A new integrable symplectic map and the lie point symmetry associated with nonlinear lattice equations

Research Article

A new integrable symplectic map and the lie point symmetry associated with nonlinear lattice equations

Huanhe Dong^a,b, Tingting Chen^a,∗, Longfei Chen^c, Yong Zhang^a

aCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China.

bKey Laboratory for Robot and Intelligent Technology of Shandong Province, Qingdao, 266510, P. R. China.

cSchool of Economics of Shanghai University, Shanghai 200444, P. R. China.

Communicated by S. S. Chang

Abstract

A discrete matrix spectral problem is proposed, the hierarchy of discrete integrable system is inferred, which are Liouville integrable. And the Hamiltonian structures of the hierarchy are constructed. A family of finite-dimensional completely integrable systems and a new integrable symplectic map are provided in terms of the binary nonlinearity of spectral problem. In particular, two explicit formulations are acquired under the condition of the bargmann constraints. After that, the symmetry of the discrete integrable systems is given on the basis of the seed symmetry and its prolongation. Moreover, the solution of the discrete lattice equation can be gained by the way of the infinitesimal generator. c

Keywords:

Symplectic map, symmetry, discrete integrable system, liouville integrability, nonlinearization.

2010 MSC:

35Q51, 37K40, 58J70, 35A35, 22E65, 70H33.

1. Introduction

In the last several years, with the deepening of theoretical research on the discrete integrable systems such as the Toda lattice, Ablowitz-Ladik lattice the differential-difference KdV equation and so on [6, 9, 10, 19, 21, 23, 26], many people have made many outstanding research results which are widely used in photology and hydromechanics. After studying the integrable systems, we find that the discrete integrable systems can better explain the natural phenomenon than the continuous integrable system from the aspects of nature. There are two very important issues, one of which is to find a new Lax integrable nonlinear lattice systems and discuss their Hamiltonian structures [1, 4, 5, 11, 14, 17, 18, 22] and the other is to obtain integrable symplectic map, which has been proposed and developed in Refs. [3, 20, 24, 25, 27]. We can

∗Corresponding author

Email addresses: [email protected](Huanhe Dong),[email protected](Tingting Chen),[email protected] (Longfei Chen),[email protected](Yong Zhang)

Received 2016-01-02

solve the discrete integrable system through various methods, such as B¨ acklund transformation, Dˆ arboux transformation, Hirota approach, the inverse scattering method, etc. Lie symmetry provides a systematic approach to the purpose of reducing the order of the differential equations. Usually, the standard method is used to solve the differential equation. However, using lie point symmetry can simplify solution, and it can also be used to solve the equation. For differential equations, it is important to study the group- invariant solution and symmetry reduction, because it provides a powerful tool for the study of differential equations. And for some equations, it can greatly reduce the calculation of the equation, which can be used to solve the equation. In this paper, we will use the symmetry theory to solve the discrete integrable systems [2, 7, 8, 12, 13, 15, 16].

General discrete integrable systems are as follows:

E

(x

, x

, x

, u

, u

, u

) = 0, a = 1, 2,

, N, (1.1) where

det( ∂(E

, E

)

∂(x

, u

) )

det( ∂(E

, E

)

∂(x

, u

) )

Set infinitesimal generator ν as follows:

ν =

p

X

i=1

ξ

(x, u) ∂

∂x

+

q

X

α=1

φ

(x, u) ∂

∂u

,

and

P r

ν = ν +

q

X

α=1

X

J

φ

∂

∂u

.

Here P r

ν is the infinitesimal generator of nth order prolonged space, where n indicates the highest order, in which

φ

(x, u

) = D

(φ

p

X

i=1

ξ

u

) +

p

X

i=1

ξ

u

,

where u

,u

satisfies u

U

(n

1), u

= ∂u

/∂x

; and operator D is the total derivative, which is the differential operator of prolonged space,

D

P = ∂P

∂x

+

q

X

α=1

∂P

∂u

·

∂u

∂x

+

q

X

α=1

X

J

∂P

∂u

u

, where J = (J

,

, J

) and D

= D

D

D

.

