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 u_xxxxxx considered, we come up with the perturbed equation

According to perturbation theory, a perturbation expansion

To study symmetry reductions of Eq. (4), we first construct the Lie point symmetry transformation of the vector form

Equations (4), (7), and (8) are to be combined to determine X, Y, T, and U_k, (k = 0, 1, …). Only the first few equations are considered in Eq. (4) for convenience. Restriction of k to k ∈ {0, 1, 2} in Eq. (4) is adequate for studying the symmetry reduction of the general form. In this case, it is easily seen that the variables of X, Y, T, U₀, U₁, and U₂ are confined to {x, y, t, u₀, u₁, u₂}. We first insert Eq. (7) into Eq. (8), then eliminate {u_0,tt, u_1,tt, u_2,tt} with Eq. (4), and vanish all derivatives of {u₀, u₁, u₂}. A large system of determining equations for {X, Y, T, U₀, U₁, U₂} will be obtained. We follow a step-by-step procedure to solve these determining equations.

For k = 0 in Eqs. (4), (7), and (8), 447 determining equations for the functions {X, Y, T, U₀} can be achieved and solved as

Similarly, for k = 1 in Eqs. (4), (7), and (8) with Eq. (9), we can obtain 95 determining equations for U₁, which hold only if U₁ = −2bu₁. Taking k = 2, in the same way we can gain 95 determining equations for U₂, from which we get U₂ = −3bu₂. It is thus inferred that

The constant b in U_k should be distinguished in the following two subsections.

The constants {a₀, a₁, a₂} in Eq. (10) that signify translational invariance are negligible. Without loss of generality, we adjust the constants a ↦ ab and b ↦ 2b. Similarity solutions to Eq. (4) can be solved as

These similarity solutions transform Eq. (4) into reduction equations with a general form, which is shown in Eq. (A1) in Appendix A. We consider particular solutions to these reduction equations such as Jacobi elliptic function solutions and make the assumption

Taking k = 0 in Eq. (A1), considering Eq. (12) and balancing the highest degree of

The determination of P_k in Eq. (12) is implemented through a step-by-step procedure.

From the coefficient of sn⁶(g(ξ),m), we obtain

From the coefficient of sn⁵(g(ξ),m) and Eq. (13), we see that

From the coefficient of sn⁴(g(ξ),m), Eqs. (13) and (14), we obtain

After the substitution of Eqs. (13), (14), and (15) into the coefficient of sn³(g(ξ),m), the unknown function g(ξ) can be determined as

Other coefficients of sn(g(ξ),m) vanish automatically due to Eqs. (13)–(16). Meanwhile, the zero-order equation in Eq. (A1) subjects to the solution

The coefficient of sn⁸(ξ^{1/(a − 2)},m) leads to

From the coefficient of sn⁶(ξ^{1/(a − 2)},m) and Eq. (18), we gain

From the coefficient of sn⁵(ξ^{1/(a − 2)},m), Eqs. (18) and (19), we see that

From the coefficient of sn⁴(ξ^{1/(a − 2)},m), Eqs. (18), (19), and (20), we obtain

Other coefficients of {sn(ξ^{1/(a − 2)},m), cn(ξ^{1/(a − 2)},m), dn(ξ^{1/(a −2)},m)} give rise to partial differential equations with only one function f_1,3 under Eqs. (18)–(21), from which we see that f_1,3 = 0. Accordingly, the one-order reduction equation in Eq. (A1) admits the solution

Other coefficients of

Other coefficients of

More solutions to reduction equations (A1) can be solved through this procedure. It is easily seen that these solutions grow more and more complicated.

For the case of Subsection 2.1 in the last section, a general form of series solutions to the perturbed Boussinesq equation (2) can be formulated from Eqs. (3), (11), (12), (17), (22), (29), and (38) as follows:

It is known that the Jacobi elliptic function in Eq. (67) is bounded in the interval [−1, 1], but as the subscript k increases, the increased number of terms of the Jacobi elliptic function and the incremental magnitude for coefficients c_{k, i} of these terms in A_k mean that the growth of A_k is uncontrollable. In order that the series solutions are convergent, two conditions for B_k should be considered. First, the perturbation parameter ε should be sufficiently small. Second, a − 2 should be negative to make sure that |[(t − y)/2]^{2(k + 1)/(a - 2)|} decreases when |t − y| or the index k grows. In this case, divergence exists when t and y get close.

For the case of Subsection 2.2 in the last section, the general form of series solutions to the perturbed Boussinesq equation (2) can also be constructed. Similarly, a series solution in a simpler form can be given from Eqs. (45), (50), (57), and (66)

The rapid growth of P_k should also be compensated by the perturbation parameter ε. Figure 1 shows one shot of the truncated series solution (68) at t = 0, where we diminish the perturbation parameter to ε = 0.01. Singularity appears when y → 0, since ln(y + t) is contained in the solution. It can be observed in Fig. 2 that the curve shape of the profile tends to straighten when it evolves, showing that the singular area around y = 0 moves faster when the whole profile propagates. The reason for the straightening is that the singularity for ln(y + t) around y = 0 fades when the time t goes away from 0. The straightening also indicates the improvement for the convergence of the series solution as it evolves.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Profile of truncated series solutions (68) at t = 0 with ε = 0.01, m = 0.5, and a = 1.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) Density plots of truncated series solutions (68) with ε = 0.01, m = 0.5, and a = 1 for (a) t = 0 and (b) t = 5.

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 εu_xxxxxx 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.