LIE SYMMETRY, FULL SYMMETRY GROUP, AND EXACT SOLUTIONS TO THE (2+1)-DIMENSIONAL DISSIPATIVE AKNS EQUATION

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1 LIE SYMMETRY FULL SYMMETRY GROUP AND EXACT SOLUTIONS TO THE (2+1)-DIMENSIONAL DISSIPATIVE AKNS EQUATION ZHENG-YI MA 12 HUI-LIN WU 1 QUAN-YONG ZHU 1 1 Department of Mathematics Lishui University Lishui PR China 2 Department of Mathematics Zhejiang Sci-Tech University Hangzhou PR China ma-zhengyi@163com (corresponding author) Received January Abstract In this paper a detailed Lie symmetry analysis of the (2+1)-dimensional dissipative AKNS equation is performed The general finite transformation group is given via a simple direct method which is in fact equivalent to Lie point symmetry group The similarity reductions are considered from the general Lie symmetry and some exact solutions of the (2+1)-dimensional dissipative AKNS equation are obtained Key words: (2+1)-dimensional dissipative AKNS equation Lie symmetry full symmetry group similarity reduction invariant solutions PACS: 0230Jr 0220Sv 0545Pq 0220Tw 1 INTRODUCTION The study of the symmetries of partial differential equations (PDEs) is a central issue in nonlinear mathematical physics During the nineteenth century Sophus Lie 1 2] proposed a standard technique the Lie symmetry method to find the classical Lie point symmetry groups and algebras for differential systems A lot of universal applications of Lie symmetry groups in differential equation were discussed in the literature 1 19] such as the reduction of order of ordinary differential equations and dimension of PDEs the construction of invariant solutions the mapping solutions to other solutions and the detection of linearizing transformations the derivation of conservation laws and so on Several generalizations of the classical Lie group method for symmetry reduction were developed Since all solutions of the classical determining equations need to satisfy the nonclassical determining equations and the solution set may be larger in the nonclassical case Bluman and Cole 3] presented the method of conditional symmetries These methods were further extended by Olver and Rosenau 4] to include weak symmetries and even more generally side conditions or differential constraints Motivated by the fact that the known symmetry reductions of the Boussinesq equation are not obtained using the classical Lie group method Clarkson and Kruskal 5] (CK) introduced a simple direct method to find Romanian Journal of Physics (2017) v20*201771#662d169c

2 Article no 114 Zheng-Yi Ma Hui-Lin Wu Quan-Yong Zhu 2 all the possible similarity reductions of a nonlinear system without using any group theory Most recently inspired by the CK direct method Lou and Ma 6] modified the CK direct method to find the generalized Lie and non-lie symmetry groups for some well-known nonlinear evolution equations Such kind of full symmetry group can obtain not only the Lie point symmetry group but also the non-lie symmetry group for the given nonlinear system In this paper we consider the following (2+1)-dimensional dissipative AKNS equation 20 23] 4u xt + u xxxy + 8u xy u x + 4u y u xx + ru xx = 0 (1) where r is a constant Especially the coefficient r 0 shows that the system has dissipative effects When r = 0 Eq (1) reduces to the (2+1)-dimensional AKNS equation 20] Based on the Bell polynomial theory Liu et al studied the complete integrability of Eq (1) such as the Bäcklund transformation the Lax pair and the infinite conservation laws 21] Some exact traveling wave solutions were derived for the (2+1)-dimensional AKNS equation by using the extended homoclinic test approach 24] and the improved tanh method 25] Wazwaz 22] investigated the soliton solutions by using the simplified form of the bilinear method By means of the multidimensional Riemann theta function the periodic wave solutions were explicitly constructed 23] The outline of this paper is as follows In Sec 2 the Lie point symmetry of the (2+1)-dimensional dissipative AKNS equation (1) and the commutator relations of the generators associated with the symmetry are provided by means of the classical method In Sec 3 with the help of a simple direct method the finite transformation groups of Eq (1) is given which is equivalent to Lie point symmetry groups generated through the standard approach Section 4 is devoted to performing the (1+1)-dimensional similar reductions from the general Lie point symmetry In Sec 5 exact solutions of the dissipative AKNS equation (1) are constructed from the reduced equations Some conclusions and a brief discussion of the results are given in the last section 2 LIE POINT SYMMETRY VIA THE CLASSICAL SYMMETRY METHOD In this Section we present the Lie symmetry analysis for the (2+1)-dimensional dissipative AKNS equation (1) In order to apply the classical method to Eq (1) (ie =rhs of (1) = 0) we consider a one-parameter Lie group of infinitesimal (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