Substituting prolongation operator P r

ν into Eq. (1.1) and handling the coefficients of all u

, u

, we gain a linear independent expression and get the extension of the solution through setting the coefficient as zero.

In this paper, we would like to consider a new hierarchy of integrable nonlinear lattice equation which is inferred from a new discrete spectrum problem. Then, we will construct its Hamiltonian structure and test its properties of Liouville integrability. After that, a new integrable symplectic map and a family of finite-dimension completely integrable systems would be given according to the binary nonlinearization of the spectral problem. At the end of the paper, we use the symmetry theory and the gˆ ateaux derivative to solve the symmetry and the infinitesimal generator of the discrete Lattice equation. We explain our results by some figures.

2. A new discrete integrable hierarchy and its Hamiltonian structure

Consider the following discrete spectrum problem,

Eϕ = U ϕ, U = U (u, λ) =

λ λq

1

p

1 +

, ϕ =

ϕ

ϕ

. (2.1)

The shift operator and the difference operator are defined as follows

(Ef )(n) = f(n + 1), (E

f )(n) = f (n

1), n

Z, (Df )(n) = f(n + 1)

f (n) = (E

1)f (n), n

Z, where f

= E

f, j

Z, λ is the spectral parameter, λ

= 0.

We give a static discrete zero curvature equation for obtaining the discrete integrable systems

(E Γ)U

U Γ = 0, (2.2)

and choose

Γ =

A λB C

.

We have four equations from Eq. (2.2) as follows:











λA

+ λ

B

λA

λqC = 0,

λqA

+ λ(1 +

)B

λ

B + λqA = 0, λC

(A

+ A)

(1 +

)C = 0, λqC

λ

B

(1 +

)(A

A) = 0.

(2.3)

Furthermore, substituting

∞

P

m=0

A_mλ^−m, B=

∞

P

m=0

B_mλ^−m, C=

∞

P

m=0

C_mλ^−m into Eq. (2.3), we have

C₀⁽¹⁾= 0, B0= 0, A⁽¹⁾₀ −A0+1

pB₀⁽¹⁾−qC0= 0, and















A⁽¹⁾m −Am+¹_pBm⁽¹⁾−qCm= 0,

q(A⁽¹⁾m +Am) + (1 +_p^q)B⁽¹⁾m −Bm+1= 0, qC_m+1⁽¹⁾ −¹_p(A⁽¹⁾m +A_m)−(1 + ^q_p)C_m= 0,

qC_m+1⁽¹⁾ −¹_pB_m+1−(1 + ^q_p)(A⁽¹⁾m −A_m) = 0, m≥0.

(2.4)

By settingA0 = 1

2, B0 = 0,the coefficientsAm, Bm, Cm,(m≥1) can be obtained according to the Eq. (2.4). A set of coefficients are as follows

A1=− q

p⁽⁻¹⁾, B1=q, C1= 1 p⁽⁻¹⁾,· · ·. For any integerm≥0, we letf = P

m∈Z

fmλ^mand denote f+= P

m≥0

fmλ^m,choose

Γ⁽ⁿ⁾₊ =

n

X

m=0

Am λBm

Cm −Am

λ^n−m,Γ⁽ⁿ⁾₋ =λⁿΓ−Γ⁽ⁿ⁾₊ , and rewrite Eq. (2.2) into

(Γ⁽ⁿ⁾₊ )U−UΓ⁽ⁿ⁾₊ =−(Γ⁽ⁿ⁾₋ )U+UΓ⁽ⁿ⁾₋ . A direct calculation reads that

(Γ⁽ⁿ⁾₊ )U−UΓ⁽ⁿ⁾₊ = 0 λB_n+1

−C_n+1^{(1) 1}_pB_n+1−qC_n+1⁽¹⁾

! .

