冉 冊 t = t + d „ t , t … CompletelistofDarbouxintegrablechainsoftheform

Complete list of Darboux integrable chains of the form t

= t

+ d „t,t

…

Ismagil Habibullin,^1,a兲 Natalya Zheltukhina,^2,b兲 and Aslı Pekcan²

1Ufa Institute of Mathematics, Russian Academy of Science, Chernyshevskii Str., 112, Ufa 450077, Russia

2Department of Mathematics, Faculty of Science, Bilkent University,

06800 Ankara, Turkey

共Received 28 July 2009; accepted 23 September 2009; published online 30 October 2009兲

We study differential-difference equation 共d/dx兲t共n+1,x兲= f共t共n,x兲,t共n + 1 , x兲,共d/dx兲t共n,x兲兲 with unknown t共n,x兲 depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, 兵t共n+k,x兲其k=−⬁ ⬁, 兵共d^k/dx^k兲t共n,x兲其_k=1^⬁ , such that D_xF = 0 and DI = I, where D_xis the operator of total differentiation with respect to x and D is the shift operator: Dp共n兲=p共n+1兲. Re- formulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The com- plete list of Darboux integrable equations is given in the case when the function f is of the special form f共u,v,w兲=w+g共u,v兲. © 2009 American Institute of Physics.

关doi:10.1063/1.3251334兴

I. INTRODUCTION

In this paper we continue investigation of integrable semidiscrete chains of the form d

dxt共n + 1,x兲 = f

冉

dxt共n,x兲

冊

started in our previous paper¹共see also Refs.2–4兲. Here t=t共n,x兲 and t1= t共n+1,x兲 are unknown.

Function f = f共t,t1, tx兲 is assumed to be locally analytic and⳵^f/⳵^txis not identically zero. Nowa- days discrete phenomena are very popular due to their applications in physics, geometry, biology, etc.共see Refs.5–8and references therein兲.

Below we use subindex to indicate the shift of the discrete argument: tk= t共n+k,x兲, k苸Z, and derivatives with respect to x: t_关1兴= tx=共d/dx兲t共n,x兲, t_关2兴= txx=共d²/dx²兲t共n,x兲, t_关m兴=共d^m/dx^m兲t共n,x兲, m苸N. Introduce the set of dynamical variables containing 兵tk其_k=−^⬁ _⬁;兵t_关m兴其_m=1^⬁ .

We denote through D and Dxthe shift operator and the operator of the total derivative with respect to x correspondingly. For instance, Dh共n,x兲=h共n+1,x兲 and Dxh共n,x兲=共d/dx兲h共n,x兲.

Functions I and F, both depending on x and a finite number of dynamical variables, are called, respectively, n- and x-integrals of共1兲, if DI = I and DxF = 0共see also Ref.9兲. One can see that any n-integral I does not depend on variables tm, m苸Z\兵0其, and any x-integral F does not depend on variables t_关m兴, m苸N.

Chain 共1兲 is called Darboux integrable if it admits a nontrivial n-integral and a nontrivial x-integral.

Note that all Darboux integrable chains of the form共1兲are reduced to the d’Alembert equation w_1x− wx= 0 by the following “differential” substitution w = F + I. Indeed, Dx共D−1兲w=共D−1兲DxF + Dx共D−1兲I=0. This implies that two arbitrary Darboux integrable chains of the form共1兲

a兲Electronic mail: habibullinismagil@gmail.com.

b兲Electronic mail: natalya@fen.bilkent.edu.tr.

50, 102710-1

0022-2488/2009/50共10兲/102710/23/$25.00 © 2009 American Institute of Physics

u_1x= f共u,u1,ux兲, v1x= f˜共v,v1,vx兲 are connected with one another by the substitution

F共x,u,u1,u₋₁, . . .兲 + I共x,u,ux,uxx, . . .兲 = F˜共x,v,v1,v₋₁, . . .兲 + I˜共x,v,vx,vxx, . . .兲, which is evidently split down into two relations,

F共x,u,u1,u₋₁, . . .兲 = F˜共x,v,v1,v₋₁, . . .兲 − h, I共x,u,ux,uxx, . . .兲 = I˜共x,v,vx,vxx, . . .兲 + h, where h is some constant.

