Nonlocal symmetries, exact solutions and conservation laws of the coupled Hirota equations

Accepted Manuscript Nonlocal symmetries, exact solutions and conservation laws of the coupled Hirota equations Xiangpeng Xin, Yutang Liu, Xiqiang Liu PII: DOI: Reference:

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Received date: 1 September 2015 Revised date: 14 November 2015 Accepted date: 14 November 2015 Please cite this article as: X. Xin, Y. Liu, X. Liu, Nonlocal symmetries, exact solutions and conservation laws of the coupled Hirota equations, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.11.009 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.

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Nonlocal symmetries, Exact solutions and Conservation laws of the coupled Hirota equations Xiangpeng Xin ∗a , Yutang Liu a , Xiqiang Liu a a

School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, People’s Republic of China

Abstract Using the Lax pair, nonlocal symmetries of the coupled Hirota equations are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are successfully localized to Lie point symmetries. With the help of Lie symmetries of the closed prolongation, exact solutions and nonlocal conservation laws of the coupled Hirota equations are studied. Keywords: Nonlocal symmetry, Exact solution, Nonlocal conservation laws.

1. Introduction The Lie symmetries[1–3] and their various generalizations have become an important subject in mathematics and physics. One can reduce the dimensions of partial differential equations(PDEs) and proceed to construct analytical solutions by using classical or non-classical Lie symmetries. However, with all its importance and power, the traditional Lie approach does not provide all the answers to mounting challenges of the modern nonlinear physics. In the 80s of the last century, there exist so-called nonlocal symmetries which entered the literature largely through the work of Olver[4]. Compared with the local symmetries, little importance is attached to the existence and applications of the nonlocal ones. The reason lies in that nonlocal symmetries are difficult to find and similarity reductions cannot be directly calculated. Many researchers[5–7] have done a lot of work in this area, references[8– 12] give a direct way to solve this problem which so-called localization method of nonlocal symmetries. I.e. the original system is prolonged to a larger system such that the nonlocal symmetry of the original model becomes a local one of the prolonged system. When we get the Lie symmetries of prolonged system, there will correspond to a family of group-invariant solutions. These symmetry group techniques[13–15] provide one method for obtaining exact and special solutions of a given PDE in terms of solutions of lower dimensional equations, in particular, ordinary differential equations. Conservation laws are used for the development of appropriate numerical methods and for mathematical analysis, in particular, existence,uniqueness and stability analysis. It can lead to some new integrable systems via reciprocal transformation. The famous Noether’s theorem[16] provides a systematic way of determining conservation laws, for Euler-Lagrange differential equations, to each Noether symmetry associated with the Lagrangian there corresponds a conservation law which can be determined explicitly by a formula. But this theorem relies on the availability of classical Lagrangians. To find conservation laws of differential equations without classical Lagrangians, researchers have made various generalizations of Noether’s theorem. Steudel[17] writes a conservation law in characteristic form, where the characteristics are the multipliers of the differential equations. In order to determine a conservation law one has to also find the related characteristics. Anco and Bluman[18] provides formulae for finding conservation laws for known characteristics. Infinitely many nonlocal conservation laws for(1+1)-dimensional evolution equations are revealed by Lou[19]. Symmetry considerations for PDEs were incorporated by Ibragimov[20] which can be computed by a formula. This paper is arranged as follows: In Sec.2, the nonlocal symmetries of the coupled Hirota equations 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, exact groupinvariant solutions of the coupled Hirota equations are obtained. In Sec.5, based on the symmetries of prolonged system, nonlocal conservation laws of the coupled Hirota equations are given out. Finally, some conclusions and discussions are given in Sec.6. ∗ Corresponding

author. School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, People’s Republic of China Email address: [email protected] ( Xiangpeng Xin ∗ )

Preprint submitted to Applied Mathematics Letters

November 14, 2015

2. Nonlocal symmetries of the coupled Hirota equations The well-known the Hirota equation[21] reads iut + α(u xx + 2 |u|2 u) + iβ(u xxx + 6 |u|2 u x ) = 0,