Let Γ⁽ⁿ⁾= Γ⁽ⁿ⁾₊ , then the discrete zero curvature equation meets the following Lax integrable system p_t=C_n+1⁽¹⁾ p²,

q_t=B_n+1. (2.5)

According to Eq. (2.4), we acquire the recurrence operatorL as follows

L= E⁻¹(1 +_p^q) +^q_p(E+ 1)(E−1)⁻¹E⁻¹ E⁻¹(E+ 1)(E−1)⁻¹

−_p^q22(E+ 1)(E−1)⁻¹ (1 +^q_p) +^q_p(E+ 1)(E−1)⁻¹

! . We rewrite System (2.5) into

q p

nt

=J

Cn+1

−^B

(1) n+1

p²

!

=J Lⁿ

C₁

−^B

(1) 1

p²

!

=J Lⁿ

1 p⁽⁻¹⁾

−^q_p⁽¹⁾2

!

, (2.6)

where

J =p²

0 −E⁻¹

E 0

. When we taken= 1, system (2.6) reduces to

( pt=_p(−1)^p2 −q⁽¹⁾,

q_t=q⁽¹⁾−_p(−1)^q2 . (2.7)

For purpose of constructing the Hamiltonian structure of system (2.6), we define

V = ΓU⁻¹= λ⁻¹(1 +^q_p)A−^B_p λB−qA λ⁻¹(1 +^q_p)C+_pλ¹A −qC−A

! . We have

∂U

∂λ =

1 q 0 0

,∂U

∂p =

0 0

−_p¹2 −_p^q2

,∂U

∂q =

0 λ 0 ¹_p

. Therefore,











V,^∂U_∂λ

=_λ¹(1 +_p^q)(A+qC)−^B_p +_pλ^q A, D

V,^∂U_∂pE

=_p¹2(2qA−λB+q²C), DV,^∂U_∂qE

=C,

andhA, Bi=T r(AB), where AandB are the same order square matrix. By applying the discrete trace identity δ

δu X

n∈Z

V,∂U

∂λ

=

λ^−ε ∂

∂λ

λ^ε V,∂U

∂uⁱ

, i= 1,2, we obtain that

δ δu

A

λ =λ^−ε ∂

∂λλ^ε C

−^B_p⁽¹⁾2

! . By comparing with the coefficient ofλ⁻ⁿ⁻¹, we get

δ

δuAn= (ε−n) C_n

−^B_p⁽¹⁾ⁿ2

! . Whenn= 1, letε= 0, then we have

δ δu

−An

n

= C_n

−^B_pⁿ⁽¹⁾2

! ,

and

C_n+1

−^B

(1) n+1

p²

!

=δHn

δu , H_n=−An+1

n+ 1,

whereJ is a Hamiltonian operator. Hence, Eq. (2.6) can be written as the Hamiltonian structure.

q p

nt

=JδHn

δu . (2.8)

Furthermore, we test the following result

(J L)^∗=−J L, where

K=J L=

q²(E+ 1)(E−1)⁻¹E⁻¹ qp(E+ 1)(E−1)⁻¹E⁻¹−(p²+qp)E⁻¹ qp(E+ 1)(E−1)⁻¹+ (p²+qp) p²(E+ 1)(E−1)⁻¹

nH˜m,H˜l

o

J= 0, m, l≥1, (2.9)

( ˜H_m)_tl=X

n∈Z

δH˜_m δu u_tl

!

(n) =X

n∈Z

(δH˜_m δu , JδH˜_l

δu ) =n

H˜_m,H˜_lo

J

= 0, m, l≥1.

Hence, the conserved densitiesn H˜m

o∞ m=1

are the involution with respect to Poisson bracket (2.9) and we conclude that each nonlinear difference-differential equation of the discrete hierarchy is Liouville integrable.