The idea of such kind integrability goes back to Laplace’s discovery of cascade method of integration of linear hyperbolic-type partial differential equation with variable coefficients made in 1773共see Ref.10兲. Roughly speaking the Laplace theorem claims that a linear hyperbolic partial differential equation admits general solution in a closed form if and only if its sequence of Laplace invariants terminates at both ends共see Ref.11兲. More than hundred years later Darboux applied the cascade method to the nonlinear case. He proved that a nonlinear hyperbolic equation is integrable 共Darboux integrable兲 if and only if the Laplace sequence of the linearized equation terminates at both ends. This result has been rediscovered very recently by Anderson and Kamran¹² and Sokolov and Zhiber.¹³

An alternative approach was suggested by Shabat and Yamilov in 1981 共see Ref.14兲. They assigned two Lie algebras, called characteristic Lie algebras, to each hyperbolic equation and proved that the equation is Darboux integrable if and only if both characteristic Lie algebras are of finite dimensions.

The purpose of the present article is to study characteristic Lie algebras of the chain 共1兲 introduced in our papers^2–4and convince the reader that in the discrete case these algebras provide a very effective classification tool.

We denote through Lxand Lncharacteristic Lie algebras in x- and n-directions, respectively.

Remind the definition of Lx. Rewrite first the chain 共1兲 in the inverse form tx共n−1,x兲

= g共t共n,x兲,t共n−1,x兲,tx共n,x兲兲. It can be done 共at least locally兲 due to the requirement 共⳵^f/⳵^tx兲 共t,t1, tx兲⫽0. An x-integral F=F共x,t,t_⫾1, t_⫾2, . . .兲 solves the equation DxF = 0. Applying the chain rule, one gets KF = 0, where

K = ⳵

⳵^{x+ tx}

⳵

⳵^{t+ f}

⳵

⳵^t1

+ g ⳵

⳵^t−1

+ f₁ ⳵

⳵^t2

+ g₋₁ ⳵

⳵^t−2

+ ¯ . 共2兲

Since F does not depend on the variable t_x, then XF = 0, where X =⳵/⳵^tx. Therefore, any vector field from the Lie algebra generated by K and X annulates F. This algebra is called the charac- teristic Lie algebra L_x of chain 共1兲 in x-direction. The notion of characteristic algebra is very important. One can prove that chain共1兲admits a nontrivial x-integral if and only if its Lie algebra L_xis of finite dimension. The proof of the next classification theorem from Ref.1is based on the finiteness of the Lie algebra L_x.

Theorem 1.1: Chain

t_1x= tx+ d共t,t1兲 共3兲

admits a nontrivial x-integral if and only if d共t,t1兲 is one of the following kinds:

共1兲 d共t,t1兲=A共t1− t兲,

共2兲 d共t,t1兲=c1共t1− t兲t+c2共t1− t兲²+ c₃共t1− t兲, 共3兲 d共t,t1兲=A共t1− t兲e^␣t,

共4兲 d共t,t1兲=c4共e^␣t1− e^␣t兲+c5共e^−␣t1− e^−␣t兲,

where A = A共t1− t兲 is an arbitrary function of one variable and␣⫽0, c1⫽0, c2, c₃, c₄⫽0, c₅⫽0 are arbitrary constants. Moreover, some nontrivial x-integrals in each of the cases are 共i兲 F = x +兰^t1−t共du/A共u兲兲, if A共u兲⫽0, F=t1− t, if A共u兲⬅0,

共ii兲

F = 1

共− c2+ c₁兲ln

冏

冏

ln

冏

冏

for c₂共c2− c₁兲⫽0,

F = ln

冏

冏

F =t₂− t₁

t₃− t₂+ ln

冏

冏

for c₂= c₁. 共iii兲

F =

冕

冕

F =共e^␣t− e^␣t2兲共e^␣t1− e^␣t3兲 共e^␣t− e^␣t3兲共e^␣t1− e^␣t2兲.

