Lie Symmetry, Full Symmetry Group, and Exact Solutions to the (2+1)-Dimensional Dissipative Akns Equation

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

ZHENG-YI MA1,2,∗, HUI-LIN WU1, QUAN-YONG ZHU1 1Department of Mathematics, Lishui University, Lishui 323000, P.R. China 2Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. China E-mail∗: [email protected] (corresponding author) Received January 13, 2017

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: 02.30Jr, 02.20.Sv, 05.45.Pq, 02.20.Tw.

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 62, 114 (2017) v.2.0*2017.7.1#662d169c 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]

4uxt + uxxxy + 8uxyux + 4uyuxx + ruxx = 0, (1) where r is a constant. Especially, the coefficient r 6= 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 Backlund¨ 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) (i.e., ∆=r.h.s. of (1) = 0), we consider a one-parameter Lie group of inﬁnitesimal (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c 3 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no. 114 transformation in (x,y,t,u) given by

x → x + X(x,y,t,u), y → y + Y (x,y,t,u), (2) t → t + T (x,y,t,u), u → u + U(x,y,t,u), where is the group parameter. The associated Lie algebra of inﬁnitesimal symmetries is the set of vector ﬁelds of the form

∂ ∂ ∂ ∂ V = X + Y + T + U . (3) ¯ ∂x ∂y ∂t ∂u

Then the invariance of Eq. (1) under the transformation (2) provides that

(4) pr V(∆) |∆=0= 0, (4) where pr(4)V is the fourth prolongation of the vector field (3). This yields an overde- termined, 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:

1 X = (c t + 2c − 2c )x + g, 4 6 4 5 1 Y = (c6y + 2c3)t + c5y + c2, 2 (5) 1 T = c2t2 + c t + c , 2 6 4 1 1 r 1 U = (c y + 2c )x + [g ˙ − (3c t + 2c + 2c )]y + f − (c t + 2c + 2c )u, 4 6 3 16 6 4 5 4 6 4 5 where f and g are arbitrary functions of t, ci (i = 1,2,...,6) 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

V = c1V + c2V + c3V + c4V + c5V + c6V + V (f) + V (g), (6) ¯ ¯ 1 ¯ 2 ¯ 3 ¯ 4 ¯ 5 ¯ 6 ¯ 7 ¯ 8 (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c Article no. 114 Zheng-Yi Ma, Hui-Lin Wu, Quan-Yong Zhu 4 where ∂ ∂ V = , V = , ¯ 1 ∂t ¯ 2 ∂y ∂ x ∂ V = t + , ¯ 3 ∂y 2 ∂u x ∂ ∂ 1 ∂ V = + t − (ry + 4u) , ¯ 4 2 ∂x ∂t 8 ∂u x ∂ ∂ 1 ∂ V = − + y − (ry − 4u) , (7) ¯ 5 2 ∂x ∂y 8 ∂u xt ∂ yt ∂ t2 ∂ 1 ∂ V = + + − (3ryt − 4xy + 4tu) , ¯ 6 4 ∂x 2 ∂y 2 ∂t 16 ∂u ∂ V (f) = f , ¯ 7 ∂u ∂ ∂ V (g) = g +gy ˙ , ¯ 8 ∂x ∂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 [V ,V ]. ¯ j ¯ k

Table 1

The commutation relation between inﬁnitesimal generators of point symmetries.