3. A new integrable symplectic map and representation of solutions for Eq. (2.7)

In the subsection, we will discuss the symmetry constraint of Eq. (2.5). Consider the adjoint spectral problem of Eq. (2.1)

E⁻¹ψ= (E⁻¹U^T(a, λ))ψ, ψ= ψ1

ψ2

, (3.1)

and the auxiliary problem

ψt_m =−(Vm(a, λ))^Tψ. (3.2)

In terms of the compatibility condition of Eq. (2.6) and Eq. (2.7) (E⁻¹ψ)t_m =E⁻¹(ψt_m), we gain E⁻¹U_t^T

m= (E⁻¹U^T)(V_m)^T−(E⁻¹(V_m)^T)(E⁻¹U^T). (3.3) It is easy to test that Eq. (3.3) andU_tm= (EV_m)U−U V_mare equivalent. Hence, Eq. (3.3) is the another kind of zero curvature representation of the discrete soliton Eq. (2.8), where (3.1) and (3.2) are regarded as the adjoint Lax pairs of discrete soliton Eq. (2.8).

Assumingλ₁, λ₂,· · ·, λ_N arendifferent eigenvalues of the spectral problem (2.1), we gain Eϕ1j

Eϕ2j

=U(a, λ_j) ϕ1j

ϕ2j

, E⁻¹ψ1j

E⁻¹ψ2j

= (E⁻¹U^T(a, λj)) ψ1j

ψ2j

,1≤j≤N,

(3.4)

ϕ1j

ϕ2j

t_m

=Vm(a, λj) ϕ1j

ϕ2j

, ψ1j

ψ2j

t_m

=−V_m^T(a, λj) ψ1j

ψ2j

, 1≤j≤N,

(3.5)

(Eϕ1j ,Eϕ2j) = (ϕ1j, ϕ2j)U(a, λj)^T, 1≤j ≤N,

(Eψ1j, Eψ2j) = (ψ1j, ψ2j)U(a , λj)⁻¹, 1≤j≤N.

According to Ref. [3], we infer δλj

δ p = q

p²ϕ1jψ1j− 1

p²ϕ1jψ2j+q²

p²ϕ2jψ1j− q

p²ϕ1jψ2j, 1≤j≤N, δλ_j

δ q =ϕ_2jψ_1j, 1≤j≤N,

whereh. , .idenotes the inner product inR^N. By making use of the discrete Bargmann constraint JδH˜0

δu =J

N

X

j=0

δλj

δu, whereαj= 1, 1≤j≤N. That is,

δH˜₀ δ p = q

p²hΦ1,Ψ1i − 1

p²hΦ1,Ψ2i+q²

p²hΦ2,Ψ1i − q

p²hΦ2,Ψ2i, δH˜₀

δ q =hΦ2,Ψ1i, where

Φ_i= (ϕ_i1, ϕ_i2,· · ·, ϕ_iN)^T, Ψ_i= (ψ_i1, ψ_i2,· · ·, ψ_iN)^T, i= 1,2.

We obtain two explict constraints







q= Λ⁻¹hΦ₁,Ψ₂i,

p= Λ²+ ΛhΦ2,Ψ2i − hΦ1,Ψ2ihΦ2,Ψ1i −ΛhΦ1,Ψ1i

ΛhΦ2,Ψ1i , (3.6)

or

q= Λ⁻¹hΦ1,Ψ2i,

p=−Λ⁻¹hΦ1,Ψ2i. (3.7)

Substituting Eqs. (3.6) and (3.7) into Eq. (3.4), we obtain a discrete Bargmann system



















































