Three types of generalized Kadomtsev Petviashvili equations arising from baroclinic potential vorticity equation

Transcription

1 Chin. Phys. B Vol. 19, No. (1 1 Three types of generalized Kadomtsev Petviashvili equations arising from baroclinic potential vorticity equation Zhang Huan-Ping( 张焕萍 a, Li Biao( 李彪 ad, Chen Yong ( 陈勇 ab, and Huang Fei( 黄菲 c a Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 1511, China b Institute of Theoretical Computing, East China Normal University, Shanghai 6, China c Department of Marine Meteorology, Ocean University of China, Qingdao 66, China d Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 119, China (Received 4 March 9; revised manuscript received 4 July 9 By means of the reductive perturbation method, three types of generalized (+1-dimensional Kadomtsev Petviashvili (KP equations are derived from the baroclinic potential vorticity (BPV equation, including the modified KP (mkp equation, standard KP equation and cylindrical KP (ckp equation. Then some solutions of generalized ckp and KP equations with certain conditions are given directly and a relationship between the generalized mkp equation and the mkp equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mkp equation, some solutions of the generalized mkp equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations. Keywords: baroclinic potential vorticity equation, generalized Kadomtsev Petviashvili equation, symmetry groups, approximate solution PACC:, 11, 1c, 4k 1. Introduction In recent years, the study of nonlinear partial differential equations (PDEs have become one of the most exciting and extremely active areas of research. Many methods can be used to obtain solutions of PDEs, such as symmetry reductions, the general direct method and the extended Jacobi elliptic function rational expansion method, and so on. [1 1] However, there are many difficulties in obtaining solutions of baroclinic potential vorticity (BPV equation [11] by these methods. Luckily, the reductive perturbation method can make an intricate equation become another new equation, then the intricate equation can be researched more succesfully with the help of a new equation. This method has been used in many fields. Lou et al. [1] derived coupled KdV equations from twolayer fluids. Tang et al. [1] obtained the variable coefficient KdV equation from the Euler equation with an earth rotation term. According to the KdV equation, they obtained the approximate solution of the Euler equation with an earth rotation term. Gao et al. [14] presented a coupled variable coefficient modified KdV equation from a two-layer fluid system. The main purpose of this paper is to obtain the approximate solutions of the BPV equation. The paper is organized as follows. In Section, we obtain generalized modified Kadomtsev Petviashvili (mkp, KP and cylindrical KP (ckp equations from the BPV equation by means of the reductive perturbation method. Some KP equations have been studied by some authors. Li et al. [15] obtained soliton-like solution and periodic form solution of a KP equation. Liu [16] gained a solution of the ckp equation by auto-backlünd transformation. Wazwaz [17] obtained