(1)

α, β are real constants. Eq.(1) is the third flow of the nonlinear Schr¨odinger (NLS)hierarchy which can be used to describe many kinds of nonlinear phenomenas or mechanisms in the fields of physics,optical fibers,electric communication and other engineering sciences. Eq.(1)reduces to NLS equation when α = 1, β = 0. In this section, we shall consider the coupled Hirota equations iut + α(u xx − 2u2 v) + iβ(u xxx − 6uvu x) = 0,

ivt − α(v xx − 2v2 u) + iβ(v xxx − 6uvv x ) = 0,

(2)

Eqs.(2)are reduced to the Eq.(1)when u = −v∗ , and ∗ denotes the complex conjugate. The Lax pair of Eq.(2) has been obtained in[22] ! −iλ u Φ x = UΦ, U = (3) v iλ and

Φt = VΦ, V = with

a b c −a

!

(4)

a = −4βiλ3 − 2αiλ2 − 2βiuvλ − αiuv + β(vu x − uv x ), b = 4βuλ2 + (2βiu x + 2αu)λ + αiu x − β(u xx − 2u2 v), c = 4βvλ2 − (2βiv x − 2αv)λ − αiv x − β(v xx − 2v2 u).

(5)

σ1t − ασ1xx i + 4iασ1 uv + 2iαu2 σ2 + βσ1xxx − 6βσ1 vu x − 6βuσ2 u x − 6βuvσ1x = 0, σ2t + ασ2xx i − 4iαuvσ2 − 2iαv2 σ1 + βσ2xxx − 6βσ2 uv x − 6βuvσ2x − 6βvσ1 v x = 0,

(6)

where u and v are two potentials, the spectral parameter λ is an arbitrary complex constant and its eigenfunction is Φ = (φ, ψ)T . To seek for the nonlocal symmetries, we adopt a direct method[23]. First of all, the symmetries σ1 , σ2 of the coupled Hirota equations are defined as solutions of their linearized equations

which means equation (2) is form invariant under the infinitesimal transformations u → u + ǫσ1 ,

v → v + ǫσ2 ,

(7)

with the infinitesimal parameter ǫ. The symmetry can be written as, e x + Teut − U, e σ1 = Xu

e x + Tevt − V, e σ2 = Xv

(8)

eT e, U, e V e dependent on the variables (x, t, u, v, φ, ψ), so one may obtain some different results from Lie where X, symmetries. Substituting Eq.(8) into Eq.(6) and eliminating ut , vt , φ x , φt , ψ x , ψt , it yields a system of determining eT e, U, e V, e which can be solved by virtue of Maple to give equations for the functions X, 2 e = c1 x + 2α c1 t + c3 , X 3 9β

e = iαc1 ux + c5 u + c4 φ2 , Te = c1 t + c2 , U 9β

2 e = ((−6c1−9c5 )v + 9c4 ψ )β − iαvc1 x . (9) V 9β

where ci (i = 1, ..., 5) are five arbitrary constants and i2 = −1. It can be seen from the results(9), the results contain a nonlocal symmetry when c4 , 0. 3. Localization of the nonlocal symmetry As we all know, the nonlocal symmetries cannot be used to construct explicit solutions directly. Hence, one need to transform the nonlocal symmetries into local ones[8, 9]. In this section, a related system which possesses a Lie point symmetry that is equivalent to the nonlocal symmetry will be found. For simplicity, we let c1 = c2 = c3 = c5 = 0, c4 = −1 in formula (9), i.e., σ1 = φ2 , σ2 = ψ2 . 2

(10)

To localize the nonlocal symmetry (10), we have to solve the following linearized equations σ3x + iλσ3 − σ1 ψ − uσ4 = 0,

σ4x − σ2 φ − vσ3 − iλσ4 = 0,

(11)

which means that Eqs.(3) invariant under the infinitesimal transformations φ → φ + ǫσ3 ,

