Approximate Similarity Reduction for the Nonlinear K ( n , 1 ) Equation with Weak Damping via Symmetry Perturbation and Direct Method

The research is financed by Asian Development Bank. No. 2006-A171(Sponsoring information) Abstract The nonlinear K(n, 1) equation with weak damping is investigated via the approximate symmetry perturbation method and approximate direct method. The approximate symmetry and similarity reduction equations of different orders are derived and the corresponding series reduction solutions are obtained. As a result, the formal coincidence for both methods is displayed.


Introduction
In this paper, we intend to investigate the approximate similarity reductions and the infinite series reduction solutions for the nonlinear K(n, 1) equation with weak damping (Biswas, 2009, p9-10) via the approximate symmetry perturbation (Zhao, Zhang and Lou, 2009, p1-4) (Jiao, Yao, Lou, 2008, p1-11 and the approximate direct methods (Clarkson and Kruskal, 1989, p2201-2212), where is a small parameter and u is a function of x and t.Hereafter, we put stress on the general case while n > 2, irrespective of the simple case of n = 2.In terms of the perturbation analysis (Cole, 1968) (Van Dyke, 1975) (Nayfeh, 2000) any solution to a perturbed PDE can be expressed as a series containing small parameter with u k functions of x and t.Substituting Eq. (2) into Eq.( 1) and vanishing the coefficients of all different powers of , we obtain the following system where 0 ≤ i m ≤ k (m = 1, . . ., n) and u −1 =0.
In Sec. 2 and 3, we apply the approximate symmetry perturbation method and approximate direct method to Eq. ( 1) respectively.Sec. 4 shows the formal coincidence for both methods on the results obtained by both methods under certain transformations.The last section is the concluding remarks

Approximate Symmetry Perturbation Method to Equation (1)
In order to study Lie symmetry reduction of Eq. ( 3), we construct the Lie point symmetry in the vector form where X, T , and U k are functions of x, t, and u k , (k = 0, 1, . ..), equivalently, Eq. ( 3) is invariant under the transformation with infinitesimal parameter ε.
Since Eq. ( 1) is not explicitly dependent upon space-time x, t, the symmetry in the vector form (4) can be written as a function form Under notation (5), the symmetry equations for Eqs.(3) read which are the linearized equations for Eqs.
It seems difficult to figure out X, T and U k , (k = 0, 1, . ..) directly because there are infinite number of equations and arguments concerning or in X, T and U k , (k = 0, 1, . ..).To make brief of it, we begin the discussion with finite number of equations.
Confining the range of k to (k = 0 − 2) in Eqs.
(3), ( 5) and ( 6), we see that X, T , U 0 , U 1 and U 2 are functions of x, t, u 0 , u 1 and u 2 .In this case, the determining equations can be derived by substituting Eq. ( 5) into Eq.( 6), eliminating u 0,t , u 1,t and u 2,t in terms of Eq. ( 3).Some of the determining equations read The general solution to Eqs. ( 7) is Using relations (8), the remaining determining equations are immediately simplified to It is straightforward to find that Likewise, restricting the range of k to {k | k = 0, 1, 2, 3} in Eqs.(3) ( 5) and ( 6), where X, T , U 0 , U 1 , U 2 and U 3 are functions of x, t, u 0 , u 1 , u 2 and u 3 , repeating the calculation process as before, then we have With more similar computation considered, we find that X, T and U k (k = 0, 1, . ..) are formally coherent, i.e., where c, x 0 and t 0 are arbitrary constants.
Subsequently, solving the characteristic equations leads to the similarity solutions to Eq. ( 3).Two subcases are distinguished as follows.
(3), we get the following related similarity reduction equations are known, since it can be rewritten as where with i m k (m = 1, . . ., n).
Case 2: When c = 0, we have the similarity solutions are thus the series reduction solution to Eq. ( 1) is where The kth (k > 0) similarity reduction equation can be rewritten as an ODE with i m k (m = 1, . . ., n).

Approximate Direct Method to Equation (1)
In this section, we develop the direct method to investigate Eq. ( 3) for its similarity solutions of the form which satisfy a system of ODEs resulting from inserting Eq. ( 16) into Eq.( 3).
On substituting Eq. ( 16) into Eq.( 3), since only one term u k,xxx in Eq. (3) generates the terms P k,zzz and P k,z P k,zz during the substitution, it is easily seen that the coefficients of P k,zzz and P k,z P k,zz are f k,P k (z x ) 3 and 3 f k,P k P k (z x ) 3 , respectively.We reserve uppercase Greek letters for undetermined functions of z hereafter.The ratios of the coefficients are functions of z, namely, with the solution where α k (x, t) and β k (x, t) are arbitrary functions.Hence, rewriting e 1 3Γ(z) P k as P k , it is sufficient to seek the similarity reduction of Eq. (3) in the special form instead of the general form Eq. ( 16).
Remark: Three freedoms in the determination of α k (x, t), β k (x, t) and z(x, t) should be notified: where Ω(z) is any invertible function, then one can take Ω(z) = z.
Then Eq. ( 3) is degenerated into From the coefficients of P 0,zzz , P 0,z and P 0 and the relations we have where A and B are arbitrary constants.
Assume that k ≥ 1, inserting Eq. ( 17) into Eq.( 3), we know that the coefficients of P k−1 , P n−2 0 P 0,z and P k,zzz are x respectively, which leads to then using remark (i) and (ii), we have We distinguish the following two subcases.
Case 1: When A 0, Eq. ( 23) has solution where t 0 and s 0 are arbitrary constants.

Analysis on Formal Coincidence for Both Methods
In the following, we discuss the formal coincidence for both methods on the basis of the results obtained by both methods.
From the above analysis of the results from both methods, we can see that approximate direct method produces more general approximate similarity reduction than the approximate symmetry perturbation method does.

Conclusion
To sum up, applying the approximate symmetry perturbation method and the approximate direct method to the nonlinear K(n, 1) equation with weak damping, we have summarized the similarity reduction equations of different orders in uniform forms and obtained the infinite series similarity reduction solutions in general formulas for Eq. ( 1).As a result, we have demonstrated the formal coincidence for both methods by relating both results.It is interesting to take both methods into account while dealing with other perturbed PDEs.Moreover, the extension of approximate Lie symmetry perturbation method to approximate nonclassical symmetry ones is likely to improve the method.