multiple-soliton solutions of the mkp equqtion by the Hirota s bilinear method. In Section, under certain conditions we obtain the solutions of the generalized ckp and KP equation directly, and then get an approximate solution of the BPV equation. In Section 4, by making use of the Project supported by National Natural Science Foundation of China (Grant Nos. 175 and 47754, Ningbo Natural Science Foundation (Grant No. 8A6117, National Basic Research Program of China (97 Program (Grant Nos. 5CB41 and 7CB8148 and K.C. Wong Magna Fund in Ningbo University. Corresponding author. biaolee@yahoo.com.cn c 1 Chinese Physical Society and IOP Publishing Ltd

2 Chin. Phys. B Vol. 19, No. (1 1 symmetry group direct method proposed by Lou et al., [18 1] we establish a relationship between the generalized mkp equation and mkp equation, and then based on this relationship we obtain an approximate solution of the BPV equation by complex and tedious calculations. In the last section, we give the conclusion of the paper.. Generalized mkp, KP and ckp equations from (+1- dimensional BPV equation The ( + 1-dimensional BPV equation may be written as q t + ψ x q y ψ y q x + βψ x =, (1 q = ψ xx + ψ yy + ψ zz, ( where β is constant, ψ ψ(x, y, z, t, q q(x, y, z, t, and subscripts x, y, z, t represent partial derivatives. First, by substituting Eq. ( into Eq. (1, BVP equation becomes: ψ txx + ψ tyy + ψ tzz + ψ x ψ xxy + ψ x ψ yyy + ψ x ψ yzz ψ y ψ xxx ψ y ψ xyy ψ y ψ xzz + βψ x =. ( Now we use the long wave approximation in the x- direction and z-direction and assume the stream functions ψ to be ψ = ψ (y, t + ψ 1 (x, y, z, t. (4 We introduce the stretched variables X = ϵ(x c t, Z = ϵ (z c 1 t, T = ϵ t, (5 where X X(x, t, Z Z(z, t, T T (t, c and c 1 are constants, ϵ is a small parameter. In general, the base field ψ (y, t is often taken only as a linear function of y, ψ 1 (x, y, z, t is expanded as ψ 1 (x, y, z, t = ϵ i ϕ i (X, y, Z, T ϵ i ϕ i. (6 i=1 We also have the expansion ψ (y, t = U (y + i=1 ϵ i U i (y, T. (7 i=1 Then substituting Eqs. (5, (6 and (7 into Eq. (, we obtain M 1 ϵ + M ϵ + M ϵ 4 + O(ϵ 5 =, (8 where M 1 = U y (yϕ 1Xyy + U yyy (yϕ 1X c ϕ 1Xyy + βϕ 1X, M = ϕ 1X ϕ 1yyy c ϕ Xyy U 1y (y, T ϕ 1Xyy ϕ 1Xyy ϕ 1y c 1 ϕ 1yyZ + U 1yyy (y, T ϕ 1X U y (yϕ Xyy + U yyy (yϕ X + βϕ X, M = ϕ X ϕ 1yyy c ϕ Xyy + U 1yyy (y, T ϕ X + βϕ X + ϕ 1T yy U y (yϕ 1XXX + U yyy (yϕ X U y (y, T ϕ 1Xyy c ϕ 1XXX U y (yϕ Xyy U 1y (y, T ϕ Xyy c 1 ϕ yyz + U yyy ϕ 1X + ϕ 1X ϕ yyy + U 1yyT (y, T ϕ Xyy ϕ 1y ϕ 1Xyy ϕ y. Vanishing the ϵ of Eq. (8 by Maple, we get a special solution ϕ 1 = A(X, Z, T G(y, T AG, (9 where G is determined by U yyy (yg U y (yg yy c G yy + βg =. (1 Then vanishing the ϵ of Eq. (8, we have ϕ = G 1 (y, T A(X, Z, T + G (y, T A(X, Z, T + G (y, T A Z (X, Z, T dx G 1 A + G A + G A Z dx, (11 where G 1, G and G should satisfy G yy G y + GG yyy + U yyy (yg 1 U y (yg 1yy c G 1yy + βg 1 =, U 1yyy (y, T G U 1y (y, T G yy + βg + U yyy (yg U y (yg yy c G yy =, U yyy (yg U y (yg yy c G yy c 1 G yy + βg =. (1 Substituting the solutions of Eqs. (1, (1 and ϕ = into M, integrating the result with respect to y from to y, then we obtain a 1 A T + a A XXX + a AA X + a 4 AA Z + a 5 A A X + a 6 A X A Z dx + a 7 A ZZ dx + a 8 A X + a 9 A Z + a 1 A + a 11 =, (1 where a i a i (T, i = 1,,..., 11 a 1 = y y1 G yy dydy 1, 1

