Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application

The homogeneous balance of undetermined coefficient method is firstly proposed to derive a more general bilinear equation of the nonlinear partial differential equation (NLPDE). By applying perturbation method, subsidiary ordinary differential equation (subODE) method, and compatible condition to bilinear equation, more exact solutions of NLPDE are obtained. The KdV equation, Burgers equation, Boussinesq equation, and Sawada-Kotera equation are chosen to illustrate the validity of our method. We find that the underlying relation among the ðG′/GÞ-expansion method, Hirota’s method, and HB method is a bilinear equation. The proposed method is also a standard and computable method, which can be generalized to deal with other types of NLPDE.

However, there is no unified direct method which can be used to deal with all types of NLPDE. And also, no literature is available to illustrate the underlying relations among the three direct methods.
In the present paper, by improving some key steps in the HB method [26], we propose a new method, HB of undetermined coefficient method, which can be used to derive the bilinear equation of NLPDE. Based on the bilinear equation, by applying the perturbation method, sub-ODE method, and compatible condition, more exact solutions of NLPDE are obtained. We illustrate the real meaning of balance numbers. We show the underlying relations among the ðG ′ /GÞ-expansion method, Hirota's method, and HB method.
This paper is organized as follows: the HB of undetermined coefficient method is described in Section 2. In Sections 3 and 4, the KdV equation and Burgers equation are chosen as examples to illustrate the method, respectively. In Section 5, the bilinear equations of Boussinesq equation and Sawada-Kotera equation are derived, respectively. Some brief conclusions are given in Section 6.

Description of the HB of Undetermined Coefficient Method
Let us consider a general NLPDE, say, in two variables where P is a polynomial function of its arguments and the subscripts denote the partial derivatives. The HB of undetermined coefficient method consists of three steps.
Step 2. Solving the set of algebraic equations and simplifying Equation (1), we can get the bilinear equation of Equation (1) directly or after integrating some times (Generally, integrating times equal to the orders of the lowest partial derivative of Equation (1).) with respect to x, t.
Step 3. Generally, in order to obtain the exact solutions of Equation (1), there are three methods to deal with the bilinear equation of Equation (1): (i) Applying the perturbation method to the bilinear equation of Equation (1), N-soliton solution of Equation (1) can be obtained.
(ii) By using traveling wave transformations the bilinear equation of Equation (1) satisfies the following ODE: where the prime denotes the derivation with respect to ξ and λ, μ, and V are constants to be determined later.
Substituting Equations (3) and (4) into Equation (1), it is converted into the following equation: where l 1 , l 2 , and l 3 are polynomial functions of V, λ, and μ.
Setting l 1 = l 2 = l 3 = 0 yields a set of algebraic equations for V, λ, and μ. Solving the set of algebraic equations and using the solutions of Equation (4)

Application to the KdV Equation
Let us consider the celebrated KdV equation in the form where δ is a constant. Suppose that the solution of Equation where m, n, and a ij ði = 0, 1, ⋯, m, j = 0, 1, ⋯, nÞ are constants to be determined later.
Balancing u xxx and uu x in Equation (6), it is required that m + 3 = 2m + 1 and n = 2n. Then, Equation (7) can be written as From Equation (8), one can calculate the following derivatives: Journal of Function Spaces Equating the coefficients of ðw x /wÞ 5 and ðw x /wÞ 4 on the left-hand side of Equation (6) to zero yields a set of algebraic equations for a 20 and a 10 as follows: −2a 2 20 + 24δa 20 = 0,: 3a 20 a 10 − 6δa 10 = 0: ð10Þ Solving the above algebraic equations, we get a 20 = 12δ and a 10 = 0. Substituting a 20 and a 10 back into Equation (8), we get where a 00 is an arbitrary constant. Substituting Equation (11) into Equation (6), we get where Simplifying Equation (12) and integrating once with respect to x, we get Equation (14) is identical to where CðtÞ is an arbitrary function of t and a 00 is an arbitrary constant.
(i) Now, we apply the perturbation method to Equation (17) to derive N-soliton solution of Equation (6). Suppose that w can be expanded as follows: where ε is a parameter and w i = w i ðx, tÞ ði = 1, 2,⋯Þ.
Substituting Equation (20) into Equation (17) and arranging it at each order of ε, we get
(ii) Now, we discuss Equation (16) by using the sub-ODE method.
Using transformations wðx, tÞ = wðξÞ, ξ = x − Vt, Equation (16) is reduced to where the prime denotes the derivation with respect to ξ and V is a constant to be determined later.
Noticing the bilinear property of Equation (16), suppose that w satisfies the following ODE: where λ and μ are parameters. Substituting Equation (30) into Equation (16), we get where Setting l 1 = l 2 = l 3 = 0 yields a set of algebraic equations for V, λ, and µ. Solving this set of algebraic equations, we get where λ, µ, and a 00 are arbitrary constants. Substituting Equation (30) into Equation (11), we get Substituting the general solutions of Equation (30) into Equation (34), we get three types of traveling wave solutions of Equation (6) as follows.
(iii) Now, we discuss Equation (16) from the compatible condition. Equation (16) can be written as Notice w x ≠ 0; otherwise, we can only get a trivial solution. Setting the second term of Equation (42) to zero and solving w t yield Substituting Equation (43) into Equation (42), we get Integrating Equation (44) once with respect to x, we get where bðtÞ is an arbitrary function of t.
Using transformations Y = w 2 xx and X = w x , Equation (45) is reduced to Solving the above equation, we get namely, where bðtÞ and cðtÞ are arbitrary functions of t.
Substituting the above equation into Equation (48), we get Setting the coefficients of x i ði = 0, 1Þ to zero in the above equation, we get 5 Journal of Function Spaces Solving the above equations, we get where C i ði = 1, 2Þ are arbitrary constants. Then, we get Substituting Equation (53) into Equation (11), we get an exact solution of Equation (6) as follows: where C i ði = 1, 2Þ and a 00 are arbitrary constants.
Case 2. When bðtÞ = 0 and cðtÞ > 0, similar to Case 1, we get and an exact solution of Equation (6) as follows: where C i ði = 1, 2, 3Þ and a 00 are arbitrary constants.
Case 3. When bðtÞ ≠ 0 and cðtÞ = 0 similar to Case 1, we get and an exact solution of Equation (6) as follows: where C i ði = 1, 2, 3Þ and a 00 are arbitrary constants.
Case 5. When bðtÞ ≠ 0 and cðtÞ < 0, similar to Case 1, we get Journal of Function Spaces and an exact solution of Equation (6) as follows: where where C i ði = 1, 2, 3, 4Þ and a 00 are arbitrary constants.