(12)

ψ → ψ + ǫσ4 ,

with σ1 , σ2 given by (10). It is not difficult to verify that the solutions of (11) have the following forms σ3 = φ f,

(13)

σ4 = ψ f,

where f is given by f x = φψ, ft = −βφ2 v x + 12βλ2 φψ + 4λαφψ − βψ2 u x + 2βφψuv + 4iλβψ2 u − 4iλβφ2 v + iαψ2 u − iαφ2 v. It is easy to obtain the following result

σ5 = σ f = f 2 .

(14) (15)

The results (13) and (15) show that the nonlocal symmetry (10) 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, φ, ψ, f } with the vector form V1 = φ2

∂ ∂ ∂ ∂ ∂ + ψ2 + φ f + ψf + f2 . ∂u ∂v ∂φ ∂ψ ∂f

(16)

After succeeding in making the nonlocal symmetry(10) equivalent to Lie point symmetry (16) of the related prolonged system, we can construct the exact solutions naturally by Lie group theory. With the Lie point symmetry(16), by solving the following initial value problem du = φ2 , u|ǫ=0 = u, dǫ

dv = ψ2 , v|ǫ=0 = v, dǫ

dφ dψ = φ f, φ|ǫ=0 = φ, = ψ f, ψ|ǫ=0 = ψ, dǫ dǫ

df = f 2 , f |ǫ=0 = f, (17) dǫ

the finite symmetry transformation can be calculated as u=

ǫ f u − εφ2 − u ǫ f v − ǫψ2 − v φ ψ f ,v = ,φ = ,ψ = ,f = , ǫf −1 ǫf −1 εf − 1 ǫf −1 ǫf −1

(18)

For a given solution u, v, φ, ψ, f of Eqs.(18), above finite symmetry transformation will arrive at another solution u¯ , v¯ . To search for more similarity reductions of Eqs.(2), one should consider Lie point symmetries of the whole prolonged system and assume the vector of the symmetries has the form ∂ ∂ ∂ ∂ ∂ ∂ ∂ V¯ = X +T +U +V +P +Q +F , ∂x ∂t ∂u ∂v ∂φ ∂ψ ∂f

(19)

which means that the closed system is invariant under the infinitesimal transformations (x, t, u, p, q, f ) → (x + ǫX, t + ǫT, u + ǫU, φ + ǫP, ψ + ǫQ, f + ǫF), with σ1 = Xu x + T ut − U, σ2 = Xv x + T vt − V, σ3 = Xφ x + T φt − P, σ4 = Xψ x + T ψt − Q, σ5 = X f x + T ft − F. (20) with σ1 , σ2 , σ3 , σ4 , σ5 satisfy the linearized equations of Eqs.(2,3,4,14),and X, T, U, V, P, Q, F dependent on the variables (x, t, u, v, φ, ψ, f ). Substituting Eq.(20) into linearized equations and eliminating ut , vt , φ x , φt , ψ x , ψt , f x , ft in terms of the closed system, one 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 = c4 , T = c3 , U = c2 φ2 +c1 u, V = c2 ψ2 −c1 v, P =

(−2c2 f + c1 − c5 )ψ (2c2 f + c1 + c5 )φ ,Q = − , F = c2 f 2 +c5 f +c6 , 2 2 (21)

where ci , i = 1, 2, ..., 6 are arbitrary constants. 3

4. Exact solutions of coupled Hirota equations In order to give more group invariant solutions, we would like to solve the symmetry constraint conditions, by setting σ1 , σ2 , σ3 , σ4 , σ5 be zeros in Eq.(20), which is equivalent to solving the characteristic equations dx dt du dv dφ dφ d f = = = = = = . X T U V P Q F

(22)