3 a = a = a 4 = a 5 = a 6 = a 7 = a 8 = a 9 = a 1 = a 11 = y y1 y y1 Chin. Phys. B Vol. 19, No. (1 1 ( U y (yg c Gdydy 1, [U 1yyy (y, T G 1 U 1y (y, T G 1yy G y G yy + G yyy G + GG yyy G yy G y ]dydy 1, y y1 y y1 (G yyy G c 1 G 1yy G y G yy dydy 1, ( G y G 1yy + G yyy G 1 + GG 1yyy G yy G 1y dydy 1, y y1 y y1 y y1 (GG yyy G yy G y dydy 1, ( c 1 G yy dydy 1, [ U y (y, T G yy + U yyy (y, T G U 1y (y, T G yy + U 1yyy (y, T G ]dydy 1, y y1 [U 1yyy (y, T G c 1 G yy U 1y (y, T G yy ]dydy 1, y y1 y y1 G yyt dydy 1, U 1yyT (y, T dydy 1. Equation (1 contains the following important cases: (i If a 4 =, a 5 =, a 6 =, a 8 =, a 9 =, a 1 =, a 11 =, equation (1 becomes a generalized KP equation a 1 A T + a A XXX + a AA X + a 7 A ZZ dx =. (14 (ii If a 4 =, a 5 =, a 6 =, a 8 =, a 9 =, a 1 =, equation (1 becomes a generalized ckp equation a 1 A T + a A XXX + a AA X + a 7 A ZZ dx + a 11 A =. (15 (iii If a =, a 4 =, a 8 =, a 9 =, a 1 =, a 11 =, equation (1 becomes a generalized mkp equation a 1 A T + a A XXX + a 5 A A X + a 6 A X A Z dx + a 7 A ZZ dx =. (16. The solutions of BPV equation from generalized KP and ckp equations In this section, we divide into two parts to obtain approximate solutions of the BPV equation with the help of the generalized KP and ckp equations. First, we set about the generalized KP equation to obtain the approximate solution of Eq. (. We know that equation (14 should satisfy a 4 = a 5 = a 6 = a 8 = a 9 = a 1 = a 11 =, U yyy (ya X G U y (ya X G yy c A X G yy =, G yy G y + GG yyy + U yyy (yg 1 U y (yg 1yy c G 1yy =, U yyy (yg U y (yg yy c G yy c 1 G yy =, U 1yyy (y, T G U 1y (y, T G yy + β 1 G + U yyy (yg U y (yg yy c G yy =. (17 According to Eq. (17, we obtain G(y, k, G 1 (y, k, G (y, k, G (y, k, U (y, U 1 (y, k and U (y, k after complicated computing process G = C 8, G 1 = C 7, G = F (T (C 1 y + C, G = c 1 F (T + F 4 (T (C 1 y + C, U (y = 1 6 βy + 1 C 4y + C 5 y + C 6, U i (y, k = (i, U 1 (y, T = 1 C 1y + C y + C + F 1 (T, (18 where C i (i = 1,,..., 8 are arbitrary constants, F 1 (T, F (T and F 4 (T are arbitrary functions with respect to time. At the same time, we can obtain a 1, a, a and a 7 : a = 1 ( 1 C 8 1 βy4 1 C 4y C 5 y c y, a 1 = C 8, a = C 1 C 7 y, a 7 = c 1 (c 1 F (T + F 4 (T (C 1 y + C. (19 Then we discuss the solution of Eq. (14, the balancing procedure yields n = for Eq. (14, therefore, we may choose A = f(z, T + f 1 (T sech(w 1 (T X + W (T Z + W (T + f (T sech(w 1 (T X + W (T Z + W (T, ( In the next section, we discuss the solutions of Eq. ( with the help of Eqs. (14, (15 and (16 by a direct method and symmetry group direct method. Substituting Eq. ( into Eq. (14, we can get f(z, T, f 1 (T, f (T, W 1 (T, W (T, and W (T by vanishing sech, 1