Eφ1j =λjφ1j+λjqφ2j= Λφ1j+hΦ1,Ψ2iφ2j, Eφ2j =1

pφ1j+ (1 +q p)φ2j

= ΛhΦ2,Ψ₁i(Λ²−ΛhΦ1,Ψ₁i − hΦ1,Ψ₂ihΦ2,Ψ₁i+ ΛhΦ2,Ψ₂i)⁻¹φ_1j

+ (1 +hΦ1,Ψ2ihΦ2,Ψ1i(Λ²−ΛhΦ1,Ψ1i − hΦ1,Ψ2ihΦ2,Ψ1i+ ΛhΦ2,Ψ2i)⁻¹)φ2j, Eψ_1j = 1

λ_j(1 +q

p)ψ_1j− 1 λ_jpψ_2j

= Λ⁻¹(1 +hΦ1,Ψ2ihΦ2,Ψ1i(Λ²−ΛhΦ1,Ψ1i − hΦ1,Ψ2ihΦ2,Ψ1i

+ ΛhΦ2,Ψ2i)⁻¹)ψ1j− hΦ2,Ψ1i(Λ²−ΛhΦ1,Ψ1i − hΦ1,Ψ2ihΦ2,Ψ1i+ ΛhΦ2,Ψ2i)⁻¹ψ2j, Eψ_2j =−qψ_1j+ψ_2j =−Λ⁻¹hΦ₁,Ψ₂iψ_1j+ψ_2j,

(3.8)

or 





















EΦ1j=λjφ1j+λjqφ2j= Λφ1j+hΦ1,Ψ2iφ2j, EΦ2j= 1

pφ1j+ (1 +q

p)φ2j=−ΛhΦ1,Ψ2i⁻¹φ1j, EΨ1j= 1

λj

(1 +q

p)ψ1j− 1

λjpψ2j=−hΦ1,Ψ2i⁻¹ψ2j, EΨ_2j=−qψ_1j+ψ_2j=−Λ⁻¹hΦ₁,Ψ₂iψ_1j+ψ_2j.

(3.9)

Letf_i=f_i(Φ₁,Φ₂,Ψ₁,Ψ₂), g_i=g_i(Φ₁,Φ₂,Ψ₁,Ψ₂), 1≤i≤2N.We present



































f_j= Λ⁻¹(1 +hΦ₁,Ψ₂ihΦ₂,Ψ₁i(Λ²−ΛhΦ₁,Ψ₁i − hΦ₁,Ψ₂ihΦ₂,Ψ₁i+ ΛhΦ₂,Ψ₂i)⁻¹)ψ_1j

− hΦ2,Ψ1i(Λ²−ΛhΦ1,Ψ1i − hΦ1,Ψ2ihΦ2,Ψ1i+ ΛhΦ2,Ψ2i)⁻¹ψ2j, fN+j=−Λ⁻¹hΦ1,Ψ2iψ1j+ψ2j,

g_j= Λφ_1j+hΦ1,Ψ₂iφ2j,

gN+j= ΛhΦ2,Ψ1i(Λ²−ΛhΦ1,Ψ1i − hΦ1,Ψ2ihΦ2,Ψ1i+ ΛhΦ2,Ψ2i)⁻¹φ1j

+ (1 +hΦ₁,Ψ₂ihΦ₂,Ψ₁i(Λ²−ΛhΦ₁,Ψ₁i − hΦ₁,Ψ₂ihΦ₂,Ψ₁i+ ΛhΦ₂,Ψ₂i)⁻¹)φ_2j, or



















fj=−hΦ1,Ψ2i⁻¹ψ2j, f_N+j=−Λ⁻¹hΦ₁,Ψ₂iψ_1j+ψ_2j,

gj= Λφ1j+hΦ1,Ψ2iφ2j,

gN+j=−ΛhΦ1,Ψ2i⁻¹φ1j, 1≤j≤N.

We define Poisson bracket as

{f , g}=

2

X

i=1 N

X

j=1

∂f

∂ψ_ij

∂g

∂ϕ_ij − ∂f

∂ϕ_ij

∂g

∂ψ_ij

=

2

X

i=1

∂f

∂Ψ_i, ∂g

∂Φ_i

− ∂f

∂Φ_i, ∂g

∂Ψ_i

on any pair of functionsf =f(Φ1,Φ2,Ψ1,Ψ2) andg=g(Φ1,Φ2,Ψ1,Ψ2). Let H(Ψ1,Ψ2,Φ1,Φ2) = (EΨ1, EΨ2, EΦ1, EΦ2), then, via a direct calculation, we infer

{f_i, f_j}={g_i, g_j}= 0, {f_i, g_j}=δ_ij, 1≤i, j≤2N,

we define Eqs. (3.8) and (3.9) as an integrable symplectic map. Then we take the binary nonlinearization of the Lax pairs and the adjoint Lax pairs into account. According to the Eq. (2.4), the following values could be selected as















A˜0= 1

2,B˜0= ˜C0= 0,A˜1= 0,

B˜_m= Λ^m−2hΦ₁,Ψ₂i,C˜_m= Λ^m−1hΦ₂,Ψ₁i, A˜m= Λ^m−1hΦ1,Ψ1i −Λ^m−1hΦ2,Ψ2i

2 , m≥1.