3 3 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no 114 transformation in (xytu) given by x x + ɛx(xytu) y y + ɛy (xytu) t t + ɛt (xytu) u u + ɛu(xytu) (2) where ɛ is the group parameter The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form = X x + Y y + T t + U u (3) Then the invariance of Eq (1) under the transformation (2) provides that pr (4) V( ) =0 = 0 (4) where pr (4) V is the fourth prolongation of the vector field (3) This yields an overdetermined linear system of differential equations for the infinitesimals X Y T and U With the help of the symbolic computation system MAPLE by solving these determining equations we find that the infinitesimals can be written as follows: X = 1 4 (c 6t + 2c 4 2c 5 )x + g Y = 1 2 (c 6y + 2c 3 )t + c 5 y + c 2 T = 1 2 c2 6t 2 + c 4 t + c 1 U = 1 4 (c 6y + 2c 3 )x + ġ r 16 (3c 6t + 2c 4 + 2c 5 )]y + f 1 4 (c 6t + 2c 4 + 2c 5 )u (5) where f and g are arbitrary functions of t c i (i = 126) are arbitrary constants and the dot over the function means its derivative with respect to time t The presence of the arbitrary function and constants leads to an infinite-dimensional Lie algebra of symmetries A general element of this algebra is written as = c c c c c c (f) + 8(g) (6) (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

4 Article no 114 Zheng-Yi Ma Hui-Lin Wu Quan-Yong Zhu 4 where 1 = t 2 = y 3 = t y + x 2 u 4 = x 2 x + t t = x 2 x + y y = xt 4 x + yt 2 y + t2 2 7 (f) = f u 8 (g) = g x + ġy u (ry + 4u) u (ry 4u) u t 1 (3ryt 4xy + 4tu) 16 u construct a basis for the vector space The associated Lie algebra among these vector fields are given by Table 1 where the entry in the j-th row and the k-th column represents the commutator j k] (7) Table 1 The commutation relation between infinitesimal generators of point symmetries j k] (r) (r) (rt) (rt) (rt) 16 7(rt2 ) j k] 7(f) 8(g) 1 7 ( f) 8 (ġ) (ġ) (tġ 1 2 g) 4 8 (tġ 1 2 g) ( 1 2 t2 f tf) 8( 1 2 t2 ġ 1 4 tg) 7 (f) (g) 0 (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

5 5 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no 114 We now consider a point transformation G : (xytu) (χζγp ) (8) From the transformation (2) we have the corresponding one-parameter group of symmetries of the (2+1)-dimensional dissipative AKNS equation G 1 : (xytu) (xyt + ɛu) G 2 : (xytu) (xy + ɛtu) G 3 : (xytu) (xy + ɛtt 1 ɛx + u) 2 G 4 : (xytu) (xe ɛ 2 yte ɛ ue ɛ r (e ɛ 2 1)y) G 5 : (xytu) (xe ɛ 2 ye ɛ tue ɛ r 2 4 (eɛ e ɛ 2 )y) ( i 2ɛt 4x 2 2y G 6 : (xytu) ɛt 2 2t ɛt 2 i i 2ɛt 4u + 2ɛt 4 2 4t 2 ri ) ln( )xy 2 ɛt 8ɛt 16 2ɛt 4(2 ɛt) + 4i]y G 7 : (xytu) (xytu + ɛf) G 8 : (xytu) (x + ɛgytu + ɛġy) So the entire symmetry group is obtained by composing one-dimensional subgroups G i (i = 128) When G is an element of this group if u(xyt) is a solution of the dissipative AKNS equation then P (χζγ) is also a solution of the dissipative AKNS equation (9) 3 SYMMETRY GROUP VIA A SIMPLE DIRECT METHOD According to the symmetry group direct method 6] we can take the simplified symmetry transformation ansatz as u = α + βu(ξητ) (10) where αβ(i = 123)ξη and τ are functions of {xyt} to be determined We require that U(ξ η τ) also satisfies the (2+1)-dimensional dissipative AKNS equation but with different independent variables {ξητ} 4U ξτ + U ξξξη + 8U ξη U ξ + 4U η U ξξ + ru ξξ = 0 (11) Substituting (10) into Eqs (1) eliminating U ξτ and their higher-order derivatives by Eqs (11) then setting the coefficients of the polynomials of U and its derivatives to be zero we obtain a huge numbers of nonlinear PDEs with respect to differentiable functions: {α β ξ η τ} After thorough analysis and quite tedious (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