4 Chin. Phys. B Vol. 19, No. (1 1 f(z, T = g 1 (T Z + g (T, f 1 (T =, f (T = 1C 1a 1/ W 1 (T = C 1a 1/ a 1/, W (T = ( c W (T = 1W (T F (T C 8 W 1 (T a 1/ a g 1 (T W 1 (T a 1 dt + C 9,, + c 1W (T F 4 (T C 1 y + c 1W (T F 4 (T C + C 8 W 1 (T C 8 W 1 (T W 1(T C 4 y W 1(T C 1 C 7 y g (T C 8 + W 1 (T c y 1 6 W 1(T βy 4 + W 1 (T C 5 y dt + C 11, (1 where g 1 (T, g (T are arbitrary functions of T, C 9, C 1 and C 11 are arbitrary constants. Then, we have A = g 1 (T Z + g (T + 1C 1a 1/ sech(w a 1/ 1 (T X + W (T Z + W (T. ( According to Eqs. ( and (1, we can obtain one possible approximate solution of Eq. ( in the form ψ U (y + ϵu 1 (y, T + ϵϕ 1 + ϵ ϕ = U (y + ϵu 1 (y, T + ϵag + ϵ ( G 1 A + G A + G A Z dx, ( where A, G, G 1, G, G, U (y and U 1 (y, k satisfy Eqs.(18 (. When we define uncertain parameters, some new soliton solutions can be obtained as shown in Figs. 1(a and 1(b. Fig. 1. Evolution of solutions ( with the parameters: C 1 = 1, β = 1, y = 8, ϵ =.1, c 1 = 1, c = 1, g 1 (T = T, C 6 =, C 9 =, g (T =, F 1 (T = T, C 8 = 1, C 4 = 1, C 5 = 1, C 7 = 1/1, C 11 =, C 1 = 6/16 /, F (T = 1, C =, F 4 (T = 1/T, C =.. Next, we discuss the solution of Eq. ( with the generalized ckp equation. In a similar way, equation (15 should satisfy a 4 = a 5 = a 6 = a 8 = a 9 = a 1 =, U yyy (ya X G U y (ya X G yy c A X G yy =, G yy G y + GG yyy + U yyy (yg 1 U y (yg 1yy c G 1yy =, U yyy (yg U y (yg yy c G yy c 1 G yy =, U 1yyy (y, T G U 1y (y, T G yy + β 1 G + U yyy (yg U y (yg yy c G yy =, (4 1 4

5 Chin. Phys. B Vol. 19, No. (1 1 We obtain the solution of Eq. (4 by tedious calculations, G 1 = P 5 (T, G = c 1 P (T + P 4 (T (D 1 y + D, G = P 5 (T P 4 (T D 1, G = P (T (D 1 y + D, U (y = 1 6 βy + 1 D 4y + D 5 y + D 6, U 1 (y, T = 1 D 1y + D y + D + P 1 (T, U i (y, T = (i, (5 where D i, i = 1,,..., 6 are arbitrary constants, P i (T, i = 1,,..., 5 are arbitrary functions of T. Then we obtain a 1 = P 5 (T P 4 (T D 1, a = 1 4 P 5(T P 4 (T D 1 y (βy 4D 4 y 1c 1D 5, a = P 5 (T (P 4 (T P (T D 1 + D 1 y D, a 7 = c 1P (T c 1 P 4 (T (D 1 y + D, a 11 = P 5T (T P 4 (T D 1 + P 5 (T P 4T (T D 1. (6 Now we discuss the solution of Eq. (15. If we let P 4 (T = (D 1y D D 1 ( 1 + P (T D 1, D 4 = 1 4 and P (T, P 5 (T satisfy ( 4 1c y + βy 4 1D 5 y y, (7 1 c 1 ( c 1 P (T D 1 + c 1 P (T D1 D1y + D = σ 1 D 1 P 5 (T (D 1 y D T, ( P 5T (T + P 5T (T P (T D 1 P 5 (T P T (T D 1 ( 1 + P (T D 1 P 5 (T = 1 T, (8 where σ 1 = ±1, which also arises in water waves. The ckp equation is known to be completely integrable. [] According to Eqs. (7 and (8, equation (15 becomes A T + A XXX + AA X + σ 1 T A ZZ dx + 1 A =, (9 T we know the solution [16] of Eq. (49: A = λ sech (ξ, [ ξ = 1 λx λσ T Z 1 T (λ4 λδz + σ δ λ + γ ], ( where σ = ±1, λ, δ, γ are integral constants. Then we obtain an approximate solution of Eq. (: ψ U (y + ϵu 1 (y, T + ϵϕ 1 + ϵ ϕ = U (y + ϵu 1 (y, T + ϵag + ϵ ( G 1 A + G A + G A Z dx, (1 where U (y, U 1 (y, T, G, G 1, G, G and A satisfy Eqs. (5 (8 and (. When we define uncertain parameters, some new soliton solutions can be obtained. The evolution of solution (1 with some different parameters are shown in Figs. (a and (b. Fig.. Evolution of solution (1 with the parameters: P 1 (T =, λ = 1, ϵ1 = 1, ϵ =.1, δ = 1, γ = 1, D =, c = 1, D =, y = 1, β =, D 1 = 1, D 6 =, D 5 =, D 4 =