In what follows we study semidiscrete chains共3兲admitting not only nontrivial x-integrals but also nontrivial n-integrals. First of all we will give an equivalent algebraic formulation of the n-integral existence problem. Rewrite the equation DI = I defining n-integral in an enlarged form, I共x,t1, f, fx, . . .兲 = I共x,t,tx,txx, . . .兲. 共4兲 The left hand side contains the variable t₁ while the right hand side does not. Hence we have D⁻¹共d/dt1兲DI=0, i.e., the n-integral is in the kernel of the operator

Y₁= D⁻¹Y₀D, where

Y₁= ⳵

⳵^{t+ D−1共Y0f兲}

⳵

⳵^tx

+ D⁻¹Y₀共fx兲 ⳵

⳵^txx

+ D⁻¹Y₀共fxx兲 ⳵

⳵^txxx

+¯ 共5兲

and

Y₀= d

dt₁. 共6兲

It can easily be shown that for any natural j the equation D^−jY₀D^jI = 0 holds. Direct calculations show that

D^−jY₀D^j= X_j−1+ Yj, jⱖ 2, where

Y_j+1= D⁻¹共Yjf兲 ⳵

⳵^tx

+ D⁻¹Y_j共fx兲 ⳵

⳵^txx

+ D⁻¹Y_j共fxx兲 ⳵

⳵^txxx

+ ¯ , jⱖ 1, 共7兲

Xj= ⳵

⳵t_−j

, jⱖ 1. 共8兲

The following theorem defines the characteristic Lie algebra Ln of the chain共1兲.

Theorem 1.2:共Ref.3兲 Equation(1)admits a nontrivial n-integral if and only if the following two conditions hold.

共1兲 Linear space spanned by the operators 兵Yj其₁^⬁is of finite dimension, denote this dimension by 共2兲 Lie algebra LN. n generated by the operators Y₁, Y₂, . . . , YN, X₁, X₂, . . . , XN is of finite dimen-

sion. We call L_n the characteristic Lie algebra of(1) in the direction of n.

We use x-integral classification Theorem 1.1 and n-integral existence Theorem 1.2 to obtain the complete list of Darboux integrable chains of the form共3兲. The statement of this main result of the present paper is given in the next theorem.

Theorem 1.3: Chain(3)admits nontrivial x- and n-integrals if and only if d共t,t1兲 is one of the kind.

共1兲 d共t,t1兲=A共t1− t兲, where A共t1− t兲 is given in implicit form A共t1− t兲=共d/d␪兲P共␪兲, t1− t = P共␪兲, with P共␪兲 being an arbitrary quasipolynomial, i.e., a function satisfying an ordinary differ- ential equation,

P^共N+1兲=␮NP^共N兲+ ¯ +␮1P⬘⁺␮0P, with constant coefficients␮k, 0ⱕkⱕN.

共2兲 d共t,t1兲=C1共t₁²− t²兲+C2共t1− t兲.

共3兲 d共t,t1兲=

冑

where␣⫽0, Ci, 1ⱕiⱕ6, are arbitrary constants. Moreover, some nontrivial x-integrals F and n-integrals I in each of the cases are the following.

共i兲 F = x −兰^t1−tds/A共s兲, I=L共Dx兲tx, where L共Dx兲 is a differential operator which annihilates 共d/d␪兲P共␪兲 where Dx␪^{= 1.}

共ii兲 F =兵共t3− t₁兲共t2− t兲其/兵共t3− t₂兲共t1− t兲其, I=tx− C₁t²− C₂t.