6 Article no 114 Zheng-Yi Ma Hui-Lin Wu Quan-Yong Zhu 6 calculations the general results read: α = σ 2s 5(s 1 t + s 2 ) s 1 s1 (y + 2s 7 ) + 2(s 1 t + s 2 ) s 1 β = σ 4s 5 (s 1 t + s 2 ) s 1 ξ = σ 4s 5 (s 1 t + s 2 ) ] 1 2 r(s 2 s 3 s 1 s 4 ) 4(s 1 t + s 2 ) 2 d dt ξ 0(t) 1 ] 4 σr y ] x + α 0 (t) ] 1 2 ] 1 2 x + ξ0 (t) η = 2s 5(s 1 s 4 s 2 s 3 ) (y + 2s 7 ) + s 6 s 1 (s 1 t + s 2 ) τ = s 3t + s 4 s 1 t + s 2 σ 2 = 1 (12) where ξ 0 ξ 0 (t) and α 0 α 0 (t) are arbitrary functions of t and s i (i = ) are arbitrary constants It is necessary to point out that the independent variable t possess invariant property under the Möbious (conformal) transformation In summary we can arrive at the following final transformation group theorem of Eq (1) Theorem 31 If U = U(xyt) is a solution of the (2+1)-dimensional dissipative AKNS equation then so is u u =σ 2s ] 1 5(s 1 t + s 2 ) 2 r(s 2 s 3 s 1 s 4 ) + s 1 s1 (y + 2s 7 ) 2(s 1 t + s 2 ) ] x + α 0 (t) + σ 4(s 1 t + s 2 ) 2 d dt ξ 0(t) 1 ] 4 σr y s 1 4s 5 (s 1 t + s 2 ) ] 1 2 U(ξητ) (13) where ξη and τ are determined by (12) In order to see the equivalence between the Lie point symmetry group obtained in Theorem 31 and the known one from classical Lie group method we need to take the arbitrary functions ξ 0 (t)α 0 (t) and arbitrary constants s i (i = ) to be different forms with respect to an infinitesimal parameter ɛ σ = 1 s 1 = ɛ 2 c 6 s 2 = 1 + ɛ(c 5 c 4 ) s 4 = ɛc 1 s 5 = ɛ 4 c 6 s 6 = 2c 3 c 6 s 7 = c ( 3 c2 + ɛ c 6 2 c ) 3c 5 ξ 0 (t) = ɛg(t) α 0 (t) = ɛf(t) c 6 (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

7 7 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no 114 then (13) can be written as u = u + ɛσ = u + ɛσ(u) ] ] 1 1 σ(u) = 4 (c 6t + 2c 4 2c 5 )x + g u x + 2 (c 6y + 2c 3 )t + c 5 y + c 2 u y + ( 1 2 c2 6t 2 + c 4 t + c 1 )u t 1 4 (c 6y + 2c 3 )x ġ r 16 (3c 6t + 2c 4 + 2c 5 )]y f (c 6t + 2c 4 + 2c 5 )u which is exactly the same as the one obtained by the standard Lie approach 4 (1+1)-DIMENSIONAL SIMILAR REDUCTION OF THE DISSIPATIVE AKNS EQUATION After determining the infinitesimal generators in Sec 2 the similarity variables can be obtained by solving the characteristic equations dx X = dy Y = dt T = du U Here we only list the following cases Case 1 For the generator 1 + a 2 one has the following invariants 1 χ = x γ = t y u = P (χγ) (14) Substituting Eq (14) into Eq( 1) yields the corresponding reduced equation P χχχγ + P χγ 8P χγ p χ 4P χχ P γ + rp χχ = 0 (15) Case 2 For the generator 1 +a 2 2 +a 3 we have the following invariants 3 χ = x γ = a 3 2 t2 + a 2 t y u = 1 2 (a 3t + a 2 )x + P (χγ) (16) Substituting Eq (16) into Eq (1) leads to the corresponding reduced equation P χχχη 8P χγ p χ 4P χχ P γ + rp χχ + 2a 3 = 0 (17) Case 3 For the generator 4 the following invariants can be derived χ = y γ = t x 2 u = 1 ry + P (χγ) (18) 4 Substituting Eq (18) into Eq (1) then the reduced equation reads 8γ 3 P χγγγ 48γ 2 P χγγ + 2(8P 27)γP χγ 8γP γγ + (32γ 2 P χγ + 56γP χ 12)P η + (16γ 2 P γγ + 16P 6)P χ = 0 Case 4 For the generator 5 one can get the following invariants (19) χ = yx 2 γ = t u = 1 4 ry + P (χγ) (20) (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

