝑡𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆⁴𝐶²₂𝐹²₁𝑦×𝜆21−tanh𝜆𝜉2²−3𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆³𝐶²₂𝐹²₁𝑦×𝜆21−tanh𝜆𝜉2+𝐶₁, (25) where 𝜉=−((𝛼𝐹₁(𝑦)+(𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂)𝐹²₁(𝑦))/2𝛽𝐶₂𝜆²𝐹₁(𝑦))𝑥+𝐶₂∫𝐹₁(𝑦)𝑑𝑦+𝐶₃. 

Consider 𝑢₈𝑥,𝑦,𝑡=6𝛽𝐶₁𝐹^′₁𝑦𝜆𝐶₁𝛼𝜆21−tanh𝜆𝜉2²−6𝛽𝐶₁𝐹^′₁𝑦𝐶₁𝛼𝜆21−tanh𝜆𝜉2,𝑣₈𝑥,𝑦,𝑡=6𝐶₁𝜆²𝜆21−tanh𝜆𝜉2²−6𝐶₁𝜆𝜆21−tanh𝜆𝜉2+𝐶₁, (26) where 𝜉=(−𝐶₁𝛼𝑥/𝜆)+𝐹₁(𝑦).

By a combination with (4) we can obtain some special trigonometric function solutions as follows: 𝑢₉𝑥,𝑦,𝑡=−6𝛽𝐶₁𝐹^′₂𝑦𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2²+6𝛽𝜆𝐶₁𝐹^′₂𝑦𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2,𝑣₉𝑥,𝑦,𝑡=6𝐶²₁𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2²−6𝐶²₁𝜆𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2+𝐹₁(𝑡), (27) where 𝜉=𝐶₁𝑥−∫𝐶₁𝛼𝐹₁(𝑡)𝑑𝑡+𝐶³₁𝜆²𝑡+𝐹₂(𝑦), 𝜆 is a real number and 𝜆=𝑖𝜆.

One has 𝑢₁₀𝑥,𝑦,𝑡=−6𝛽𝐶₁𝐶₂𝐹₂𝑦𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2²+6𝛽𝜆𝐶₁𝐶₂𝐹₂𝑦𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2+𝐹₂𝑦,𝑣₁₀𝑥,𝑦,𝑡=6𝐶²₁𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2²−6𝐶²₁𝜆𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2+𝐹₁(𝑡), (28) where 𝜉=𝐶₁𝑥+∫((−𝛼𝛽𝐶₁𝐶₂𝐹₁(𝑡)+𝐶²₁(𝛼+𝐶₁𝐶₂𝛽𝜆²))/𝛽𝐶₂)𝑑𝑡+𝐶₂∫𝐹₂(𝑦)𝑑𝑦+𝐶₃.  

One has 𝑢₁₁=3𝑥,𝑦,𝑡𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦𝜆²𝛼×𝜆2−𝜆𝑖2tan𝜆𝜉2²−3𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦×𝜆𝜆𝛼2−𝜆𝑖2tan𝜆𝜉2+𝐹₁𝑦,𝑣₁₁=3𝑥,𝑦,𝑡𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆⁴𝐶²₂𝐹²₁𝑦×𝜆2−𝜆𝑖2tan𝜆𝜉2²−3𝛼𝐹₁𝑦+𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂𝐹²₁𝑦²2𝛼𝛽²𝜆³𝐶²₂𝐹²₁𝑦×𝜆2−𝜆𝑖2tan𝜆𝜉2+𝐶₁, (29) where 𝜉=−((𝛼𝐹₁(𝑦)+(𝛼²+4𝛼𝛽²𝜆²𝐶₁𝐶²₂)𝐹²₁(𝑦))/2𝛽𝐶₂𝜆²𝐹₁(𝑦))𝑥+𝐶₂∫𝐹₁(𝑦)𝑑𝑦+𝐶₃.𝑢₁₂𝑥,𝑦,𝑡=6𝛽𝐶₁𝐹^′₁𝑦𝜆𝐶₁𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2²−6𝛽𝐶₁𝐹^′₁𝑦𝐶₁𝛼𝜆2−𝜆𝑖2tan𝜆𝜉2,𝑣₁₂𝑥,𝑦,𝑡=6𝐶₁𝜆²𝜆2−𝜆𝑖2tan𝜆𝜉2²−6𝐶₁𝜆𝜆2−𝜆𝑖2tan𝜆𝜉2+𝐶₁, (30) where 𝜉=−(𝐶₁𝛼𝑥/𝜆)+𝐹₁(𝑦).

