† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11505094) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20150984).
In this paper, the (2+1)-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional (2D) similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished.
Lie group theory[1,2] is one of the most effective tools for investigating nonlinear equations. By reducing the number of independent variables, the classical or nonclassical Lie group method[3] and the Lie symmetry method[4–6] can be effectively used to simplify partial differential equations (PDEs). The partial differential equations can also be dealt with by the direct reduction method,[7] although it is not directly concerned with group theory. Another extended direct method[8,9] produces various kinds of transform groups including the Lie symmetry group and discrete group.
For the nonclassical Lie group method, invariant curve conditions are appended to the original equation, then the classical Lie group method applies for the enlarged system. An enlarged system can be generated through other ways. By introducing auxiliary variables, nonlocal symmetries can be localized and a prolonged nonlinear system can be constructed and simplified by the Lie symmetry method. This scheme benefits the research into interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves, and Painlevé waves.[10–14]
Combined with perturbation theory,[15,16] the Lie group method facilitates the construction of approximate solutions to perturbed nonlinear equations through the following two ways. The first was to generalize the symmetry group generators to perturbation forms in Ref. [17] and the second was to apply the Lie symmetry method to the approximate equations from the perturbation expansion in Ref. [18]. The second one was further developed to achieve reduced equations with general forms in Refs. [19]–[22]. A further extension of the second method is to introduce a homotopy model to produce the approximate homotopy symmetry method,[23,24] which is superior to the original method for several aspects such as adjusting the convergence of the series solutions. These applications of the approximate symmetry method were only confined to (1+1)-dimensional perturbed equations and some problems such as convergence of series solutions remain to be studied. In this paper, a (2+1)-dimensional perturbed Boussinesq equation will be investigated towards these subjects.
The (2+1)-dimensional Boussinesq equation
With higher order nonlinear perturbation terms such as uxxxxxx considered, we come up with the perturbed equation
According to perturbation theory, a perturbation expansion
To study symmetry reductions of Eq. (
Equations (
For k = 0 in Eqs. (
Similarly, for k = 1 in Eqs. (
The constant b in Uk should be distinguished in the following two subsections.
The constants {a0, a1, a2} in Eq. (
These similarity solutions transform Eq. (
Taking k = 0 in Eq. (
![](cpb_27_10_100202/cpb_27_10_100202_ieqn1.gif)
The determination of Pk in Eq. (
From the coefficient of sn6(g(ξ),m), we obtain
From the coefficient of sn5(g(ξ),m) and Eq. (
From the coefficient of sn4(g(ξ),m), Eqs. (
After the substitution of Eqs. (
Other coefficients of sn(g(ξ),m) vanish automatically due to Eqs. (
The coefficient of sn8(ξ1/(a − 2),m) leads to
From the coefficient of sn6(ξ1/(a − 2),m) and Eq. (
From the coefficient of sn5(ξ1/(a − 2),m), Eqs. (
From the coefficient of sn4(ξ1/(a − 2),m), Eqs. (
Other coefficients of {sn(ξ1/(a − 2),m), cn(ξ1/(a − 2),m), dn(ξ1/(a −2),m)} give rise to partial differential equations with only one function f1,3 under Eqs. (
Other coefficients of
Other coefficients of
More solutions to reduction equations (
In this case, it is inferred from Eq. (
Similar to the last subsection, the assumption (
Other coefficients of sn(g(ξ),m) vanish due to Eqs. (
Other coefficients of sn(ξ, m), cn(ξ,m), and dn(ξ,m) give rise to f1,3 = 0. The one-order solution to Eq. (
Other coefficients of sn(ξ,m), cn(ξ,m), and dn(ξ,m) give rise to f2,3 = 0. The two-order solution to Eq. (
Other coefficients of sn(ξ,m), cn(ξ,m), and dn(ξ,m) give rise to f3,3 = 0. The three-order solution to Eq. (
Finally, truncated series solutions to the perturbed Boussinesq equation (
For the case of Subsection
It is known that the Jacobi elliptic function in Eq. (
For the case of Subsection
The rapid growth of Pk should also be compensated by the perturbation parameter ε. Figure
![]() | Fig. 1. (color online) Profile of truncated series solutions ( |
The combination of perturbation theory and the symmetry method achieve the construction of truncated series solutions to the (2+1)-dimensional perturbed Boussinesq equation. The reduction equations and the related reduction solutions are regular and accordant, from which two types of tidy series solutions are classified.
A trial and summary methodology is applied to solve both the approximate equations by the Lie symmetry method and the reduction equations by the Jacobi elliptic function expansion method. Although the computation is concerned with rapidly growing complexity, the truncated series solutions are finally constructed and displayed visually.
Aside from some particular area, convergence for the series solutions can generally be guaranteed when the perturbation parameter is diminished. Perturbation terms improve the accuracy of the model equations to reality, so the approximate solutions truncated from the convergent series solutions are more reasonable to get the exact solutions to unperturbed equations.
For the first time, the perturbation term εuxxxxxx is added to the (2+1)-dimensional Boussinesq equation in order to investigate the truncated series solutions. In fact, various forms of perturbation terms can similarly be considered to produce more suitable models on different occasions, where the approximate symmetry method could be more significative. Furthermore, the method is also applicable to three-dimensional perturbed nonlinear equations.
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