8 Article no 114 Zheng-Yi Ma Hui-Lin Wu Quan-Yong Zhu 8 Substituting Eq (20) into Eq (1) it follows the reduced equation 8χ 3 P χχχχ + 24χ 2 P χχχ 2χ(8P 3)P χχ 8χP χγ + 48χ 2 P χχ P χ 4P γ = 0 (21) Case 5 For the generator 6 the corresponding invariants are χ = y x 2 γ = t x 2 u = 1 xy ry + + P (χγ) (22) 4 2t Substituting Eq (22) into Eq (1) yields the reduced equation 8χ 3 P χχχχ 8γ 3 P χγγγ 24χ 2 γp χχχγ 24χγ 2 P χχγγ 72χ 2 P χχχ + 2γ(32χP χ + 16γP γ + 8P 75)P χγ + 2χ(24χP χ + 16γP γ + 8P 75)P χχ + 88χP 2 χ + (88γP γ + 32P 60)P χ + 16γ 2 P χ P γγ 144χγP χχγ 72γ 2 P χγγ = 0 read Case 6 For the generator 1 + 7(f) + 8(g) the corresponding invariants χ = x g γ = y u = d dt gy + f + P (χγ) g = d dt g f = d dt f (23) Substituting Eq (23) into Eq (1) the initial equation becomes the reduced equation P χχχγ + 8P χγ P χ + 4P χχ P γ + rp χχ = 0 (24) 5 EXACT SOLUTIONS OF THE DISSIPATIVE AKNS EQUATION In this Section we provide some exact solutions of the dissipative AKNS equation (1) from three kinds of reduction transformation given in the previous Section A Solutions from reduction Here we are looking for the traveling wave solutions for the dissipative AKNS equation (1) therefore we consider a similarity variable θ = χ + k 1 γ where k 1 is a constant The reduced equation becomes an ordinary differential equation k 1 φ θθθ 6k 1 φ 2 θ + (r + 4k 1 )φ θ + c 0 = 0 (25) where c 0 is an integral constant In fact the ordinary differential equation that the function φ θ in Eq(25) need to satisfy is an elliptic equation With different choices of the constants we have exact solutions listed below Hereafter sn cn and dn represent the Jacobi elliptic functions and Ei is the second kind incomplete elliptic integral Case 1 u 1 = A h ω + hcn(ωm)dn(ωm) + hei(sn(ωm)m) (26) sn(ω m) (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

9 9 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no 114 with ω = h x k ] A = 1 3 (m2 2)h 2 + (r + 4k 1 ) c 0 = 2 3 k 1(m 4 m 2 + 1)h 4 a2 1 (r + 4k 1) 2 (27) with u 2 = A h ω + hcn(ωm)dn(ωm) sn(ω m) + 2hEi(sn(ωm)m) (28) ω = h x k ] A = 1 3 (m2 5)h 2 + (r + 4k 1 ) c 0 = 2 3 k 1(m m 2 + 1)h 4 a2 1 (r + 4k 1) 2 Especially when m = 1 the elliptic functions degenerate to the hyperbolic functions then the above solutions become ũ 1 = A x k ] { + htanh 1 h x k ]} A = h2 3 + (r + 4k 1 ) c 0 = 2 3 k 1h 4 a2 1 (r + 4k 1) 2 (30) and ũ 2 = A x k ] { + 2htanh 1 2h x k ]} A = 4h2 3 + (r + 4k 1 ) c 0 = 32 3 k 1h 4 a2 1 (r + 4k 1) 2 Case 2 (29) (31) with u 3 = A h ω + hsn(ωm)dn(ωm) cn(ω m) + hei(sn(ωm)m) (32) ω = h x k ] A = 1 3 (m2 2)h 2 + (r + 4k 1 ) c 0 = 2 3 k 1(m 4 m 2 + 1)h 4 a2 1 (r + 4k 1) 2 (33) u 4 = A h ω + hsn(ωm)dn(ωm) + 2hEi(sn(ωm)m) (34) cn(ω m) (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

