Nonlocal symmetries and explicit solutions of the AKNS system

Accepted Manuscript Nonlocal symmetries and explicit solutions of the AKNS system Qian Miao, Xiangpeng Xin, Yong Chen PII: DOI: Reference:

S0893-9659(13)00265-6 http://dx.doi.org/10.1016/j.aml.2013.09.002 AML 4436

To appear in:

Applied Mathematics Letters

Received date: 8 July 2013 Revised date: 7 September 2013 Accepted date: 7 September 2013 Please cite this article as: Q. Miao, X. Xin, Y. Chen, Nonlocal symmetries and explicit solutions of the AKNS system, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.09.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript Click here to view linked References

Nonlocal symmetries and explicit solutions of the AKNS system Qian Miao a , Xiangpeng Xin a , Yong Chen a

a,∗

Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, People’s Republic of China

Abstract In this paper, based on the lax pair of the Ablowitz-Kaup-Newell-Segur(AKNS) system, the nonlocal symmetry is obtained and successfully localized to a Lie point symmetry by introducing an appropriate auxiliary dependent variable. For the closed prolongation, the construction for one-dimensional optimal system is presented in detail. Furthermore, using the obtained optimal system, we give out the reductions and the explicit analytic interaction solutions between cnoidal waves and solitary waves. For some interesting solutions, the figures are given out to show their properties. Keywords: Nonlocal symmetry, Auxiliary system, Optimal system, Lie point symmetry.

1. Introduction Lie group method[1–10] is one of the most important methods for constructing exact solutions of nonlinear partial differential equations (PDEs). In addition to the classical and non-classical Lie symmetries, there exists socalled nonlocal symmetries[11–18] which enlarge the class of symmetries and connected with integrable models. However, it is difficult to find the nonlocal symmetries of nonlinear PDEs. Recently, Lou et al.[19, 20] obtain nonlocal symmetries which are related to the Darboux transformation(DT) and give explicit analytic solutions through the localization procedure. Different from above method, we can obtain the nonlocal symmetries through a direct assumption method which can give both local and nonlocal symmetries. For the full symmetry group that leaves the PDE invariant, there is no need to list all possible group-invariant solutions. One can minimize these solutions to find nonequivalent branches, which leads to the concept of the optimal systems[21–25]. In this paper, we obtain the nonlocal symmetry by using a direct method and localize it to a Lie point symmetry by introducing an appropriate auxiliary dependent variable. For the prolonged system, the one-dimensional optimal system is presented. Based on the optimal system, some reductions and explicit solutions of AKNS system are derived. This paper is arranged as follows: In Sec.2, the nonlocal symmetries of the AKNS system are obtained by using the Lax pair. In Sec.3, we transform the nonlocal symmetries into Lie point symmetries. Then, the finite symmetry transformations are obtained by solving the initial value problem. In Sec.4, an optimal system is constructed to classify the group-invariant solutions of AKNS system. In Sec.5, based on the optimal system, some symmetry reductions and explicit solutions of the AKNS system are given out by using the Lie point symmetry of extending system. Finally, some conclusions and discussions are given in Sec.6. 2. Nonlocal symmetries of AKNS system The well-known AKNS system[14] reads ut = 21 iu xx + iu2 v, vt = − 21 iv xx − iv2 u.

(1)

∗ Corresponding author. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, People’s Republic of China Email address: [email protected] ( Yong Chen )

Preprint submitted to Elsevier

September 7, 2013

The Lax pair of Eq.(1) has the form ! p = q !x p = q t

! ! 0 v p , −u 0 q ! 1 1 − 2 iuv − 2 iv x − 21 iu x 21 iuv

p q

!

(2) .

To seek for the nonlocal symmetries, we adopt a method which can obtain directly the nonlocal symmetries. Practice shows that this method can obtain not only the nonlocal symmetries but also the general Lie point symmetries of the given equations. First of all, the symmetries σ1 , σ2 of the AKNS system are defined as solutions of their linearized equations σ1t − 21 iσ1xx − iu2 σ2 − 2iuvσ1 = 0, σ2t + 12 iσ2xx + 2iuvσ2 + iv2 σ1 = 0,

(3)

which means equation (1) is form invariant under the infinitesimal transformations u → u + ǫσ1 ,

v → v + ǫσ2 ,

(4)

with the infinitesimal parameter ǫ. The symmetry can be written as σ1 = X(x, t, u, v, p, q)u x + T (x, t, u, v, p, q)ut − U(x, t, u, v, p, q), σ2 = X(x, t, u, v, p, q)v x + T (x, t, u, v, p, q)vt − V(x, t, u, v, p, q).