(3.10)

Let

Γ =˜

A˜ λB˜ C˜ −A˜

=

∞

X

m=0

A˜_m λB˜_m C˜_m −A˜_m

λ^−m and

F˜_m= det ˜Γ = 1

2tr˜Γ²= ˜A²+ Λ ˜BC˜ =

m

X

0

( ˜A_iA˜_m−i+ Λ ˜B_iC˜_m−i), (3.11)

then we have F˜0= 1

4,F˜1= 0,F˜m= Λ^m−1hΦ1,Ψ₁i −Λ^m−1hΦ2,Ψ₂i

2 +

m−1

X

i=1

(Λⁱ⁻²hΦ1,Ψ2iΛ^m−ihΦ2,Ψ1i +Λⁱ⁻¹hΦ1,Ψ1i −Λⁱ⁻¹hΦ2,Ψ2i

2

Λ^m−i−1hΦ1,Ψ1i −Λ^m−i−1hΦ2,Ψ2i

2 ).

We could obtain a family of finite-dimensional integrable systems and an integrable symplectic map via the binary nonlinearity of the isospectral problem. Bringing Eq. (3.10) into Eq. (3.5), we have

ϕ1j

ϕ2j

t_m

= V^(m)(u, λj) _B

ϕ1j

ϕ2j

, ψ1j

ψ2j

t_m

=−V^(m)T(u, λj) _B

ψ1j

ψ2j

, j= 1,2,· · ·, N,

(3.12)

where subscriptB stands for Eqs. (3.6) and (3.7).

Rewriting Eq. (3.12) as the Hamilton systems, we infer D( ˜A²+ ˜BC) = 0,˜ Ψit_m =−∂( ˜Fm+1)

∂Φi

, Φit_m =∂( ˜Fm+1)

∂Ψi

, i= 1,2.

Setting

Fj=ϕ1jψ1j+ϕ2jψ2j, 1≤j≤N, we have

{ F˜_m+1, F¯j}= dF¯_j dtm

= 0, 1≤j≤N, m≥0; { F¯_i, F¯j}= 0, 1≤i, j≤N.

Therefore, we get

∂F˜m

∂Φ₁ = Λ^m−1Ψ1

2 +

m−1

X

i=1

C˜_m+1−iΛⁱ⁻²Ψ2+

m−1

X

i=1

A˜iΛ^m−i−1Ψ1,

∂F˜_m

∂Φ2

=−Λ^m−1Ψ₂

2 +

m−1

X

i=1

B˜m−iΛ^m−iΨ1+

m−1

X

i=1

A˜iΛ^m−i−1Ψ2. According to Eq. (3.11), the following conclusions can be drawn directly

∂F˜m+1

∂Φ1

_Φ

1=Φ₂=0

=Λ^m−1Ψ1

2 ,

∂F˜m+1

∂Φ_{2 Φ}

1=Φ2=0

=−Λ^m−1Ψ2

2 .

Set

N

P

j=1

ψ²_1j6= 0 , we suppose that Φ1= (ϕ_11,ϕ₁₂,· · ·ϕ_1N)^T is satisfied with the following nonlinear equation



















ϕ11ψ11+ϕ12ψ12+· · ·ϕ1Nψ1N = 0,

λ₁ϕ₁₁ψ₁₁+λ₂ϕ₁₂ψ₁₂+· · ·+λ_Nϕ_1Nψ_1N = 0, ...

λ^N−1₁ ϕ₁₁ψ₁₁+λ^N₂⁻¹ϕ₁₂ψ₁₂+· · ·+λ^N_N⁻¹ϕ_1Nψ_1N = 0.