As an example, we will discuss the case c2 = c5 = 0, c1 = 2, c3 = 1, c4 = w, c6 = a0 in the following part, where w, a0 are arbitrary constants. By solving the characteristic equation(22) with above constraints, one can obtain the group invariant solution, f = F1 (ξ) − a0 t, φ = F2 (ξ)et , ψ = F3 (ξ)e−t , u = F4 (ξ)e2t , v = F5 (ξ)e−2t ,

(23)

with ξ = x − wt. Substituting Eqs.(23) into the prolonged system (2),(3),(4) and (14) yields, F3 (ξ) =

−iλF2 (ξ)F1′ (ξ) + F2 (ξ)F1′′ (ξ) − F1′ (ξ)F2′ (ξ) iλF22 (ξ) + F2 (ξ)F2′ (ξ) F1′ (ξ) , F5 (ξ) = , F4 (ξ) = . ′ F2 (ξ) F1 (ξ) F23 (ξ)

(24)

where F2 (ξ) is arbitrary functions of ξ and ′ means ∂ξ∂ . For the sake of simplicity, let F2 (ξ) = a1 , 0. By calculating, F1 (ξ) satisfy the following second order ordinary differential equation, (iαλ2 + 1)(F1′ (ξ))2 + 3iλβ(F1′′ (ξ))2 − 2αλF1′ (ξ)F1′′ (ξ) = 0.

(25)

That is to say, if one can get the solutions of Eq.(25) then some new solutions of coupled Hirota equations (2) can be obtained by using Eqs.(23) and Eqs.(18). It is straightforward to prove that the second order ODE (25) has following solutions, q F1 (ξ) = C1 + C2 e

±i/3(∓αλ+∆)ξ βλ

,∆ =

λ(3αβλ2 + α2 λ − 3iβ),

(26)

where C1 , C2 are arbitrary constants. By substituting (26) and (24) into (23) leads to the following exact solutions of coupled Hirota equations, ±i/3(±αλ+∆)(x−wt)+2tβλ

βλ 3λ2 a21 βe( u=∓ C2 (±αλ + ∆)

∓i/3(±αλ+∆)(x−wt) ) βλ

)

,

−C2 (±αλ + ∆)e( v= 9a21 β2 λ2 e2t

,

(27)

p where ∆ = 3αβλ3 + α2 λ2 − 3iβλ. Remark 1: Using the Euler formula, one can known that exact solutions (27) are periodic function solutions. If we take (27) as a seed solution of (18), one can obtain some new solutions of coupled Hirota equations with the help of (23) and (24). Other types of solutions can also be obtained by choosing different parameters in (21). This kinds of solutions can explain the phenomenon of period propagation of the waves, the phenomenon of the light propagation in optical fiber, etc. So they have some important applications in physics. 5. Nonlocal conservation law of C-H equations In this section, we briefly present the notations and theorems used in this paper firstly. Consider a system F{x; u} of N partial differential equations of order s with n independent variables x = (x1 , ..., xn ) and m dependent variables u(x) = (u1 (x), ..., um(x)), given by Fα [u] = Fα (x, u, u(1) , ..., u(s) ) = 0, α = 1, ..., N

(28)

where u(1) ,...u(s) denote the collection of all first,..., sth-order partial derivatives. ui = Di (u), u(i j) = D j Di (u), .... ∂ + ui j ∂u∂ j + ..., i = 1, 2, ..., n. Here Di = ∂x∂ i + ui ∂u Definition 1: A conservation law of PDE system (28) is a divergence expression Di Φi [u] = D1 Φ1 [u] + ... + Dn Φn [u] = 0 holding for all solutions of PDE system (28). 4

(29)

It is easy to see that Eqs.(14) determine a nonlocal conservation law, i.e. Dt ( f x ) + D x (− ft ) = 0.