[Vj,Vk] V1 V2 V3 V4 V5 V6 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 V1 0 0 V2 V1 0 V4 + 2 V5 ¯ ¯ 1 ¯ 1 ¯ 3 ¯ V2 0 0 − 8 V7(r) V2 − 8 V7(r) V3 − 16 V7(rt) ¯ ¯1 ¯ 1 ¯ ¯ 3 ¯ 2 V3 0 −V3 − 8 V7(rt) V3 − 8 V7(rt) − 16 V7(rt ) V¯ ¯ 0¯ ¯ 0¯ V¯ ¯ 4 ¯ 6 V5 0 0 V¯ 0 ¯ 6 [V ,V ] V (f) V (g) ¯ j ¯ k ¯ 7 ¯ 8 V V (f˙) V (g ˙) ¯ 1 ¯ 7 ¯ 8 V2 0 V7(g ˙) ¯ ¯ 1 V3 0 V7(tg˙ − 2 g) ¯ 1 ¯ V4 V8(tg˙ − 2 g) V¯ ¯ − 1 V 1 V ¯ 5 2 ¯ 7 2 ¯ 8 V V ( 1 t2f˙ + 1 tf) V ( 1 t2g˙ − 1 tg) ¯ 6 ¯ 7 2 4 ¯ 8 2 4 V7(f) 0 0 V¯ (g) 0 ¯ 8 (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c 5 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no. 114

We now consider a point transformation G :(x,y,t,u) 7→ (χ,ζ,γ,P ). (8) From the transformation (2), we have the corresponding one-parameter group of symmetries of the (2+1)-dimensional dissipative AKNS equation

G1 :(x,y,t,u) 7→ (x,y,t + ,u),

G2 :(x,y,t,u) 7→ (x,y + ,t,u), 1 G :(x,y,t,u) 7→ (x,y + t,t, x + u), 3 2 − r − G :(x,y,t,u) 7→ (xe 2 ,y,te ,ue 2 + (e 2 − 1)y), 4 4 − r G :(x,y,t,u) 7→ (xe 2 ,ye ,t,ue 2 − (e − e 2 )y), 5 4 (9) i √ 2y 2t i √ i √ G :(x,y,t,u) 7→ 2t − 4x,− ,− , 2t − 4u + 2t − 4 6 2 t − 2 t − 2 2 4t 2 ri √ ln( )xy − [ 2t − 4(2 − t) + 4i]y , 2 − t 8t − 16 G7 :(x,y,t,u) 7→ (x,y,t,u + f),

G8 :(x,y,t,u) 7→ (x + g,y,t,u + gy˙ ). So the entire symmetry group is obtained by composing one-dimensional subgroups Gi (i = 1,2,...,8). When G is an element of this group, if u(x,y,t) is a solution of the dissipative AKNS equation, then P (χ,ζ,γ) is also a solution of the dissipative AKNS equation.

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(ξ,η,τ), (10) where α,β(i = 1,2,3),ξ,η and τ are functions of {x,y,t} to be determined. We re- quire that U(ξ,η,τ) also satisﬁes the (2+1)-dimensional dissipative AKNS equation but with different independent variables {ξ,η,τ}

4Uξτ + Uξξξη + 8UξηUξ + 4UηUξξ + rUξξ = 0. (11)

Substituting (10) into Eqs. (1), eliminating Uξτ 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) v.2.0*2017.7.1#662d169c Article no. 114 Zheng-Yi Ma, Hui-Lin Wu, Quan-Yong Zhu 6 calculations, the general results read:

1 2s5(s1t + s2) 2 r(s2s3 − s1s4) d 1 α = σ − 2 − ξ0(t) − σr y s1 4(s1t + s2) dt 4 s1(y + 2s7) + x + α0(t), 2(s1t + s2) 1 s 2 β = σ − 1 , 4s5(s1t + s2) (12) 1 s1 2 ξ = σ − x + ξ0(t), 4s5(s1t + s2) 2s5(s1s4 − s2s3) η = (y + 2s7) + s6, s1(s1t + s2) s t + s τ = 3 4 , σ2 = 1, s1t + s2 where ξ0 ≡ ξ0(t) and α0 ≡ α0(t) are arbitrary functions of t and si(i = 1,2,3,4,5,6,7) are arbitrary constants. It is necessary to point out that the independent variable t possess invariant property under the Mobious¨ (conformal) transformation. In summary, we can arrive at the following ﬁnal transformation group theorem of Eq. (1). Theorem 3.1 If U = U(x,y,t) is a solution of the (2+1)-dimensional dissipative AKNS equation then so is u