(5)

Substituting Eq.(5) into Eq.(3) and eliminating ut , vt , p x , pt , q x , qt in terms of the closed system, it yields a system of determining equations for the functions X, T, U, V, which can be solved by virtue of Maple to give X(x, t, u, v, p, q) = c3 x + c1 t + c2 , T (x, t, u, v, p, q) = 2c3 t + c4 , U(x, t, u, v, p, q) = (c1 ix − 2c3 − c6 )u − c6 q2 , V(x, t, u, v, p, q) = (−c1 ix + c6 )v + c5 p2 ,

(6)

where ci (i = 1, ..., 6) are six arbitrary constants and i2 = −1. In Ref.[26], Kazuhiro gave the nonlocal symmetry of AKNS hierarchy with implicit form and the symmetry algebra was isomorphic to a loop algebra. Here, we construct the nonlocal symmetry with explicit form using an algebra method. Compare the results with Ref.[14], we not only give the nonlocal symmetries but also the Lie point symmetries. 3. Localization of the nonlocal symmetry As we all know, the nonlocal symmetries cannot be used to construct directly explicit solutions of differential equations(DEs). Hence, we need to transform the nonlocal symmetries into local ones[19, 20]. In this section, we will find a related system which possesses a Lie point symmetry that is equivalent to the nonlocal symmetry. For simplicity, we let c1 = c2 = c3 = c4 = 0, c5 = c6 = 1 in formula (6), i.e., σ1 = q2 ,

σ2 = −p2 .

(7)

To localize the nonlocal symmetry (7), we have to solve the following linearized equations σ3x − vσ4 − σ2 q = 0,

σ4x + uσ3 + σ1 p = 0,

(8)

with σ1 , σ2 given by (7). It is not difficult to verify that the solutions of (8) have the following forms σ3 = p f,

σ4 = q f,

(9)

where f is given by f x = −pq,

ft =

1 i(up2 + vq2 ). 2

(10)

It is easy to obtain the following result σ5 = σ f = f 2 . 2

(11)

The results (9) show us that the nonlocal symmetry (7) in the original space x, t, u, v has been successfully localized to a Lie point symmetry in the enlarged space x, t, u, v, p, q, f . Another interesting point one can see is that the introduced auxiliary dependent variable f just satisfies the Schwartzian form of the AKNS system ! ! 3φ2t 1 φt − {φ; x} , = (12) φx t 2φ2x 4 x where {φ; x} = (p xxx /p x ) − 3/2(p xx/p x )2 is the Schwartzian derivative. After succeeding in making the nonlocal symmetry(7) equivalent to Lie point symmetry (10) of the related prolonged system, we can construct the explicit solutions naturally by Lie group theory. With the Lie point symmetry(10), by solving the following initial value problem d u¯ dε d q¯ dε

= q2 , u¯ |ε=0 = u; = q f, q¯ |ε=0 = q;

d¯v 2 dε = −p , d f¯ 2 dε = f ,

v¯ |ε=0 = v; f¯ |ε=0 = f,

d p¯ dε

= p f, p¯ |ε=0 = p;

(13)

the finite symmetry transformation can be calculated as −εq2 +εu f −u , ε f −1

u¯ =

v¯ =

εp2 +εv f −v ε f −1 ,

p¯ = − ε fp−1 ,

q¯ = − ε fq−1 ,

f¯ = − ε ff−1 .

