The homogeneous balance of undetermined coefficients method and its application

1 Introduction

Nonlinear PDEs are known to describe a wide variety of phenomena not only in physics, but also in biology, chemistry and several other fields. In recent years, many powerful methods to construct the exact solutions of nonlinear PDEs have been established and developed, which are one of the most excited advances of nonlinear science and theoretical physics. The exact solutions of nonlinear PDEs play an important role since they can provide much physical information and more insight into the physica1 aspects of the problems and thus lead to further applications. Seeking the exact solutions of nonlinear PDEs has long been an interesting topic in the nonlinear mathematical physics. With the development of soliton theory, various methods for obtaining the exact solutions of nonlinear PDEs have been presented, such as the inverse scattering method [1], the Bäcklund and Darboux transformation method [2], the homotopy perturbation method [3], the first integral method [4], the variational iteration method [5], the Riccati-Bernoulli sub-ODE method [6], the Jacobi elliptic function method [7], the tanhsech method [8], the (G'/G)-expansion method [9, 10], the Hirota’s method [11], the homogeneous balance method (HBM) [12, 13], the differential transform method (DTM) [14–17] and so on.

In these traditional methods, the HBM and the DTM are straightforward and popular tools for handling many types of functional equations. Recently, the Adomian’s decomposition method (ADM) for solving differential and integral equations, linear or nonlinear, has been the subject of extensive analytical and numerical studies because the ADM provides the solution in a rapid convergent series with elegantly computable components [18–23]. The Jacobi pseudo spectral approximation method [24], the fully spectral collocation approximation method [25, 26], the Jacobi tau approximation method [27], and the homotopy perturbation Sumudu transform method [28,29] are powerful and effective tools for solving nonlinear PDEs.

As a direct method, the HBM provides a convenient analytical technique to construct the exact solutions of nonlinear PDEs and has been generalized to obtain multiple soliton (or multiple solitary-wave) solutions [12, 13]. Fan improved this method to investigate the Bäcklund transformation, Lax pairs, symmetries and exact solutions for some nonlinear PDEs [30]. He also showed that there are many links among the HBM, Weiss-Tabor-Carnevale method and Clarkson-Kruskal method [31].

However, the HBM usually encounters complicated and tedious algebraic calculation. The exact solutions are fixed and single types when we deal with nonlinear PDEs by using the HBM. Moreover, the balance numbers of all nonlinear PDEs dealt with by this method usually are limited to positive integers.

Based on these problems, the homogeneous balance of undetermined coefficients method is proposed to improve the HBM, to derive more general bilinear equation of the KdV equation and the generalized Boussinesq equation, and to the exact solution of the Eckhaus equation, the balance numbers of which is not a positive integer.

The remainder of this paper is organized as follows: the homogeneous balance of undetermined coefficients method is described in Section 2. In Section 3, the exact solutions of the Eckhaus equation are obtained by using the homogeneous balance of undetermined coefficients method. In Section 4, the bilinear equation of the KdV equation and the generalized Boussinesq equation are derived respectively. A brief conclusion is given in Section 5.

2 Description of the homogeneous balance of undetermined coefficients method

Let us consider a general nonlinear PDE, say, in two variables,

where P is a polynomial function of its arguments, the subscripts denote the partial derivatives. The homogeneous balance of undetermined coefficients method consists of three steps.

Step 1 Suppose that the solution of Eq. (1) is of the form

where u = (u(x,t), w = w(x,t), (In w)k,j=∂k+j(In w(x,t))∂xk∂tj, and m, n, b (balance numbers) and a_kj (k = 0, 1, ...,m, j = 0, 1, ...,n) (balance coefficients) are constants to be determined later.

By balancing the highest nonlinear terms and the highest order partial derivative terms, m, n and b can be determined.

Step 2 Substituting Eq. (2) into Eq. (1) and arranging it at each order of w yield an equation as follows:

where f_l (l = 0, 1, ...) are differential-algebraic expressions of w and a_kj. Setting f_l = 0 and using compatible condition (w_xt = w_xt) yield a set of differential-algebraic equations (DAEs).

Step 3 Solving the set of DAEs, w and a_kj (k = 0, 1, ...,m, j = 0, 1, ...,n) can be determined. By substituting w, m, n, b and a_kj into Eq. (2), the exact solutions of Eq. (1) can be obtained.

In the following section, the exact solutions of Eckhaus equation can be obtained by using the homogeneous balance of undetermined coefficients method.