(30)

Definition 2:[20]. The adjoint equations of Eq. (28) is defined by Fα∗ (x, u, v, u(1) , m(1) , ..., u(s), m(s) ) = Euα (mβ Fβ ) = 0, α = 1, 2, ..., N, where Euα =

∂ ∂uα

+

s P

µ=1 2

(−)µ Di1 , ..., Diµ ∂uα∂

i1 ,...,iµ

(31)

denotes the Euler-Lagrange operator, m = m(x) is a new dependent

variable m = (m1 , m , ..., mN ). Theorem 1 :[20]. The system consisting of Eqs.(28) and their adjoint Eqs.(31) ( Fα (x, u, u(1) , ..., u(s)) = 0, Fα∗ (x, u, m, u(1), m(1) , ..., u(s), m(s) ) = 0,

(32)

has a formal Lagrangian L = mβ Fβ (x, u, u(1) , ..., u(s)).

(33)

Theorem 2:[20] Any Lie point, Lie-B¨acklund and non-local symmetry V = ξi

∂ ∂ + ηα α , ∂xi ∂u

(34)

of Eqs.(28)provides a conservation law Di (T i ) = 0 for the system comprising Eqs.(28) and its adjoint Eqs.(31). The conserved vector is given by T i = ξi L + W α Euαi (L) + D j (W α )Euαi j (L) + D j Dk (W α )Euαi jk (L) + ...,

(35)

where W α is the Lie characteristic function W α = ηα − ξ j uαj , and L is determined by Eqs.(33). Next, we construct the conservation laws Eqs(2). Let us consider the prolonged system, one can obtain the following results from the above, ξ1 = c3 , ξ2 = c4 , η1 = c2 φ2 + c1 u, η2 = c2 φ2 − c1 v, η3 =

(2c2 f +c1 +c5 )φ 4 ,η 2

=

(−2c2 f +c1 −c5 )ψ 5 ,η 2

= c2 f 2 + c5 f + c6 .

and L = m1 (iut + α(u xx − 2u2 v) + iβ(u xxx − 6uvu x )) + m2 (ivt − α(v xx − 2v2 u) + iβ(v xxx − 6uvv x)) + m3 (−φ x − iλφ + uψ) +m4 (−ψ x + vφ + iλψ) + m5 (−φt + aφ + bψ) + m6 (−ψt + cφ − aψ) + m7 (− f x + ϕφ) + m8 (− ft − βϕ2 v x + 12βλ2 ϕφ +4λαϕφ − βφ2 u x + 2βϕφuv + 4iλβφ2 u − 4iλβϕ2 v + iαφ2 u − iαϕ2 v) and a, b, c are determined by Eqs.(5). Using the theorem 2, one can get the following results by calculation, T 1 = m1 (c3 βu xxx + 2iαc3 u2 v + c2 φ2 − c4 u x + 2c1 u − c3 iαu xx − 6c3 βuvu x ) + m2 (−6c3 βuvv x + c2 ψ2 − c4 v x − 2c1 v + c3 βv xxx − 2ic3 αuv2 + c3 αiv xx ) + m4 (c3 iαφuv + c3 βuφv x − 2c3 βu2 vψ − 4c3 βλ2 ψu − c4 ψu − 2c3 αλuψ + c3 βψu xx + c4 iλφ − c3 iαψu x − c3 βφvu x + 2ic3 βλuvφ + 2ic3 αφλ2 c2 f φ + 4ic3 βφλ3 − 2ic3 βλψu x + c5 φ + c1 φ) + m6 (−c4 iλψ − 2ic3 βλψuv + c3 βvψu x − 4c3 βφvλ2 + c5 ψ + c2 f ψ + 2ic3 βλφv x − c3 iαuvψ − c4 vφ + c3 βφv xx − 2c3 αλφv − 2c3 βφuv2 − 4ic3 βλ3 ψ + c3 iαφv x − c3 βuψv x − 2c3 iαλ2 ψ − c1 ψ) + m8 (−2c3 βφuvψ − 4ic3 βλuψ2 − c4 φψ − 12c3 βλ2 φψ − 4c3 αλφψ − c3 iαuψ2 + c2 f 2 + 2c5 f + c6 + c3 βφ2 v x + c3 βψ2 u x + 4ic3 βλvφ2 + c3 iαvφ2 ) T 2 = m7 (c2 f 2 + 2c5 f + c6 − c3 ft − c4 ψφ) + m5 (−c3 ψt + c5 ψ − c1 ψ − c4 vφ − c4 iλψ + c2 f ψ) + m8 (2c2 βψ2 φ2 c3 βφ2 vt + 2c1 βuψ2 − 2c1 βvφ2 + c4 ft − 2c4 βψφuv − 4ic4 βλuψ2 + 4ic4 βλvφ2 + c4 iαvφ2 − 12c4 βλ2 ψφ − 4c4 λαψφ − c4 iαuψ2 − c3 βψ2 ut )+m6 (c4 ψt −4ic1 βλvφ−c4 iαuvψ−2ic3 βλφvt +2ic2 βλψ2 φ+c2βvψφ2 +4c1 βuvψ+c2 iαψ2 φ−c3 βvψut +c3 βuψvt + 2c2 βψφψ x − 2c4 βuv2 φ − 4c4 βλ2 vφ − 2c4 αλvφ − 2ic1 αvφ − iαc3 φvt − 4ic4 βλ3 ψ − 2ic4 λ2 αψ − 2ic4 λβuvψ − c2 βψ2 φ x + c3 βvt φ x + c4 βv x φ x − 2c1 βv x φ − c2 βuψ3 − c3 βφv xt ) + m4 (c4 φt − 4c1 βuvφ + c2 βuψ2 φ − c3 βuφvt + 2c2 βφψφ x − 2c4 βu2 vψ − 4c4 βλ2 uψ − 2c4 λαuψ + c3 βvφut + c3 iαψut − c2 iαψφ2 − 2c1 iαuψ + 4ic4 βλ3 φ + 2ic4 αλ2 φ + ic4 αuvφ + 2ic3 βλψut − 2ic2 βλψφ2 − 4ic1 βλψu + 2ic4 βλuvφ − c3 βψu xt − c2 βφ2 ψ x − 2c1 βuψ x + c3 βψ x ut + c4 βψ x u x + 2c1 βψu x − c2 βφ3 v) + 5