(14)

For a given solution u, v of Eq.(1), above finite symmetry transformation will arrive at another solution u¯ , v¯ . To search for more similarity reductions of Eq.(1), we study Lie point symmetries of the whole prolonged equation system instead of the single Eq.(1) and assume that the vector of the symmetries has the form V=X

∂ ∂ ∂ ∂ ∂ ∂ ∂ +T +U +V +P +Q +F , ∂x ∂t ∂u ∂v ∂p ∂q ∂f

(15)

where X, T, U, P, Q, F are the functions with respect to x, t, u, p, q, f , which means that the closed system is invariant under the infinitesimal transformations (x, t, u, p, q, f ) → (x + ǫX, t + ǫT, u + ǫU, p + ǫP, q + ǫQ, f + ǫF), with

σ1 σ2 σ3 σ4 σ5

= = = = =

X(x, t, u, v, p, q, X(x, t, u, v, p, q, X(x, t, u, v, p, q, X(x, t, u, v, p, q, X(x, t, u, v, p, q,

f )u x + T (x, t, u, v, p, q, f )ut − U(x, t, u, v, p, q, f ), f )v x + T (x, t, u, v, p, q, f )vt − V(x, t, u, v, p, q, f ), f )p x + T (x, t, u, v, p, q, f )pt − P(x, t, u, v, p, q, f ), f )q x + T (x, t, u, v, p, q, f )qt − Q(x, t, u, v, p, q, f ), f ) f x + T (x, t, u, v, p, q, f ) ft − F(x, t, u, v, p, q, f ).

(16)

And σ1 , σ2 , σ3 , σ4 , σ5 satisfy the following equations σ1t − 12 iσ1xx − iu2 σ2 − 2iuvσ1 = 0, σ2t + 21 iσ2xx + 2iuvσ2 + iv2 σ1 = 0, σ3x − vσ4 − qσ2 = 0, σ3t + 12 iuvσ3 + 12 iupσ2 + 12 ivpσ1 + 21 iv x σ4 + 21 iqσ2x = 0, σ4x + uσ3 + pσ1 = 0, σ4t + 12 iu x σ3 + 12 ipσ1x − 21 iuvσ4 − 12 iuqσ2 − 21 iqvσ1 = 0, σ5x + pσ4 + qσ3 = 0, σ5t − 12 iq2 σ2 − iqvσ4 − iupσ3 − 21 ip2 σ1 = 0.

(17)

Substituting Eq.(16) into symmetry equations (17) and eliminating ut , vt , p x , pt , q x , qt , f x , ft in terms of the closed system, we arrive at a system of determining equations for the functions X, T, U, V, P, Q, and F, which can be solved by using Maple to give X(x, t, u, v, p, q, f ) = c21 x + c3 , T (x, t, u, v, p, q, f ) = c1 t + c2 , U(x, t, u, v, p, q, f ) = c5 u + c4 q2 , V(x, t, u, v, p, q, f ) = (−c1 − c6 )v − c4 p2 , P(x, t, u, v, p, q, f ) = − 2p (c5 + c1 − c6 − 2c4 f ), Q(x, t, u, v, p, q, f ) = 2q (c5 + c6 + 2c4 f ), F(x, t, u, v, p, q, f ) = c4 f 2 + c6 f + c7 ,

(18)

where ci , i = 1, 2, ..., 6 are arbitrary constants. In general, to each s-parameter subgroup of the full symmetry group, there will correspond a family of groupinvariant solutions. Because there are always an infinite number of subgroups, there is no need to list possible group-invariant solutions to the system. Different to the Ref.[27], we will construct an optimal system to classify the group-invariant solutions of Eq.(1)in the next section. 3

4. Optimal system of the prolonged system As it is said in Ref.[2], the problem of finding an optimal system of subgroups is equivalent to finding an optimal system of subalgebras. In this section, we will construct the optimal system of one-dimensional subalgebras of Eq.(1) by using the method presented in Refs.[2, 3]. From Eqs.(18), the associated vector fields for the oneparameter Lie group of infinitesimal transformations are seven generators given by ∂ ∂ ∂ ∂ ∂ + 21 q ∂q + f ∂∂f , v2 = 21 x ∂x + t ∂t∂ − v ∂v − 12 p ∂p , v1 = 21 p ∂p ∂ ∂ ∂ 2 ∂ 2 ∂ 2 ∂ v3 = q ∂u − p ∂v + p f ∂p + q f ∂q + f ∂ f , v4 = ∂ f , v5 = ∂ ∂ ∂ ∂ − v ∂v − 21 p ∂p + 12 q ∂q . v7 = u ∂u

∂ ∂t ,

v6 =

∂ ∂x ,

(19)