1 2s (s t + s ) 2 r(s s − s s ) d 1 u =σ − 5 1 2 2 3 1 4 − ξ (t) − σr y s 4(s t + s )2 dt 0 4 1 1 2 (13) 1 s1(y + 2s7) s1 2 + x + α0(t) + σ − U(ξ,η,τ), 2(s1t + s2) 4s5(s1t + s2) where ξ,η, and τ are determined by (12). In order to see the equivalence between the Lie point symmetry group obtained in Theorem 3.1 and the known one from classical Lie group method, we need to take the arbitrary functions ξ0(t),α0(t) and arbitrary constants si(i = 1,2,3,4,5,6,7) to be different forms with respect to an inﬁnitesimal parameter

2c3 σ = 1, s1 = − c6, s2 = 1 + (c5 − c4), s4 = c1, s5 = c6, s6 = − , 2 4 c6 c3 c2 c3c5 s7 = + − , ξ0(t) = g(t), α0(t) = −f(t), c6 2 c6 (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c 7 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no. 114 then (13) can be written as u = u + σ = u + σ(u), 1 1 σ(u) = (c t + 2c − 2c )x + g u + (c y + 2c )t + c y + c u 4 6 4 5 x 2 6 3 5 2 y 1 1 r + ( c2t2 + c t + c )u − (c y + 2c )x − [g ˙ − (3c t + 2c + 2c )]y 2 6 4 1 t 4 6 3 16 6 4 5 1 − f + (c t + 2c + 2c )u, 4 6 4 5 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 dy dt du = = = . X Y T U Here we only list the following cases. Case 1. For the generator V1 + a1V2, one has the following invariants ¯ ¯ y χ = x, γ = t − , u = P (χ,γ). (14) a1 Substituting Eq. (14) into Eq.( 1) yields the corresponding reduced equation

−Pχχχγ + a1Pχγ − 8Pχγpχ − 4PχχPγ + a1rPχχ = 0. (15)

Case 2. For the generator V +a2V +a3V , we have the following invariants ¯ 1 ¯ 2 ¯ 3 a 1 χ = x, γ = 3 t2 + a t − y, u = (a t + a )x + P (χ,γ). (16) 2 2 2 3 2 Substituting Eq. (16) into Eq. (1) leads to the corresponding reduced equation

−Pχχχη − 8Pχγpχ − 4PχχPγ + rPχχ + 2a3 = 0. (17) Case 3. For the generator V , the following invariants can be derived ¯ 4 t 1 χ = y, γ = , u = − ry + P (χ,γ). (18) x2 4 Substituting Eq. (18) into Eq. (1), then the reduced equation reads 3 2 − 8γ Pχγγγ − 48γ Pχγγ + 2(8P − 27)γPχγ − 8γPγγ 2 2 (19) + (32γ Pχγ + 56γPχ − 12)Pη + (16γ Pγγ + 16P − 6)Pχ = 0. Case 4. For the generator V , one can get the following invariants ¯ 5 1 χ = yx2, γ = t, u = − ry + P (χ,γ). (20) 4 (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c Article no. 114 Zheng-Yi Ma, Hui-Lin Wu, Quan-Yong Zhu 8

Substituting Eq. (20) into Eq. (1), it follows the reduced equation 3 2 2 8χ Pχχχχ + 24χ Pχχχ − 2χ(8P − 3)Pχχ8χPχγ + 48χ PχχPχ − 4Pγ = 0. (21) Case 5. For the generator V , the corresponding invariants are ¯ 6 y t 1 xy χ = , γ = , u = − ry + + P (χ,γ). (22) x2 x2 4 2t Substituting Eq. (22) into Eq. (1) yields the reduced equation 3 3 2 2 2 − 8χ Pχχχχ − 8γ Pχγγγ − 24χ γPχχχγ − 24χγ Pχχγγ − 72χ Pχχχ

+ 2γ(32χPχ + 16γPγ + 8P − 75)Pχγ + 2χ(24χPχ + 16γPγ + 8P − 75)Pχχ 2 2 2 + 88χPχ + (88γPγ + 32P − 60)Pχ + 16γ PχPγγ − 144χγPχχγ − 72γ Pχγγ = 0.