m2 (2c2 βψψ xx −6c2 βuvψ2 −2c1 βv xx +12c1 βuv2 −ic3 αv xt +2ic2 αψψ x −2ic4 αuv2 +c4 vt +6c3 βuvvt −2ic1 αv x +2c2 βψ2x − c3 βv xxt ) + m1 (−2ic1 αu x + c4 ut + ic3 αu xt + 2c1 βu xx + 2c2 βφφ xx + 6c3 βuvut − 12c1βvu2 − 2ic2 αφφ x + 2c2 βφ2x − c3 βu xxt − 6c2 βuvφ2 +2ic4 αu2 v)+m3 (−c4 uψ+c5 φ+ic4 λφ+c1 φ+c2 φ f −c3 φt )−2c1 βu x m1x +2c1 βv x m2x +ic2 αφ2 m1x −c2 βψφ2 m4x − 2c1 βuψm4x − ic4 αu x m1x − ic2 αψ2 m2x + 2iαc1 vm2x + 2iαc1 um1x + c4 βψu x m4x + ic4 αv x m2x + ic3 αvt m2x − c2 βψ2 φv x m6x + c4 βφv x m6x + c3 βφvt m6x + 2c1 βφvm6x − 2c2 βφφ x m1x − 2c2 βψψ x m2x − c3 βvt m2xx + c3 βv xt m2x − c3 βut m1xx + c3 βu xt m1x + c2 βφ2 m1xx + 2c1 βum1xx − c4 βu x m1xx + c2 βψ2 m2xx − c4 βv x m2xx − 2c1 βvm2xx + c4 βm1x u xx + c4 βm2x v xx − ic3 αm1x ut + c3 βψm4x ut , where m1t , m2t , m3x , m5x , m7x can be obtained by using (31) and (33). Because the formulas are complicated, so we put them in the appendix. Through the verification, T 1 , T 2 satisfy the equation(29),i.e. Dt (T 1 ) + D x (T 2 ) = 0,