共iii兲 F =兰^t1−te^−␣sds/

冑

冑

Equation of the form ␶x= A共␶兲, where ␶^{= t}1− t, is integrated in quadratures. But to get the final answer one should evaluate the integral and then find the inverse function. The general solution is given in an explicit form,

t共n,x兲 = t共0,x兲 +

兺

j=0 n−1

P共x + cj兲, 共9兲

where t共0,x兲 and cjare arbitrary functions of x and j, respectively, and A共␶兲= P⬘共␪兲, t1− t = P共␪兲.

Actually we have␶x= P_␪共␪兲␪x= P_␪共␪兲, which implies ␪x= 1, so that␶共n,x兲= P共x+cn兲. By solving the equation t共n+1,x兲−t共n,x兲= P共x+cn兲 one gets the answer above. Requirement for ␶x= A共␶兲 to be Darboux integrable induces condition on function P to satisfy a linear ordinary differential equation with constant coefficients.

The x-integrals in the cases共2兲 and 共4兲 given in Theorem 1.3 are written as cross ratios of four points t, t₁, t₂, t₃and, respectively, points e^t, e^t1, e^t2, e^t3. Due to the well known theorem, given four points z₁, z₂, z₃, z₄in the projective complex planeCP can be mapped to other given four points w₁, w₂, w₃, w₄by one and the same Möbius transformation,

z = R共w兲 ªa₁₁w + a₁₂

a₂₁w + a₂₂, 共10兲

such that zj= R共wj兲, where j=1,2,3,4, if and only if z₄− z₂

z₄− z₃ z₃− z₁

z₂− z₁=w₄− w₂ w₄− w₃

w₃− w₁

w₂− w₁. 共11兲

Evidently function F for the case共2兲 关as well as for the case 共4兲兴 can immediately be found from the equation I = c共x兲 which is equivalent to Riccati equation tx= C₁t²+ C₂t + c共x兲. It is well known that cross ratio of four different solutions of the Riccati equation does not depend on x.

Studying the examples below we briefly discuss connection between discrete models and their continuum analogs. The case共3兲 with C3= 1 and␣= 1 leads in the continuum limit to the equation

u_xy= e^u

冑

found earlier in Ref.13. Indeed set t共n,x兲=u共y,x兲 and C4= −2 +⑀², where y = n⑀. Then substitute

␶⁼⑀^uy+ O共⑀²兲 as⑀→0 into the equation t1x− tx= e^t

冑

The chain t_1x− t_x=共e^t1− e^t兲/2 goes to the equation uxy=¹₂e^uu_y in the continuum limit. Its n-integral I = tx−¹₂e^t coincides with the corresponding y-integral of the continuum analog. The Darboux integrable chain t_1x− tx= Ce^共t1+t兲/2关it comes from the case 共3兲 for appropriate choice of the parameters兴 being a discrete version of the Liouville equation uxy= e^u, also has a common integral I = 2txx− t_x² with its continuum limit equation. Note that the chain defines the Bäcklund transform for the Liouville equation.

Let us comment the list of the equations in Theorem 1.3. Case共1兲 is degenerate, it is reduced to a first order ordinary differential equation and easily integrated. Equation 共2兲 with C₂= 0 is given in Ref. 9. Case 共3兲 for C4=⫾C3 is found in Ref. 15. To the best of our knowledge Eqs.

共2兲–共4兲are new except these two cases.

The article is organized as follows. In Sec. II general results related to the Lie algebra L_nof Eq. 共1兲 are given. Section III is split into four subsections. Theorem 1.1 from Sec. I gives a complete list of Eq.共3兲admitting nontrivial x-integral. This list consists of four different types of equations共3兲. In each subsection of Sec. III one of these four different types from Theorem 1.1 is treated by imposing additional condition for an equation to possess nontrivial n-integrals. The conclusion is provided in Sec. IV.