Case 6. For the generator V1 + V7(f) + V8(g), the corresponding invariants read ¯ ¯ ¯ d d d χ = x − g,˜ γ = y, u = gy˜ + f˜+ P (χ,γ), g = g,˜ f = f.˜ (23) dt dt dt Substituting Eq. (23) into Eq. (1), the initial equation becomes the reduced equation

Pχχχγ + 8PχγPχ + 4PχχPγ + rPχχ = 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 V1 + a1V2 reduction Here we are looking¯ for the¯ traveling wave solutions for the dissipative AKNS equation (1), therefore we consider a similarity variable θ = χ + k1γ, where k1 is a constant. The reduced equation becomes an ordinary differential equation 2 −k1φθθθ − 6k1φθ + a1(r + 4k1)φθ + c0 = 0, (25) where c0 is an integral constant. In fact, the ordinary differential equation that the function φθ 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

A hcn(ω,m)dn(ω,m) u = ω + + hEi(sn(ω,m),m), (26) 1 h sn(ω,m) (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c 9 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no. 114 with k (y − a t) 1 a (r + 4k ) ω = h x − 1 1 ,A = (m2 − 2)h2 + 1 1 , a1 3 12k1 2 2 (27) 2 4 2 4 a1(r + 4k1) c0 = k1(m − m + 1)h − . 3 24k1

A hcn(ω,m)dn(ω,m) u = ω + + 2hEi(sn(ω,m),m), (28) 2 h sn(ω,m) with k (y − a t) 1 a (r + 4k ) ω = h x − 1 1 ,A = (m2 − 5)h2 + 1 1 , a1 3 12k1 2 2 (29) 2 4 2 4 a1(r + 4k1) c0 = k1(m + 14m + 1)h − . 3 24k1 Especially, when m = 1, the elliptic functions degenerate to the hyperbolic functions, then the above solutions become k1(y − a1t) −1 k1(y − a1t) u˜1 = A x − + htanh h x − , a1 a1 2 2 2 (30) h a1(r + 4k1) 2 4 a1(r + 4k1) A = − + , c0 = k1h − , 3 12k1 3 24k1 and k1(y − a1t) −1 k1(y − a1t) u˜2 = A x − + 2htanh 2h x − , a1 a1 2 2 2 (31) 4h a1(r + 4k1) 32 4 a1(r + 4k1) A = − + , c0 = k1h − . 3 12k1 3 24k1 Case 2

A hsn(ω,m)dn(ω,m) u = ω + + hEi(sn(ω,m),m), (32) 3 h cn(ω,m) with k (y − a t) 1 a (r + 4k ) ω = h x − 1 1 ,A = (m2 − 2)h2 + 1 1 , a1 3 12k1 2 2 (33) 2 4 2 4 a1(r + 4k1) c0 = k1(m − m + 1)h − . 3 24k1

A hsn(ω,m)dn(ω,m) u = ω + + 2hEi(sn(ω,m),m), (34) 4 h cn(ω,m) (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c Article no. 114 Zheng-Yi Ma, Hui-Lin Wu, Quan-Yong Zhu 10 with k (y − a t) 1 a (r + 4k ) ω = h x − 1 1 ,A = (4m2 − 5)h2 + 1 1 , a1 3 12k1 2 2 (35) 2 4 2 4 a1(r + 4k1) c0 = k1(m + 14m + 1)h − . 3 24k1 Especially, when m = 1, the elliptic functions degenerate to the hyperbolic functions, then the above solutions become k1(y − a1t) u˜3 = A x − , a1 2 2 2 (36) h a1(r + 4k1) 2 4 a1(r + 4k1) A = − + , c0 = k1h − , 3 12k1 3 24k1 and k1(y − a1t) k1(y − a1t) u˜4 = A x − + htanh h x − , a1 a1 2 2 2 (37) h a1(r + 4k1) 2 4 a1(r + 4k1) A = − + , c0 = k1h − . 3 12k1 3 24k1 Case 3