For simplicity, we omit the commutator table and the adjoint representation here. One can know that v7 is the center of the group through calculating, so we don’t have to consider it. Following Ref.[2], two subalgebras v2 and v1 of a given Lie algebra are equivalent if one can find an element g in the Lie group so that Adg(v1 ) = v2 where Adg is the adjoint representation of g on v. Given a nonzero vector, for example, V = a 1 v1 + a 2 v2 + a 3 v3 + a 4 v4 + a 5 v5 + a 6 v6 , where a j , j = 1, 2, ...6 are arbitrary constants. The key task is to simplify as many of the coefficients ai as possible though judicious applications of adjoint maps to v. In this way, one can get the following results with massive complex computations in Table 1 where α1 , α2 , α3 are arbitrary constants. Table 1: Optimal Systems

Cases

Optimal systems a3 = a4 = a5 = a6 = 0, a1 , 0, a2 , 0, a2 = a3 = a4 = a5 = 0, a1 , 0, a1 = a2 = 0, a3 , 0, a4 , 0 , a1 = a3 = a4 = 0, a2 , 0 , a1 = a3 = a4 = a5 = a6 = 0, a2 , 0 , a1 = a2 = a3 = 0, a4 , 0, a3 = a4 = 0,

(a1) (a2) (b1) (c1) (c2) (d1) (e1)

v1 + a 2 v2 v1 + α1 v5 + α2 v6 v3 + a4 v4 + α1 v5 + α2 v6 v5 or v6 v2 v4 + α1 v5 + α2 v6 α1 v5 + α2 v6

5. Symmetry reduction of the AKNS system In this section, we take the case (b1) as example, other cases can be solved in the same way. In this case, without loss of generality, we let α2 = α1 k, a4 = −α3 , V = α1

∂ ∂ ∂ ∂ ∂ ∂ ∂ + kα1 + q2 − p2 + p f + qf + ( f 2 − α3 ) . ∂t ∂x ∂u ∂v ∂p ∂q ∂f

(20)

By solving the following characteristic equation dt du dv d p dq df dt = = 2 = 2 = = = 2 , α1 α1 k pf qf q p f − α3

(21)

we have q q √ √ √ √ −α3 (t+F(ξ)) −α3 (t+F(ξ)) −α3 (t+F(ξ)) tanh( tanh( f = − −α3 tanh( ), p = P(ξ) ) − 1 )+1 a1 a1 a1 q q √ √ √ −α3 (t+F(ξ)) −α3 (t+F(ξ)) −α3 (t+F(ξ)) −Q2 (ξ) q = Q(ξ) tanh( ) − 1 tanh( ) + 1 u = √−α3 tanh( ) + U(ξ), a1 a1 a1 v=

P2 (ξ) √ −α3

tanh(

√

−α3 (t+F(ξ)) ) a1

(22)

+ V(ξ),

where ξ = x − kt. Substituting Eqs.(22) into the prolonged system yields P(ξ) = e

− 21

Rξ

ξ0

−2i+2ikFξ −Fξξ Fξ

dξ

, Q(ξ) =

−α3 Fξ a1 P(ξ) ,

U(ξ) =

−α3 (−2i+2ikFξ +Fξξ ) , 2a1 P2 (ξ)

4

V(ξ) =

a1 P2 (ξ)(−2i+2ikFξ −Fξξ ) , −2α3 Fξ2

(23)

where F(ξ) satisfies the following equation 3 a21 Fξ2 Fξξξξ − 4a21 Fξξξ Fξξ Fξ − 8ka21 Fξξ Fξ + 12a21 Fξξ + 4α3 Fξξ Fξ4 + 3a21 Fξξ = 0.

(24)

One can simplify the Eq.(24) by replacing Fξ with W(ξ), and the reduction equation is a21 W 2 Wξξξ − 4a21 Wξξ Wξξ W − 8ka21 Wξ W + 12a21 Wξ + 4α3 Wξ W 4 + 3a21 Wξ3 = 0.