(36)

thus, Eqs.(34),(35) define the corresponding components of a non-local conservation law for the system of Eqs. (2). Remark 2: For the coupled Hirota equations (2), we get its conservation laws by making use of the explicit solution of the adjoint equations of Eqs. (2). But the number of the adjoint equations is less than the number of variables, so T 1 , T 2 have arbitrary functions, we can conclude that Eqs.(2) have infinitely many conservation laws. 6. Summary and Discussion In this paper, the nonlocal symmetry of coupled Hirota equations is obtained by using the Lax pair and localized by introducing an auxiliary dependent variable. Then, the primary nonlocal symmetry is equivalent to a Lie point symmetry of a prolonged system. On the basis of this system, exact solution of coupled Hirota equations are given out by using the associated vector fields. On the one hand, non-trivial nonlocal conservation laws of the coupled Hirota equations are obtained by means of these formulas. For general evolution equation, the nonlocal conservation laws may be not enough to guarantee the integrability. However, it must be useful to help to understand the special solutions of the nonlinear systems. This method would be possible to extend to many other interesting integrable models. However, there is not a universal way to estimate what kind of nonlocal symmetries can be localized to some related prolonged system, so there are still many questions worth to study. Moreover, one can construct infinitely many nonlocal symmetries by introducing some internal parameters from the seed symmetry and infinitely many nonlocal conservation laws of the completely integrable finite-dimensional systems. Above topics will be discussed in the future series research works. Acknowledgments: This work is supported by National Natural Science Foundation of China under Grant (Nos.11505090, 11405103, 11447220),Research Award Foundation for Outstanding Young Scientists of Shandong Province (No.BS2015SF009). References ¨ [1] S. Lie, Uber die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differential gleichungen, Arch. Math. 328 (1881). [2] L.V. Ovsiannikov, Group Analysis of Differential Equations, New York: Academic, 1982. [3] N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Boston, MA: Reidel, 1985. [4] P.J. Olver, Applications of Lie Groups to Differential Equations, Berlin: Springer, 1986. [5] I.S. Akhatov, R.K. Gazizov and N.K. Ibragimov, Nonlocal symmetries. Heuristic approach, J. Sov. Math. 55(1991) 1401-1450. [6] G.W. Bluman, A.F. Cheviakov and S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer New York, 2010. [7] S.Y. Lou and X.B. Hu, Non-local symmetries via Darboux transformations, J. Phys. A: Math. Gen. 30(1997) L95-L100. [8] F. Galas, New non-local symmetries with pseudopotentials, J. Phys. A: Math. Gen. 25(1992) L981-L986. [9] 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. [10] Q. Miao, X.P. Xin, Y. Chen, Nonlocal symmetries and explicit solutions of the AKNS system, Appl. Math. Lett. 28(2014) 7-13. [11] S.Y. Lou, X.R. Hu and Y. Chen, Nonlocal symmetries related to Backlund transformation and their applications, J. Phys. A: Math. Theor. 45(2012) 155209-155209. [12] J.C. Chen and Y. Chen, Nonlocal symmetry constraints and exact interaction solutions of the (2+1) dimensional modified generalized long dispersive wave equation, J. Nonlinear Math. Phys. 21 (2014) 454-472. [13] X.R. Hu, Y.Q. Li, Nonlocal symmetry and soliton-cnoidal wave solutions of the Bogoyavlenskii coupled KdV system, Appl. Math. Lett. 51(2016) 20-26. [14] H.Z. Liu, J.B. Li, L. Liu, Y. Wei, Group classifications, optimal systems and exact solutions to the generalized Thomas equations, J. Math. Anal. Appl., 383(2011) 400-408.