10 Article no 114 Zheng-Yi Ma Hui-Lin Wu Quan-Yong Zhu 10 with ω = h x k ] A = 1 3 (4m2 5)h 2 + (r + 4k 1 ) c 0 = 2 3 k 1(m m 2 + 1)h 4 a2 1 (r + 4k 1) 2 Especially when m = 1 the elliptic functions degenerate to the hyperbolic functions then the above solutions become ũ 3 = A x k ] A = h2 3 + (r + 4k 1 ) c 0 = 2 3 k 1h 4 a2 1 (r + 4k 1) 2 (36) and Case 3 ũ 4 = A x k ] { + htanh h x k ]} A = h2 3 + (r + 4k 1 ) c 0 = 2 3 k 1h 4 a2 1 (r + 4k 1) 2 (35) (37) with with u 5 = A h ω + hm2 sn(ωm)cn(ωm) dn(ω m) + hei(sn(ωm)m) (38) ω = h x k ] A = 1 3 (m2 2)h 2 + (r + 4k 1 ) c 0 = 2 3 k 1(m 4 m 2 + 1)h 4 a2 1 (r + 4k 1) 2 u 6 = A h ω + hm2 sn(ωm)cn(ωm) dn(ω m) (39) + 2hEi(sn(ωm)m) (40) ω = h x k ] A = 1 3 (m2 2)h 2 + (r + 4k 1 ) c 0 = 2 3 k 1(m 4 16m )h 4 a2 1 (r + 4k 1) 2 (41) Especially when m = 1 the elliptic functions degenerate to the hyperbolic (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

11 11 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no 114 functions then the above solutions become ũ 5 = A x k ] and A = h2 3 + (r + 4k 1 ) c 0 = 2 3 k 1h 4 a2 1 (r + 4k 1) 2 ũ 6 = A x k ] { + htanh h x k ]} A = h2 3 + (r + 4k 1 ) c 0 = 2 3 k 1h 4 a2 1 (r + 4k 1) 2 B Solutions from 1 + a a 3 3 reduction For the reduced equation (17) it is straightforward to verify that it has one special solution P (χγ) = a 3χ 2 4λ 1 r + λ 2χ + λ 1 γ + λ 3 (44) where λ i (i = 123) are arbitrary constants Therefore we can get the following exact solution of the dissipative AKNS equation u 7 = 1 2 (a 3t + a 2 )x + a 3x 2 (42) (43) 4λ 1 r + λ 2x + λ 1 ( a 3 2 t2 + a 2 t y) (45) C Solutions from 1 + 7(f) + 8(g) reduction For the reduced equation (24) we still seek for traveling wave reduction with a similarity variable θ = χ + k 3 γ where k 3 is a constant Then the reduced equation (24) becomes an ordinary differential equation k 3 φ θθθ + 6k 3 φ 2 θ + rφ θ + c 0 = 0 (46) where c 0 is an integral constant This equation is same as Eq (25) thus we only list the final results as follows: Case 1 u 8 = d dt gy + f (m2 2)h + hei(sn(ωm)m) u 9 = d dt gy + f (m2 5)h r hcn(ωm)dn(ωm) ]ω + 12k 3 h sn(ω m) r 12k 3 h ]ω (47) + hcn(ωm)dn(ωm) + 2hEi(sn(ωm)m) sn(ω m) (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