A hm2sn(ω,m)cn(ω,m) u = ω + + hEi(sn(ω,m),m), (38) 5 h dn(ω,m) with k (y − a t) 1 a (r + 4k ) ω = h x − 1 1 ,A = (m2 − 2)h2 + 1 1 , a1 3 12k1 2 2 (39) 2 4 2 4 a1(r + 4k1) c0 = k1(m − m + 1)h − . 3 24k1

A hm2sn(ω,m)cn(ω,m) u = ω + + 2hEi(sn(ω,m),m), (40) 6 h dn(ω,m) with k (y − a t) 1 a (r + 4k ) ω = h x − 1 1 ,A = (m2 − 2)h2 + 1 1 , a1 3 12k1 2 2 (41) 2 4 2 4 a1(r + 4k1) c0 = k1(m − 16m + 16)h − . 3 24k1 Especially, when m = 1, the elliptic functions degenerate to the hyperbolic (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c 11 Symmetries and exact solutions to (2+1)-D dissipative AKNS equation Article no. 114 functions, then the above solutions become k1(y − a1t) u˜5 = A x − , a1 2 2 2 (42) h a1(r + 4k1) 2 4 a1(r + 4k1) A = − + , c0 = k1h − , 3 12k1 3 24k1 and k1(y − a1t) k1(y − a1t) u˜6 = A x − + htanh h x − , a1 a1 2 2 2 (43) h a1(r + 4k1) 2 4 a1(r + 4k1) A = − + , c0 = k1h − . 3 12k1 3 24k1

B. Solutions from V1 + a2V2 + a3V3 reduction For the reduced equation¯ (17),¯ it is¯ straightforward to verify that it has one special solution, 2 a3χ P (χ,γ) = + λ2χ + λ1γ + λ3, (44) 4λ1 − r where λi(i = 1,2,3) are arbitrary constants. Therefore, we can get the following exact solution of the dissipative AKNS equation, 2 1 a3x a3 2 u7 = (a3t + a2)x + + λ2x + λ1( t + a2t − y). (45) 2 4λ1 − r 2

C. Solutions from V1 + V7(f) + V8(g) reduction For the reduced equation¯ ¯ (24), we¯ still seek for traveling wave reduction with a similarity variable θ = χ + k3γ, where k3 is a constant. Then the reduced equation (24) becomes an ordinary differential equation 2 k3φθθθ + 6k3φθ + rφθ + c0 = 0, (46) where c0 is an integral constant. This equation is same as Eq. (25), thus we only list the ﬁnal results as follows: Case 1

d 1 2 r hcn(ω,m)dn(ω,m) u8 = gy˜ + f˜+ [ (m − 2)h − ]ω + dt 3 12k3h sn(ω,m) + hEi(sn(ω,m),m),

d 1 2 r (47) u9 = gy˜ + f˜+ [ (m − 5)h − ]ω dt 3 12k3h hcn(ω,m)dn(ω,m) + + 2hEi(sn(ω,m),m), sn(ω,m) (c) RJP 62(Nos. 5-6), id:114-1 (2017) v.2.0*2017.7.1#662d169c Article no. 114 Zheng-Yi Ma, Hui-Lin Wu, Quan-Yong Zhu 12

2 with ω = h(x+k y −g˜), c = − 2 k (m4 −m2 +1)h4 + r , and c = − 2 k (m4 + 3 0 3 3 24k1 0 3 3 2 14m2 + 1)h4 + r , respectively. 24k1 Case 2

d 1 2 r hsn(ω,m)dn(ω,m) u10 = gy˜ + f˜+ [ (m − 2)h − ]ω − dt 3 12k3h cn(ω,m) + hEi(sn(ω,m),m), (48) d 1 2 r hsn(ω,m)dn(ω,m) u11 = gy˜ + f˜+ [ (4m − 5)h − ]ω − dt 3 12k3h cn(ω,m) + 2hEi(sn(ω,m),m),