Remark 2. In [20–22], 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.

3.2. Kaup-Kupershmidt Equation

We consider the following Kaup-Kupershmidt equation [23–25]: 𝑢_{𝑥𝑥𝑥𝑥𝑥}+𝑢_𝑡+45𝑢_𝑥𝑢²−752𝑢_𝑥𝑢_𝑥𝑥−15𝑢𝑢_𝑥𝑥𝑥=0. (31) Suppose that 𝑢(𝑥,𝑡)=𝑈(𝜉), 𝜉=𝜉(𝑥,𝑡), and, furthermore, by balancing the order in (31), we can suppose that 𝑈(𝜉)=𝑎₂(𝑥,𝑡)𝐺²+𝑎₁(𝑥,𝑡)𝐺+𝑎₀(𝑥,𝑡), (32) where 𝐺=𝐺(𝜉) satisfies (1). Similar to the process mentioned before, we obtain 𝜉()=2𝑥,𝑡𝐶𝜆𝑖arctan¹𝑥+𝐶₂,𝑎2𝜆𝑖₂(𝑥,𝑡)=128𝐶²₁𝐶²₁𝑥+𝐶₂²−4𝜆²²,𝑎₁(𝑥,𝑡)=−64𝐶³₁𝑥+𝐶₂𝐶²₁𝑥+𝐶₂²−4𝜆²²,𝑎₀(𝑥,𝑡)=8𝐶⁴₁𝑥+𝐶₂²𝐶²₁𝑥+𝐶₂²−4𝜆²²+𝜆2. (33) Then we can obtain a kind of exact solutions of (31) which is unrelated to time variable as follows: 𝑢₁()=𝑥,𝑡128𝐶²₁𝐶²₁𝑥+𝐶₂²−4𝜆²²×𝜆1+𝜆𝑑𝑒^{(2/𝑖)arctan(𝐶1(𝑥+𝐶2)/2𝜆𝑖)2}−64𝐶³₁𝑥+𝐶₂𝐶²₁𝑥+𝐶₂²−4𝜆²²×𝜆1+𝜆𝑑𝑒^{(2/𝑖)arctan(𝐶1(𝑥+𝐶2)/2𝜆𝑖)}+8𝐶⁴₁𝑥+𝐶₂²𝐶²₁𝑥+𝐶₂²−4𝜆²²+𝜆2. (34) In particular, by the combination with (3)-(4) we can obtain hyperbolic function and trigonometric function solutions as follows: 𝑢₂(𝑥,𝑡)=128𝐶²₁𝐶²₁𝑥+𝐶₂²−4𝜆²²𝜆21−tanh𝜆𝜉2²−64𝐶³₁𝑥+𝐶₂𝐶²₁𝑥+𝐶₂²−4𝜆²²𝜆21−tanh𝜆𝜉2+8𝐶⁴₁𝑥+𝐶₂²𝐶²₁𝑥+𝐶₂²−4𝜆²²+𝜆2,𝑢₃(𝑥,𝑡)=128𝐶²₁𝐶²₁𝑥+𝐶₂²−4𝜆²²𝜆2−𝜆𝑖2tan𝜆𝜉2²−64𝐶³₁𝑥+𝐶₂𝐶²₁𝑥+𝐶₂²−4𝜆²²𝜆2𝜆2−𝜆𝑖2tan𝜆𝜉2+8𝐶⁴₁𝑥+𝐶₂²𝐶²₁𝑥+𝐶₂²−4𝜆²²+𝜆2, (35)(36) where 𝜉=(2/𝜆𝑖)arctan(𝐶₁(𝑥+𝐶₂)/2𝜆𝑖), 𝜆 is a real number and 𝜆=𝑖𝜆.

Remark 3. The previous established solutions for the Kaup-Kupershmidt equation cannot be obtained by the methods in [23–25] and are new exact solutions to our best knowledge.

4. 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.

Acknowledgments

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.