3 Application to the Eckhaus equation

Let us consider the Eckhaus equation in the form

where i=−1 and Ψ = Ψ(x, t) is a complex-valued function of two real variables x, t.

The Eckhaus equation was found as an asymptotic multiscale reduction of certain classes of nonlinear Schrödinger type equations [32]. In [33], a lot of the properties of the Eckhaus equation were obtained. In [34], the Eckhaus equation is linearized by making some transformations of dependent and independent variables. Exact traveling wave solutions of the Eckhaus equation can be obtained by the (G'/G)-expansion method [10] and the first integral method [4].

In this section, by applying the homogeneous balance of undetermined coefficients method to the Eckhaus equation, new exact solutions of the Eckhaus equation can be obtained.

Suppose that the solution of Eq. (4) is of the form

where m, n, b, c, d and a_kj (k = 0, 1, ...,m, j = 0, 1, ...,n) are constants to be determined later.

Balancing |Ψ|⁴Ψ and Ψ_xx in Eq. (4), it is required that 3mb + 1 = 5mb, 3nb = 5nb. Solving this set of equations, we get m = 1, n = 0, b=12. Then Eq. (5) can be written as

where a, b, c(a₁₀ = a, a₀₀ = b) and d are real constants to be determined later.

Substituting Eq. (6) into Eq. (4), we get

where

Obviously, Ψ is a solution of Eq. (4) provided that f_k = 0 (k = 0, 1, 2, 3, 4) and g_j = 0 (j = 1, 2, 3). Firstly, suppose that w_x = 0, from Eqs. (6) and (7), we get an exact solution of Eq. (4) as follows:

where C₀ and c are arbitrary real constants.

Secondly, suppose that w_x ≠ 0, setting f₀ = 0 and g₁ = 0, we get a=±22, or a=±62

Substituting Eq. (9) into Eqs. (7g) and (7h), we get g₂ = g₃ = 0.

Case A When a=±62, substituting Eq. (9) and a=±62 into Eqs. (7b), (7c), (7d), and (7e), we get

Case A-1 When b = 0, substituting b = 0 into Eqs. (10), we get

Setting f₁ = f₂ = 0 and noting that w_x ≠ 0, we get

From Eqs. (9) and (12), we get

where C₀, C₁ and c are arbitrary real constants.

Substituting a=±62, b = 0, Eqs. (12) and (13) into Eq. (6), we get an exact solution of Eq. (4) as follows:

where C₀, C₁ and c are arbitrary real constants.

Case A-2 When b ≠ 0, setting f₁ = f₄ = 0 and noting that w_x ≠ 0, we get

Substituting Eqs. (15) into Eqs. (10b) and (10c), we get

Setting f₂ = f₃ = 0, we get bw_x = 0 from Eqs. (16), which contradicts with b ≠ 0 and w_x ≠ 0. Therefore, Eq. (4) has no solution in this case.

Case B When a=±22, substituting a=±22 and Eq. (9) into Eqs. (7b), (7c), (7d) and (7e), we get

Case B-1 When b = 0, substituting b = 0 into Eqs. (17), we get

Setting f₂ = 0 yields an equation as follows:

where α = 4(c² + d).

Using transformations Z = w_x and Y=wxxwx, Eq. (19) is reduced to

Solving Eq. (20), we get

namely

where β(t) is an arbitrary function of t.

(1) When α = β(t) = 0, from Eq. (21) we get

where C₁(t) and C₂(t) are arbitrary functions of t.

Substituting the above equation into Eq. (9), we get

Setting the coefficients of x^j (j = 0, 1) to zero in the above equation, we get

Solving the above equations, we get

where C_k (k = 0, 1) are arbitrary real constants.

Then we get

Substituting a=±22, d = −c² and Eq. (22) into Eq. (6), we get an exact solution of Eq. (4) as follows:

where C_k (k = 0, 1) and c are arbitrary real constants.

(2) When β(t) = 0, α > 0, similar to (1), we get

and an exact solution of Eq. (4) as follows:

where ε = ±1, C_k (k = 0, 1), c and d are arbitrary real constants.

Especially, if C₀C₁ > 0, then Eq. (24) can be reduced to

where C_k (k = 0, 1), c, d and ξ₀ are arbitrary real constants.

If C₀C₁ < 0, then Eq. (24) can be reduced to

where C_k (k = 0, 1), c, d and ξ₀ are arbitrary real constants.