2 with ω = h(x+k y−g˜), c = − 2 k (m4 −m2 +1)h4 + r , and c = − 2 k (16m4 − 3 0 3 3 24k1 0 3 3 2 16m2 + 1)h4 + r , respectively. 24k1 Case 3

2 d 1 2 r hm sn(ω,m)cn(ω,m) u12 = gy˜ + f˜+ [ (m − 2)h − ]ω − dt 3 12k3h dn(ω,m) + hEi(sn(ω,m),m), 2 (49) d 1 2 r hm sn(ω,m)cn(ω,m) u13 = gy˜ + f˜+ [ (m − 2)h − ]ω − dt 3 12k3h dn(ω,m) + 2hEi(sn(ω,m),m),

2 with ω = h(x+k y −g˜), c = − 2 k (m4 −m2 +1)h4 + r , and c = − 2 k (m4 − 3 0 3 3 24k1 0 3 3 2 16m2 + 16)h4 + r , respectively. 24k1 In addition, with the help of Theorem 3.1, 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) v.2.0*2017.7.1#662d169c 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. 11447017), 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. P.J. Olver, Application of Lie Group to Differential Equation, Springer, New York, 1986. 2. G.W. Bluman and S.C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002. 3. G.W. Bluman and J.D. Cole, J. Math. Mech. 18, 1025–1042 (1969). 4. P.J. Olver and P. Rosenau, Phys. Lett. A 114, 107–112 (1986). 5. P.A. Clarkson and M.D. Kruskal, J. Math. Phys. 30, 2201–2213 (1989). 6. S.Y. Lou and H.C. Ma, J. Phys. A: Math. Gen. 38, L129–L137 (2005). 7. L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. 8. B. Li, W.C. Ye, and Y. Chen, J. Math. Phys. 49, 103503 (2008). 9. Y. Chen and Z.Z. Dong, Nonlinear Anal. 71, 810–817 (2009). 10. X.R. Hu and Y. Chen, Appl. Math. Comput. 215, 1141–1145 (2009). 11. E. Buhe, G. Wang, and X. Bai, Rom. J. Phys. 60, 1361–1373 (2015). 12. G. Wang, A.H. Kara, E. Buhe, and K. Fakhar, Rom. J. Phys. 60, 952–960 (2015). 13. S. Jamal and A.H. Kara, Rom. J. Phys. 60, 1328–1336 (2015). 14. E. Buhe, K. Fakhar, G. Wang, and T. Chaolu, Rom. Rep. Phys. 67, 329–339 (2015). 15. R. Kumar, R.K. Gupta, and S.S. Bhatia, Rom. J. Phys. 60, 15–21 (2015). 16. R. Constantinescu, Rom. J. Phys. 61, 77–88 (2016). 17. R. Cimpoiasu, Rom. J. Phys. 59, 617–624 (2014). 18. M.S. Hashemi, S. Abbasbandy, M.S. Alhuthali, and H.H. Alsulami, Rom. J. Phys. 60, 904–917 (2015). 19. A. Biswas, A.H. Kara, L. Moraru, A.H. Bokhari, and F.D. Zaman, Proc. Romanian Acad. A 15, 123–129 (2014). 20. P. A. Clarkson and E. L. Mansﬁeld, Nonlinearity 7, 915–1000 (1994). 21. N. Liu and X.Q. Liu, Chin. Phys. Lett. 29, 120201 (2012). 22. A.M. Wazwaz, Appl. Math. Comput. 217, 8840–8845 (2011). 23. Z.L. Cheng and X.H. Hao, Appl. Math. Comput. 234, 118–126 (2014). 24. N. Liu and X.Q. Liu, Chin. Phys. Lett. 29, 040202 (2012). 25. T. Ozer,¨ Chaos, Solitons and Fractals 42, 577–583 (2009).