12 Article no 114 Zheng-Yi Ma Hui-Lin Wu Quan-Yong Zhu 12 with ω = h(x + k 3 y g) c 0 = 2 3 k 3(m 4 m 2 + 1)h 4 + r2 and c 0 = 2 3 k 3(m m 2 + 1)h 4 + r2 respectively Case 2 u 10 = d dt gy + f (m2 2)h + hei(sn(ωm)m) u 11 = d dt gy + f (4m2 5)h + 2hEi(sn(ωm)m) r hsn(ωm)dn(ωm) ]ω 12k 3 h cn(ω m) r hsn(ωm)dn(ωm) ]ω 12k 3 h cn(ω m) (48) with ω = h(x+k 3 y g) c 0 = 2 3 k 3(m 4 m 2 +1)h 4 + r2 and c 0 = 2 3 k 3(16m 4 16m 2 + 1)h 4 + r2 respectively Case 3 u 12 = d dt gy + f (m2 2)h + hei(sn(ωm)m) u 13 = d dt gy + f (m2 2)h + 2hEi(sn(ωm)m) r 12k 3 h ]ω hm2 sn(ωm)cn(ωm) dn(ω m) r 12k 3 h ]ω hm2 sn(ωm)cn(ωm) dn(ω m) (49) with ω = h(x + k 3 y g) c 0 = 2 3 k 3(m 4 m 2 + 1)h 4 + r2 and c 0 = 2 3 k 3(m 4 16m )h 4 + r2 respectively In addition with the help of Theorem 31 one can obtain more new types of exact solutions for the (2+1)-dimensional dissipative AKNS equation from the solutions (26)-(43) (45) and (47)-(49) However we do not list them here 6 SUMMARY AND DISCUSSIONS In summary we have carried out a detailed Lie symmetry analysis of the (2+1)- dimensional dissipative AKNS equation and we have investigated the algebraic structure of the symmetry groups Meanwhile we have applied a simple direct method to derive the general finite transformation groups of the dissipative AKNS equation which is equivalent to Lie point symmetry groups generated via the standard approaches Based on general Lie point symmetry we have derived the corresponding similarity reduction by solving the characteristic equations and have obtained some exact solutions of the (2+1)-dimensional dissipative AKNS equation The other (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c

13 13 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no 114 important properties such as the Hamiltonian structure the conservation laws and the generalized (nonlocal) symmetry of the the (2+1)-dimensional dissipative AKNS equation deserves a further study Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No ) the Natural Science Foundation of Zhejiang Province (Grant No LY14A010005) and the Applied Nonlinear Science and Technology from the Most Important Among all the Top Priority Disciplines of Zhejiang Province REFERENCES 1 PJ Olver Application of Lie Group to Differential Equation Springer New York GW Bluman and SC Anco Symmetry and Integration Methods for Differential Equations Springer New York GW Bluman and JD Cole J Math Mech (1969) 4 PJ Olver and P Rosenau Phys Lett A (1986) 5 PA Clarkson and MD Kruskal J Math Phys (1989) 6 SY Lou and HC Ma J Phys A: Math Gen 38 L129 L137 (2005) 7 LV Ovsiannikov Group Analysis of Differential Equations Academic Press New York B Li WC Ye and Y Chen J Math Phys (2008) 9 Y Chen and ZZ Dong Nonlinear Anal (2009) 10 XR Hu and Y Chen Appl Math Comput (2009) 11 E Buhe G Wang and X Bai Rom J Phys (2015) 12 G Wang AH Kara E Buhe and K Fakhar Rom J Phys (2015) 13 S Jamal and AH Kara Rom J Phys (2015) 14 E Buhe K Fakhar G Wang and T Chaolu Rom Rep Phys (2015) 15 R Kumar RK Gupta and SS Bhatia Rom J Phys (2015) 16 R Constantinescu Rom J Phys (2016) 17 R Cimpoiasu Rom J Phys (2014) 18 MS Hashemi S Abbasbandy MS Alhuthali and HH Alsulami Rom J Phys (2015) 19 A Biswas AH Kara L Moraru AH Bokhari and FD Zaman Proc Romanian Acad A (2014) 20 P A Clarkson and E L Mansfield Nonlinearity (1994) 21 N Liu and XQ Liu Chin Phys Lett (2012) 22 AM Wazwaz Appl Math Comput (2011) 23 ZL Cheng and XH Hao Appl Math Comput (2014) 24 N Liu and XQ Liu Chin Phys Lett (2012) 25 T Özer Chaos Solitons and Fractals (2009) (c) RJP 62(Nos 5-6) id:114-1 (2017) v20*201771#662d169c