The homogeneous balance of undetermined coefficients method and its application

Abstract The homogeneous balance of undetermined coefficients method is firstly proposed to solve such nonlinear partial differential equations (PDEs), the balance numbers of which are not positive integers. The proposed method can also be used to derive more general bilinear equation of nonlinear PDEs. The Eckhaus equation, the KdV equation and the generalized Boussinesq equation are chosen to illustrate the validity of our method. The proposed method is also a standard and computable method, which can be generalized to deal with some types of nonlinear PDEs.


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 0 =G/-expansion method [9,10], the Hirota's method [11], the homogeneous balance method (HBM) [12,13], the differential transform method (DTM) [14][15][16][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][19][20][21][22][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.

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 .2/ where u D u .x; t/, w D w .x; t/, .ln w/ k;j D @ kCj .ln w.x;t // @x k @t j , and m; n; b (balance numbers) and a kj .k D 0; 1; ; m; j D 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 D 0; 1; / are differential-algebraic expressions of w and a kj . Setting f l D 0 and using compatible condition .w xt D w tx / yield a set of differential-algebraic equations (DAEs).
Step 3 Solving the set of DAEs, w and a kj .k D 0; 1; ; m; j D 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.

Application to the Eckhaus equation
Let us consider the Eckhaus equation in the form where i D p 1 and D .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 0 =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 .5/ where m; n; b; c; d and a kj .k D 0; 1; ; m; j D 0; 1; ; n/ are constants to be determined later. Balancing j j 4 and xx in Eq. (4), it is required that 3mb C 1 D 5mb; 3nb D 5nb. Solving this set of equations, we get m D 1; n D 0; b D 1 2 . Then Eq. (5) can be written as .6/ where a; b; c .a 10 D a; a 00 D b/ and d are real constants to be determined later. Substituting Eq. (6) into Eq. (4), we get .7a/ .7b/ Obviously, is a solution of Eq. (4) provided that f k D 0 .k D 0; 1; 2; 3; 4/ and g j D 0 .j D 1; 2; 3/. Firstly, suppose that w x D 0, from Eqs. (6) and (7), we get an exact solution of Eq. (4) as follows: where C 0 and c are arbitrary real constants.
(4) has no solution in this case.
Case B When a D˙p 2 2 , substituting a D˙p 2 2 and Eq. (9) into Eqs. (7b), (7c), (7d) and (7e), we get .17c/ (17), we get .18/ Setting f 2 D 0 yields an equation as follows: (1) When˛Dˇ.t/ D 0, from Eq. (21) we get where C 1 .t / and C 2 .t / are arbitrary functions of t . Substituting the above equation into Eq. (9), we get Setting the coefficients of x j .j D 0; 1/ to zero in the above equation, we get Solving the above equations, we get where C k .k D 0; 1/ are arbitrary real constants. Then we get w D C 1 .x 2ct / C C 0 : .22/ Substituting a D˙p 2 2 ; b D 0; d D c 2 and Eq. (22) into Eq. (6), we get an exact solution of Eq. (4) as follows: 3 D˙s .23/ where C k .k D 0; 1/ and c are arbitrary real constants.
If C 0 C 1 < 0, then Eq. (24) can be reduced to 7 D˙ where C k .k D 0; 1/ ; c; d and 0 are arbitrary real constants.

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 .36d / Equating the coefficients of w x  Solving the above algebraic equations, we get a 20 D 12ı; a 10 D 0. Substituting a 20 and a 10 back into Eq. (35), we get u D 12ı.ln w/ xx C a 00 ; .37/ where a 00 is an arbitrary constant. Substituting Eq. (37) into Eq. (34) and simplifying it, we get Simplifying Eq. (38) and integrating once with respect to x, we get @ @x .40/ where C .t/ is an arbitrary function of t, and a 00 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 Applying the Hirota's method [11], the bilinear equation of Eq. (34) can be written as  Applying the perturbation method [11] to Eq. (42), we can get 1-soliton solution and 2-soliton solution of Eq. (42) as follows:  .47/ where a 00 is an arbitrary constant. Substituting Eq. (47) into Eq. (45), we get Simplifying Eq. (48) and integrating twice with respect to x, we get .50/ where C 1 .t / and C 2 .t / are arbitrary functions of t , and a 00 is an arbitrary constant. Especially

Conclusions
The homogeneous balance of undetermined coefficients method is successfully used to solve such nonlinear PDEs, the balance numbers of which are not positive integers. By applying our method to the Eckhaus equation, more exact solutions are obtained. The proposed method can also be used to derive the bilinear equation of nonlinear PDEs. More general bilinear equation of the KdV equation and the generalized Boussinesq equation can be obtained by applying our method.
Once the bilinear equation of nonlinear PDE is obtained, the perturbation method can be employed to obtain the N -soliton solution for the nonlinear PDE. Many exact solutions of the nonlinear PDE are obtained by using the threewave method and the homoclinic test approach. Generally, some nonlinear PDE can be linearized or homogenized by using the homogeneous balance of undetermined coefficients method.
Many well-known nonlinear PDEs can be handled by our method. The performance of our method is found to be simple and efficient. The availability of computer systems like Maple facilitates the tedious algebraic calculations. Our method is also a standard and computable method, which allows us to solve complicated and tedious algebraic calculations.