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[15] Z.Y Ma, J.X. Fei, Y.M. Chen, The residual symmetry of the (2+1)-dimensional coupled Burgers equation, Appl. Math. Lett. 37(2014) 54-60. [16] E. Noether, Invariante variationsprobleme, Nachr. K¨onig. Gesell. Wissen., G¨ottingen, Math.-Phys. Kl. Heft 2(1918) 235-257. [17] H. Steudel, Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen, Z. Naturforsch. 17a(1962) 129-132. [18] S.C. Anco, G.W. Bluman, Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, Eur. J. Appl. Math. 13(2002) 545-566. [19] S.Y. Lou, Nonlocal conservation laws and related B¨acklund transformations via reciprocal transformations, arXiv:1402.7231v2. [20] N.H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333(2007) 311-328. [21] M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H.Segur, Nonlinear-evolution equations of physical significance, Phys. Rev. Lett. 31(1973) 125-127. [22] Y. Zhang, K.H. Dong, and R.J. Jin, The Darboux transformation for the coupled Hirota equation, AIP Conf. Proc.1562, 249(2013) 249-256. [23] X.P. Xin, Y. Chen, A Method to Construct the Nonlocal Symmetries of Nonlinear Evolution Equations, Chin. Phys. Lett. 30(2013) 100202.

Appendix m1t = 6βuvm1x + 6βm1 uv x − βvψm6x − 2βψm6 v x − βm6 vψ x − 2βm8 ψψ x − 4βλ2 m4 ψ − 2λαm4 ψ + 2βφm4 v x + iαm4 ψ x + βvm4 φ x − 2βv2 m6 φ + iαm4x ψ + βvm4x φ − 6βvm2 v x − iαm8 ψ2 − 2iαm2 v2 − βm1xxx − iαm1xx + βψm4xx + βψ xx m4 + 2βψ x m4x − βψ2 m8x −m3 ψ+iαφvm4 −4βuvψm4 −2βvψφm8 +2iλβm4 ψ x +4iαuvm1 −iαvψm6 −4iλβm8 ψ2 +2iλβψm4x +2iλβvφm4 − 2iλβvψm6 , m2t = iαm2xx + βφm6xx + βm4 φ xx − βm2xxx + 2βm6x φ x − 4βφuvm6 + 4iαuφm4 − 2βφψum8 − iαuψm6 − 2iβλm6 φ x + 4iβλm8 φ2 − 2iβλφm6x − 4iαuvm2 − m5 φ − 6βm1 uu x − 4λ2 βm6 φ + 2βm6 ψu x − 2λαφm6 + βm6 uψ x − 2βm4 u2 ψ + iαφ2 m8 + βuψm6x − iαm6 φ x + 2iαm1 u2 − iαφm6x + 6βvum2x + 6βvm2 u x − βuφm4x − βum4 φ x − 2βφm4 u x − 2βφm8 φ x − βφ2 m8x − 2iλβuψm6 + 2iλβuφm4 , m3x = 2iαλ2 m4 + iαm6 v x + iαuvm4 + 8iλβvφm8 + 2iαvφm8 + βum4 v x − βvm4 u x − vm5 + 2iλβuvm4 − 2βuv2 m6 − 4βλ2 vm6 +iλm3 −2αλvm6 +βm6 v xx −ψm7 +4iβλ3 m4 +2iλβm6 v x −2βuvψm8 −12βλ2 ψm8 −4λαψm8 +2βφv x m8 −m4t , m5x = −m3 u − iαuvm6 − 2βu2 vm4 − 4βλ2 um4 − iλm5 − 2αλum4 + βm4 u xx − 2iαλ2 m6 − 2iλβuvm6 − 8iλβuψm8 − 2iαuψm8 −2iλβm4 u x −βm6 uv x +βm6 vu x −m7 φ−iαm4 u x −4iβλ3 m6 −2βuvφm8 −12βλ2 φm8 +2βψu x m8 −4λαφm8 −m6t , m7x = −m8t .

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