(3.13)

Through the observation of Vandermode determinant

1 1 · · · 1

λ1 λ2 · · · λN

· · · · λ^N₁⁻¹ λ^N−1₂ · · · λ^N_N⁻¹

6= 0,

we know Φ₁= (ϕ₁₁, ϕ_12,· · ·ϕ_1N)^T is only decided by Eq. (3.13). In addition, we obtain the following proposition by direct calculation

∂F¯j

∂ϕ_il =ψijδjl, i= 1,2, j, l= 1,2· · ·N.

det

∂F¯₁

∂Φ1 · · · ^∂_∂Φ^F¯N

1

∂F˜₁

∂Φ1 · · · ^∂_∂Φ^F˜N

1

∂F¯1

∂Φ₂ · · · ^∂_∂Φ^F¯N

2

∂F˜1

∂Φ₂ · · · ^∂_∂Φ^F˜N

2

!

= det





















ψ11 · · · 0 ¹₂λ1ψ11 · · · ¹₂λ^N₁ ψ11

... . .. ... ... . .. ... 0 · · · ψ1N 1

2λNψ1N · · · ¹₂λ^N_Nψ1N

ψ21 · · · 0 −¹₂λ1ψ21 · · · −¹₂λ^N₁ψ21

... . .. ... ... . .. ...

0 · · · ψ2N −¹₂λNψ2N · · · −¹₂λ^N_Nψ2N





















= ((

N

Y

j=1

ψ1jψ2j)(

N

Y

k=1

λk)

1 λ₁ · · · λ^N₁⁻¹ 1 λ₂ · · · λ^N₂⁻¹

· · · · 1 λ_N · · · λ^N_N⁻¹

.

That is, ˜Fm+1,1≤m≤N,F¯j,1≤j≤N are functionally independent in some region ofR^4N.

4. The symmetry of the discrete integrable system

In this section we use the symmetry theory to find the solution to Eq. (2.7). The vector fields of the general form are

ν1=τ(t)∂t+φn(t, p⁽ⁿ⁾)∂_p(n), ν2=τ(t)∂t+ψn(t, q⁽ⁿ⁾)∂_q(n).

So as to obtain the Lie algebra of local Lie point symmetries of Eq. (2.7), we present the first prolongation ofν1, ν2, that is,

P r⁽ⁿ⁾ν1=τ(t)∂t+

1

X

i=−1

φi(t, p⁽ⁱ⁾)∂_p(i)+φ⁽¹⁾∂p˙, φ⁽¹⁾ =Dtφ(t, p)−[Dtτ(t)] ˙p,

P r⁽ⁿ⁾ν₂=τ₁(t)∂_t+

1

X

i=−1

ψ_i(t, q⁽ⁱ⁾)∂_q(i)+ψ⁽¹⁾∂_q˙, ψ⁽¹⁾=−Dtψ(t, q) + [D_tτ₁(t)] ˙q,

whereD_t stands for the total derivative operator. SubstitutingP r⁽ⁿ⁾ν into Eq. (2.7), we have ϕ₍₋₁₎ p²

p²⁽⁻¹⁾

−ϕ 2p

p⁽⁻¹⁾+ϕ_t+ (ϕ_p−τ)(˙ p²

p⁽⁻¹⁾ −q⁽¹⁾) +ψ 2q

p⁽⁻¹⁾−ψ₍₁₎−ψ_t−(ψ_q−τ˙₁)(q⁽¹⁾− q²

p⁽⁻¹⁾) = 0. (4.1) Substituting the second derivative∂p∂_q(−1) and∂q∂_q(1) into Eq. (4.1), we infer

−ϕpp

1 p²⁽⁻¹⁾

= 0, ψqq= 0.