(25)

The Eq.(25) can be solved in terms of solutions of equation Wξ2 =

1 (−4α3 W 4 − 2a21 W 3 + 2a21 W 2 + 8ka21 W + 4a21 ). a21

(26)

After summarizing the above formulas, we get the explicit solution of the AKNS system: u= v=

R ξ −2i+2ikFξ −Fξξ dξ Fξ

−α23 Fξ2 e ξ0 e

−

√ a21 −α3 R ξ −2i+2ikFξ −Fξξ dξ Fξ ξ0 √

−α3

tanh(

tanh(

√ −α3 (t+F(ξ)) ) a1

√ −α3 (t+F(ξ)) ) a1

+

−

a1 e

−

R ξ −2i+2ikFξ −Fξξ dξ Fξ

α3 (−2i+2ikFξ +Fξξ )e ξ0 2a1

R ξ −2i+2ikFξ −Fξξ dξ Fξ ξ0 (−2i+2ikF

−2α3 Fξ2

ξ −Fξξ )

,

(27)

,

where Fξ = W, and W satisfies Eq.(26). Remark 1: We know that the general solution of Eq. (26) can be written out in terms of Jacobi elliptic functions. Hence, the solution expressed by Eq. (27) is just the explicit exact interaction between the soliton and the cnoidal periodic wave. To show more clearly this kind of solution, we offer a special case of Eq. (27) by solving Eq. (26). A simple solution of Eq. (26) is given as W = b0 + b1 sn(l1 ξ, m) + b2 sn2 (l1 ξ, m),

(28)

substituting Eq. (28) into Eq. (26) or Eq. (25) yields a1 = C1 , b0 = C2 , b1 = C3 , m = C4 , b2 = 0, α3 =

a21 m2 , m2 b40 −b20 b21 −b20 b21 m2 +b41

k=

b20 (2b20 m2 −b21 −b21 m2 ) , m2 b40 −b20 b21 −b20 b21 m2 +b41

l= √

2b1 −m2 b40 +b20 b21 +b20 b21 m2 −b41

(29)

with C1 , C2 and C3 being three arbitrary constants and 0 < C4 < 1. Here sn, cn, and dn are usual Jacobian elliptic functions with modulus m. Substituting Eqs.(28),(29) and Fξ = W into Eq.(27), one can obtain the solutions of u, v. In order to study the properties of these solutions of AKNS system, we give some pictures of u, v as shown below –0.2

–0.2

–0.15 –0.2 uv–0.25 –0.3 –0.35

u(0,t) –0.3

u(x,0) –0.3

–10

(a)

0 x

10

20

–60 –40

–20

0 t

20

40

60

20

–60 –40 –20 0 20 t 40

(b)

–10 60

10 0 x

–20

(c)

Figure 1: Interaction solutions to the AKNS system

In Fig. 1, we plot the interaction solutions between solitary waves and cnoidal waves expressed by (27) with parameters C1 = 2, C2 = 0.2, C3 = 1, C4 = 0.2. (a) Time-sliced view at t = 0; (b) space-sliced view at x = 0; (c) the corresponding two-dimensional image. We can see that the component uv exhibits a soliton propagates on a cnoidal wave background. In fact, it is of interest to study these types of solutions, for example, in describing localized states in optically refractive index gratings. In the ocean, there are some typical nonlinear waves such as the solitary waves and the cnoidal periodic waves. Remark 2: If setting the module m degenerates to 1, the soliton + cnoidal wave solution degenerate to soliton solution which the amplification of the amplitude has been experimentally observed and has practical applications in maritime security and coastal engineering. Some other types of solutions can be given by using the Jacobian elliptic equation, in order to save space, we don’t do a detailed discussion. 5