Let us consider the Burgers equation in the form
where m, n, and a ij ði = 0, 1, ⋯, m, j = 0, 1, ⋯, nÞ are constants to be determined later.
Balancing u xx and uu x in Equation (65), it is required that m + 2 = 2m + 1 and n = 2n. Then, Equation (66) can be written as From Equation (67), one can calculate the following derivatives: u = a 10 w x w + a 00 , Equating the coefficients of ðw x /wÞ 3 on the left-hand side of Equation (65) to zero yields an algebraic equation for a 10 as follows: −a 2 10 + 2a 10 δ = 0: ð69Þ Solving the above algebraic equation, we get a 10 = 2δ. Substituting a 10 back into Equation (67), we get where a 00 is an arbitrary constant. Substituting Equation (70) into Equation (65), we get where Simplifying Equation (71), we get Equation (73) can be written concisely in terms of Doperators as Equation (74) is identical to where CðtÞ is an arbitrary function of t and a 00 is an arbitrary constant.
Substituting Equation (79) into Equation (74) and arranging it at each order of ε, we get ε : The order-ε equation can be rewritten as a linear differential equation for w 1 as follows: Solving Equation (81), we get w 1 = e P 1 x− a 00 P 1 +δP 2 where P 1 is an arbitrary constant.
(ii) Now, we discuss Equation (73) by using the sub-ODE method.
Using transformations wðx, tÞ = wðξÞ, ξ = x − Vt, Equation (73) is reduced to where the prime denotes the derivation with respect to ξ and V is a constant to be determined later.
Noticing the bilinear property of Equation (87), suppose that w satisfies the following ODE: where λ and μ are parameters. Substituting Equation (88) into Equation (87), we get Journal of Function Spaces where l 1 = μ δλ + V − a 00 ð Þ , Setting l 1 = l 2 = l 3 = 0 yields a set of algebraic equations for V, λ, and μ. Solving the set of algebraic equations, we get where λ, μ, and a 00 are arbitrary constants. Substituting the general solutions of Equation (88) into Equation (70), we get three types of traveling wave solutions of Equation (65) as follows.
(iii) Now, we discuss Equation (73) from the compatible condition.
Using the compatible condition, we can get nothing but Equation (76). Using transformations wðx, tÞ = wðξÞ, Substituting the general solutions of Equation (99) into Equation (70), we get three types of traveling wave solutions of Equation (65) as follows.
So far, applying the HB of undetermined coefficient method to the Burgers equation, we get the bilinear equation of Burgers equation. Moreover, we reduce the Burgers to a linear equation. Based on them, many exact solutions of the Burgers equation are obtained by applying the perturbation method, sub-ODE method, and compatible condition. Our results can compare with the ðG ′ /GÞ-expansion method, Hirota's method, and HB method [20][21][22][23][24][25][26][27][28][29][30].