It is easy to get that

ϕpp = 0, ψqq = 0, which infer the following result

ϕ=ap, ψ=bq. (4.2)

Substituting Eq. (4.2) into Eq. (4.1), we gain a₍₋₁₎ p²

p⁽⁻¹⁾−2a p²

p⁽⁻¹⁾ + (2a−τ)(˙ p²

p⁽⁻¹⁾ −q⁽¹⁾) + 2b q²

p⁽⁻¹⁾ −b(1)q⁽¹⁾−(2b−τ˙1)(q⁽¹⁾− q²

p⁽⁻¹⁾) = 0, (4.3) inferring the special coefficient relation of the Eq. (4.3), we have







˙

τ =a₍₋₁₎,

˙ τ1= 4b,

b₍₁₎= 2b−2a+a₍₋₁₎, where the coefficientsa₍₁₎, b₍₁₎ are constants.

Since the solution of the Eq. (2.7) could be obtained on the basis of the symmetry theory with the help of the infinitesimal generator, we can rewrite Eq. (2.7) into the following form.























p_t= p²

p⁽⁻¹⁾−q⁽¹⁾, qt=q⁽¹⁾− q²

p⁽⁻¹⁾, prX₁[p] =τ(t)p_t+ϕ= 0, prX2[q] =τ1(t)qt+ψ= 0.

(4.4)

If we take some appropriate initial values

a= 3b, b₍₁₎=−2b, a₍₋₁₎= 2b, τ˙1= 4b, τ˙ = 2b, (4.5) and set the step lengthp−p⁽⁻¹⁾=q⁽¹⁾−q=h, we obtain the solution of Eq. (2.7)











q= (ap τ + p²

p−h)−h, b

τ1

(ap τ + p²

p−h−h) =(^ap_τ −_p−h^p2 −h)² p−h −(ap

τ + p² p−h),

(4.6)

according to Eqs. (4.4) and (4.5).

5. Conclusion

In this paper, we prove that Eqs. (3.8) and (3.9) are an integrable symplectic map through the binary nonlin- earization method, and we know that ˜Fm+1,1≤m≤N,F¯j,1 ≤j ≤N are functionally independent in some region ofR^4N. According to the symmetry theory, we gain the seed symmetry and the infinitesimal generator by the seed symmetry and the recursion operator. And we gain the infinitesimal generator of the discrete lattice equation based on the Lie point symmetry theory.

The explanation of the p and q with the time variable t is given in Figs. 1–3. According to the figures, by adjusting the step size, we find that the longer the step length, the greater the minimum, and the smaller the maximum. Meanwhile, the convergence rate of the graphics is becoming slower.

−1.5 −1 −0.5 0 0.5 1 1.5

−5

−4

−3

−2

−1 0 1 2 3 4 5

t

p

−1.5 −1 −0.5 0 0.5 1 1.5

−30

−20

−10 0 10 20 30

t

q

Figure 1: the left figure indicates the solution of theq and the right is the solution of thep, based on the Eq. (4.6), with the step lengthh= 0.1.

−1.5 −1 −0.5 0 0.5 1 1.5

−5

−4

−3

−2

−1 0 1 2 3 4 5

t

p

−1.5 −1 −0.5 0 0.5 1 1.5

−30

−20

−10 0 10 20 30

t

q

Figure 2: the left figure indicates the solution of theq and the right is the solution of thep, based on the Eq. (4.6), with the step lengthh= 0.5.

−1.5 −1 −0.5 0 0.5 1 1.5

−5

−4

−3

−2

−1 0 1 2 3 4 5

t

p

−1.5 −1 −0.5 0 0.5 1 1.5

−30

−20

−10 0 10 20 30

t

q

Figure 3: the left figure indicates the solution of theq and the right is the solution of thep, based on the Eq. (4.6), with the step lengthh= 1.

Our plan of the future work is that using B¨acklund transformation or Dˆarboux transformation to research these discrete integrable systems. Moreover, we would try to create the potential of the spectral problem rt to get more explicit equation. It can be better to research the essence of nature.

Acknowledgment

This work was supported by National Natural Science Foundation of China (No.11271007), SDUST Research Fund (Nos.2012KYTD105, 2014TDJH102), Young Teachers Support Program of SDUST.

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