(3) When β(t) ≠ 0, α = 0, similar to (1), we get

and an exact solution of Eq. (4) as follows:

where C_k (k = 0, 1) and c are arbitrary constants.

(4) When β(t) ≠ 0, α > 0, similar to (1), we get

and an exact solution of Eq. (4) as follows:

where C32−4(c2+d)=0,4C1C42C2+C32=0,(k=0,1,2,3,4), c and d are arbitrary real constants.

(5) When β(t) ≠ 0, α < 0, similar to (1), we get

and an exact solution of Eq. (4) as follows:

where C12+4(c2+d)=0,ε=±1,Ck(k=0,1,2), c and d are arbitrary real constants.

Case B-2 When b ≠ 0, setting f₂ = f₃ = f₄ = 0 in Eqs. (17b), (17c) and (17d), we get

Solving Eqs. (32) and noting that Eq. (9), we get

Substituting a=±22, b=±2c2+d and Eq. (33) into Eq. (6), we find that this case is identical to Case B-1-(2).

In this section, by applying the homogeneous balance of undetermined coefficients method to the Eckhaus equation, new exact solutions of the Eckhaus equation are obtained. Among the solutions of the Eckhaus equation, Ψ_j (j = 3, 4, 5, 6, 7, 8) are the same as the results of [4, 10]. To our knowledge, other solutions of Eckhaus equation, Ψ_j (j = 2, 9, 10, 11) have not been reported in any literature.

4 Application to derive the bilinear equation of nonlinear PDEs

In this section, firstly, we modify the homogeneous balance of undetermined coefficients method to derive the bilinear equation of nonlinear PDEs. Then, more general bilinear equation of the KdV equation and the generalized Boussinesq equation are obtained by using the proposed method.

Let us still consider Eq. (1). Suppose that the solution of Eq. (1) is Eq. (2). By balancing the highest nonlinear terms and the highest order partial derivative terms, m, n and b can be determined. Substituting Eq. (2) into Eq. (1) and balancing the terms with (wxw)i(wtw)j yield a set of algebraic equations for a_ij (i = 1, ...,m, j = 1, ...,n). Solving the set of algebraic equations and simplifying Eq. (1), we can get the bilinear equation of Eq. (1) directly or after integrating some times (generally, integrating times equal to the orders of lowest partial derivative of Eq. (1).) with respect to x, t.

Next, the KdV equation and the generalized Boussinesq equation are chosen as examples to illustrate our method.

Suppose that the solution of Eq. (34) is Eq. (2). Balancing u_xxx and uu_x in Eq. (34), it is required that mb + 3 = 2mb + 1, nb = 2nb. Choosing m = 2, n = 0 and b = 1, Eq. (2) can be written as

where a_j0(j = 0, 1, 2) are constants to be determined later.

From Eq. (35), one can calculate the following derivatives:

Equating the coefficients of (wxw)5 and (wxw)4 on the left-hand side of Eq. (34) to zero yields a set of algebraic equations for a₂₀ and a₁₀ as follows:

Solving the above algebraic equations, we get a₂₀ = 12δ, a₁₀ = 0. Substituting a₂₀ and a₁₀ back into Eq. (35), we get

where a₀₀ is an arbitrary constant.

Substituting Eq. (37) into Eq. (34) and simplifying it, we get

where

Simplifying Eq. (38) and integrating once with respect to x, we get

Eq. (39) is identical to

where C(t) is an arbitrary function of t, and a₀₀ is an arbitrary constant.

Especially, taking C(t) as zero in Eq. (40), we get the bilinear equation of Eq. (34) as follows:

Eq. (41) can be written concisely in terms of D-operator as

where

Suppose that the solution of Eq. (45) is Eq. (2). In order to balance u_xxxx and uu_x in Eq. (45), it is required that mb + 4 = 2mb + 2,nb = 2nb. Choosing m = 2, n = 0 and b = 1, Eq. (2) can be written as

where a_j0 (j = 0, 1, 2) are constants to be determined later.

Substituting Eq. (46) into Eq. (45) and equating the coefficients of(wxw)6 and (wxw)6 on the left-hand side of Eq. (45) to zero yield a set of algebraic equations for a₂₀ and a₁₀. Solving the algebraic equations, we get a20=6δγ, a₁₀ = 0. Substituting a₂₀ and a₁₀ back into Eq. (46), we get

where a₀₀ is an arbitrary constant.

Substituting Eq. (47) into Eq. (45), we get

where

Simplifying Eq. (48) and integrating twice with respect to x, we get