6 Chin. Phys. B Vol. 19, No. ( The solution of BPV equation from generalized mkp equation by symmetry group method generalized mkp equation, we have a 1 v XT + v XXXX + a 5 v XX vx + a 6 v XX v Z + a 7 v ZZ =. ( Secondly, let As is well known, the generalized mkp equation is a universal model for the propagation of weakly nonlinear dispersive long waves which are essentially one directional with weak transverse effects. There are many difficulties in obtaining the solution of the generalized mkp equation, but it is easier to find the solution of the mkp equation than that of the generalized mkp equation. So we give a theorem to change the generalized mkp equation into the mkp equation by a generalized symmetry group method first, then we can find a more preferable research BPV equation with the help of the theorem. First, let A(X, Z, T = v X, ( where v v(x, Z, T. Substituting Eq. ( into the v = α + γv (ξ, η, τ = α + γv, (4 where α, γ, ξ, η, and τ are functions of {X, Z, T }, V satisfies the following equation V ξξξξ + d 1 V ξτ + d V ξξ V ξ + d 4 V ξξ V η + d 5 V ηη =, (5 where d 1, d, d 4 and d 5 are non-zero arbitrary constants. Substituting Eq. (4 into Eq. (, then eliminating V ξξξξ by using Eq. (5, from that, the remained determining equations of the functions ξ, η, τ, α, γ can be obtained by vanishing the coefficients of V and its derivatives. Then the general solution of the determining equations are found out by tedious calculations. The result reads ξ = c ( a1 τ T d 1 1/ X + ξ, τ = τ, γ = δ d a 5, η = ( / a5 c a1 τ T d 4 Z d d 1 a 6 δ + η, ξ = ( a 1a 5 a 1T τ T a 1a 5 τ T T + a 1a 5T τ T 1 d 5 d a 6 a/ 1 τ / T d1/ 1 c d 4Z + ξ 1 Z + ξ, 1 α = 1 ( a 1/ 1 τ 1/ a 6ξZ d 4d 1/ 1 + 4c ξ T a 4/ 1 τ 1/ T a 5d 4 T 4 τ T a 5 d 4 a + 4a 6ξZ d d 5 d 1/ 1 1a 6 τ T a 5 d 4 a d 1/ 1 c dz 1a 6 1 a 6 ξ Z d 1/ 1 X c a 1/ 1 τ 1/ T a + α, (6 5 where τ τ (T, ξ ξ (Z, T, ξ 1 ξ 1 (T, ξ ξ (T, η η (T, α α (T, δ = ±1, and c = 1, 1 + i, 1 i. At the same time, it should satisfy four conditions: a 7 a 5 d 4 d 5 d a 6 =, 6a 1 η T d a 5 a 6δ(a 1 τ T 1/ d / 1 d 4 + ( 6a 1a 5 τ T d 4a 6T + 4a 1 a 5 d 4a 6 a 1T τ T + 4a 1a 5 d 4a 6 τ T T + a 1τ T d 4a 5T a 6 c Z + 1d 5 d a 6ξ Z c a / 1 τ / T d1/ 1 =, 4τ T a 1T a 5 d 5 d a 1 d 4τ T a 5T + d 4τ T a 5 a 1T + a 1 d 4a 5 τ T T + 4a 1 τ T T a 5 d 5 d =, a 6ξ Z d 4d 5 d a / 1 τ / T d1/ 1 ξ ZZ + 1d 5d a 6ξ Z a / 1 τ / T d1/ 1 ξ ZZ + τ T d 4 4a 1c a 6T ξ Z a 5 + τ T d 4 4a 1c a 6 ξ ZT a 5 τ T d 4 4a 1 c a 6 ξ Z a 5 a 1T d 4 4a 1c a 6 ξ Z a 5 τ T T τ T d 4 4a 1c a 6 ξ Z a 5T + 6c τ T ξ ZT a 1a 5 d 4d 5 d a 6 =. (7 In summery, we obtain the following theorem. Theorem If U U(X, Z, T is a solution of the mkp equation (5, then u = α + γu(ξ, η, τ 1 6

7 Chin. Phys. B Vol. 19, No. (1 1 is the solution of the generalized mkp equation ( on the conditions of Eq. (7, where α, γ, ξ, η, τ, δ and c are given by Eq. (6. According to the Theorem, we will research the BPV equation easier with the help of the mkp and generalized mkp equation. Next, we will write down some types of solutions of Eq. ( with the help of the known solutions for the (+1-dimensional mkp equation. As known to all, mkp equation is u T + u XXX 6u X u 6u X u Z dx + u ZZ dx =, (8 where u u(x, Z, T. Let d 1 = 1, d = 6, d 4 = 6, d 5 =, (9 and replace V (X, Z, T with V (ξ, η, τ, equation (5 becomes V XXXX V XT 6V XX V X 6V XX V Z + V ZZ =, (4 equation (4 is equivalent to Eq. (8 under the transformation