6. Summary and Discussion In summary, the nonlocal symmetry of AKNS system is obtained by using the lax pair and successfully localized. On the basis of prolonged system, one-dimensional subalgebras of a Lie algebra have been classified and the reductions of AKNS system are given out by using the associated vector fields. Using the reduction of AKNS system, we successfully obtain the moving direction of a soliton on a cnoidal background wave which can be applicable to explain some physical processes. This method can be applied to other interesting integrable models. Moreover, infinitely many nonlocal symmetries and conservation laws can be constructed using the seed symmetry. From local and nonlocal conservation laws, one can seek new integrable systems. Above topics will be discussed in the future series research works. Acknowledgment We would like to express our sincere thanks to Professor S Y Lou and other members of our discussion group for their valuable comments. This work is supported by the National Natural Science Foundation of China (Nos. 11275072, 61021004 and 11075055), Research Fund for the Doctoral Program of Higher Education of China (No. 20120076110024), National High Technology Research and Development Program (No. 2011AA010101) and Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213). References ¨ [1] S. Lie, Uber die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differential gleichungen, Arch. Math. 6 328 (1881). [2] L.V. Ovsiannikov, Group Analysis of Differential Equations (New York: Academic, 1982). [3] P.J. Olver, Applications of Lie Groups to Differential Equations (Berlin: Springer, 1986). [4] N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics (Boston, MA: Reidel, 1985). [5] G.W. Bluman and S. Kumei, Symmetries and Differential Equations (Springer-Verlag, New York 1989). [6] Z.Z. Dong, F. Huang and Y. Chen, Symmetry reductions and exact solutions of the two-Layer model in atmosphere, Z.Naturforsch. 66a(2011) 75-86. [7] X.R. Hu, F. Huang and Y. Chen, Symmetry reductions and exact solutions of the (2+1)-dimensional navier-stokes equations, Z. Naturforsch. 65a(2010) 1-7. [8] A.G.Johnpillai, A.H.Kara and Anjan Biswas, Symmetry reduction,exact group-invariant solutions and conservation laws of the BenjaminBona-Mahoney equation, Appl. Math. Lett. 26(2013) 376-381. [9] A.H. Bokhari, A.H.Kara and F.D.Zaman, A note on a symmetry analysis and exact solutions of a nonlinear fin equation, Appl. Math. Lett. 19(2006) 1356-1360. [10] X.R. Hu, Y. Chen, B¨acklund transformations and explicit solutions of the (2+1)-dimensional barotropic and quasi-geostrophic potential vorticity equation, Commun. Theor. Phys. 53 (2010) 803-808. [11] G.W. Bluman, A.F. Cheviakov and S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations (Springer New York, 2010). [12] G.W. Bluman, G.J. Reid, S.Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806. [13] G.W. Bluman, Temuerchaolu, and R. Sahadevan, Local and nonlocal symmetries for nonlinear telegraph equation, J. Math. Phys. 46(2005) 023505 1-12. [14] F. Galas, New non-local symmetries with pseudopotentials, J. Phys. A: Math. Gen. 25(1992) L981-L986. [15] I.S. Akhatov, R.K. Gazizov and N.K. Ibragimov, Nonlocal symmetries. Heuristic approach, J. Sov. Math. 55(1991) 1401-1450. [16] S.Y. Lou, Conformal invariance and integrable models, J. Phys. A: Math. Phys. 30(1997) 4803-4813. [17] S.Y. Lou and X.B. Hu, Non-local symmetries via Darboux transformations, J. Phys. A: Math. Gen. 30(1997) L95-L100. [18] G.A. Guthrie, M.S. Hickman, Nonlocal symmetries of the KdV equation, J. Math. Phys. 34(1993) 193-205. [19] S.Y. Lou, X.R. Hu and Y. Chen, Nonlocal symmetries related to B¨acklund transformation and their applications, J. Phys. A: Math. Theor. 45(2012) 155209-155209. [20] X.R. Hu, S.Y. Lou and Y. Chen, Explicit solutions from eigenfunction symmetry of the Korteweg-deVries equation, Phys. Rev. E 85(2012) 056607 1-8. [21] J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Math. Phys. 17(1976) 986-994. [22] D. David, N. Kamran, D. Levi, and P. Winternitz, J. Math. Phys. 27(1986) 1225-1237. [23] J. Patera and P. Winternitz, Subalgebras of real three- and four-dimensional Lie algebras, J. Math. Phys. 18(1977) 1449-1455. [24] O.O. Vaneeva, A.G. Johnpillai, R.O. Popovych and C. Sophocleous, Group analysis of nonlinear fin equations, Appl. Math. Lett. 21(2008) 248-253. [25] M. Pakdemirli ,A.Z.Sahin, Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. 19(2006) 378-384. [26] K. Kazuhiro, Symmetries of the Ablowitz-Kaup-Newell-Segur hierarchy, J. Math. Phys. 35(1994) 284-293. [27] S.Y. Lou, X.P. Cheng, C.L. Chen and X.Y. Tang,Interactions Between Solitons and Other Nonlinear Schr¨odinger Waves, arXiv:1208.5314v2.

6