u = V X. So we discuss Eq. (4 firstly in the following. We know that the mkp equation possesses the following solution: u = b(1 tanh(b(x bz 16b T, (41 where b is a non-zero arbitrary constant. According to the transformation u = V X and Eq. (41, we obtain the solution of Eq. (4: V = bx + ln(sech(b( X + bz + 16b T, (4 according to the Theorem and Eq. (, we obtain the solution of the generalized mkp equation, A = 1 6a 6 a 1 τ c a 5 {6 6δ abc a 1 τ ( a 1 τ 1/ a 6 [tanh(b( X + bz + 16b T + 1] c a 1 a 5 Z(a 1T τ + a 1 τ T T ξ 1 a 6( a 1 τ / }, (4 where a 5 is a non-zero arbitrary constant. In order to obtain the solution of Eq. (, we need to know U (y, U 1 (y, T, G, G 1, G and G, which should satisfy a =, a 4 =, a 8 =, a 9 =, a 1 =, a 11 =, Eqs. (1, (1 and (7. After tedious calculations, we obtain: G = K 1 y + K + Q 5 (T, G 1 = Q 6 (T, G = Q (T K 6, G = c 1 Q (T + Q 4 (T K 6, U (y = 1 6 βy + 1 K y + K 4 y + K 5, U 1 (y, T = K 6 y + K 7 + Q 1 (T, U i (y, T = (i, (44 where Q i (T, i = 1,, 4, 5, 6 are arbitrary function of T, K i, i = 1,,..., 7 are arbitrary constants. Then we get a = 1 4 βk 1y y4 (βk K K y (c K 1 + K 4 K 1 + K K 1 K y (c + K 4, a 1 = K 1 y + K, a 5 = 4K 1, a 6 = K 1 (c 1 Q (T + Q 4 (T K 6, a 7 = c 1 (c 1 Q (T + Q 4 (T K 6. (45 From that we obtain an approximate solution of Eq. ( ψ U (y + ϵu 1 (y, T + ϵϕ 1 + ϵ ϕ = U (y + ϵu 1 (y, T + ϵag + ϵ ( G 1 A + G A + G A Z dx, (46 where A, G, G 1, G, G, U (y and U 1 (y, k satisfy Eqs. (7, (4 ( Summary and discussion In summary, by the reductive perturbation method the generalized (+1-dimensional modified Kadomtsev Petviashvili (mkp, Kadomtsev Petviashvili (KP and cylindrical Kadomtsev Petviashvili (ckp equations are derived from the baroclinic potential vorticity (BPV equation. Some solutions of the KP, mkp and ckp equations are derived by a direct method and symmetry group direct method. Thus the solution of the generalized mkp 1 7

8 Chin. Phys. B Vol. 19, No. (1 1 equation can be obtained from the mkp equation. Furthermore, some approximate solutions of the BPV equation are obtained from three types of generalized KP equations. References [1] Chen Y and Fan E 7 Chin. Phys [] Zhang S L, Lou S Y and Qu C Z 6 Chin. Phys [] Yang P, Chen Y and Li Z B 8 Chin. Phys. B [4] Wang J and Li B 9 Chin. Phys. B [5] Li J H and Lou S Y 8 Chin. Phys. B [6] Yao R X, Jiao X Y and Lou S Y 9 Chin. Phys. B [7] Qian S P and Tian L X 7 Chin. Phys. 16 [8] Zha Q L and Li Z B 8 Chin. Phys. B 17 [9] Zhang S Q, Xu G Q and Li Z B Chin. Phys [1] Xia T C, Zhang H Q and Yan Z Y 1 Chin. Phys [11] Pedlosky J 1979 Geophysical Fluid Dynamics (New York: Springer [1] Lou S Y, Tong B, Hu H C and Tang X Y 6 J. Phys. A: Math. Gen [1] Tang X Y, Huang F and Lou S Y 6 Chin. Phys. Lett [14] Gao Y and Tang X Y 7 Commun. Theor. Phys. (Beijing, China [15] Li B, Chen Y and Zhang H Q 5 Acta Mech [16] Tang X Y and Lou S Y Chin. Phys. Lett [17] Abdul-Majid Wazwaz 8 Appl. Math. Comput. 4 7 [18] Lou S Y and Ma H C 5 J. Phys. A 8 L19 [19] Lou S Y and Ma H C 6 Chaos, Solitons and Fractals 84 [] Lou S Y, Jia M, Tang X Y and Huang F 7 Phys. Rev. E [1] Lou S Y and Tang X Y 4 J. Math. Phys [] Hereman W and Zhuang W 1995 Acta Appl. Math