II. GENERAL RESULTS

Define a class F of locally analytic functions each of which depends only on a finite number of dynamical variables. In particular, we assume that f共t,t1, tx兲苸F. We will consider vector fields given as infinite formal series of the form

Y =

兺

k=0

⬁

yk

⳵

⳵^t_关k兴 ^共13兲

with coefficients y_k苸F. Introduce notions of linearly dependent and independent sets of the vector fields共13兲. Denote through PNthe projection operator acting according to the rule

PN共Y兲 =

兺

k=0 N

yk

⳵

⳵^t关k兴. 共14兲

First we consider finite vector fields as

Z =

兺

k=0 N

zk

⳵

⳵^t关k兴. 共15兲

We say that a set of finite vector fields Z₁, Z₂, . . . , Zmis linearly dependent in some open region U, if there is a set of functions ␭1,␭2, . . . ,␭m苸F defined on U such that the function 兩␭1兩²+兩␭2兩² +¯+兩␭m兩² does not vanish identically and the condition

␭1Z₁+␭2Z₂+ ¯ + ␭mZm= 0 共16兲

holds for each point of region U.

We call a set of the vector fields Z₁, Z₂, . . . , Zm of the form 共13兲 linearly dependent in the region U if for each natural N the following set of finite vector fields PN共Z1兲, PN共Z2兲, ... , PN共Zm兲 is linearly dependent in this region. Otherwise we call the set Z₁, Z₂, . . . , Zmlinearly independent in U.

Now we give some properties of the characteristic Lie algebra introduced in Theorem 1.2. The proof of the first two lemmas can be found in Ref.3. However, for the reader’s convenience we still give the proof of the second lemma.

Lemma 2.1: If for some integer N the operator Y_N+1is a linear combination of the operators Yiwith iⱕN: YN+1=␣1Y₁+␣2Y₂+ . . . +␣NYN, then for any integer j⬎N, we have a similar expres- sion Yj=␤1Y₁+␤2Y₂+¯+␤NYN.

Lemma 2.2: The following commutativity relations take place: 关Y0, X₁兴=0, 关Y0, Y₁兴=0, and 关X1, DX₁D⁻¹兴=0.

Proof: We have

关Y0,X₁兴 =

冋

, d dt₋₁

册

关Y0,Y₁兴 = D⁻¹关DY0D⁻¹,Y₀兴D = D⁻¹

冋

, d

dt₁

册

关X1,DX₁D⁻¹兴 = D关D⁻¹X₁D,X₁兴D⁻¹= D关X2,X₁兴D⁻¹= 0.

䊐 Note that

Y_k+1= D⁻¹YkD, kⱖ 2, D⁻¹Y₁D = X₁+ Y₂. 共17兲 The next three statements turned out to be very useful for studying the characteristic Lie algebra L_n.

Lemma 2.3: (Reference 1) If the Lie algebra generated by the vector fields S₀=兺^⬁_j=−⬁⳵/⳵^wjand S₁=兺j=−⬁ ⬁c共wj兲⳵/⳵^wj is of finite dimension then c共w兲 is one of the forms

共1兲 c共w兲=a1+ a₂e^␭w+ a₃e^−␭w,

共2兲 c共w兲=a1+ a₂w + a₃w², where␭⫽0, a1, a₂, and a₃are some constants.

Lemma 2.4:

共1兲 Suppose that the vector field

Y =␣共0兲⳵

⳵^t+␣共1兲 ⳵

⳵^tx

+␣共2兲 ⳵

⳵^txx

+ ¯ ,

where␣x共0兲=0 solves the equation 关Dx, Y兴=兺_{k=−⬁,k⫽0}^⬁ ␤共k兲⳵/⳵^tk, then Y =␣共0兲⳵/⳵^t.

共2兲 Suppose that the vector field

Y =␣共1兲 ⳵

⳵^tx

+␣共2兲 ⳵

⳵^txx

+␣共3兲 ⳵

⳵^txxx

+¯

solves the equation关Dx, Y兴=hY +兺_k=−^⬁ _⬁,k⫽0␤共k兲⳵/⳵^tk, where h is a function of variables t, tx, txx, . . ., t_⫾1, t_⫾2, . . ., then Y = 0.

Lemma 2.5: For any mⱖ0, we have

关Dx,Ym兴 = −

兺

j=1 m

D^−j共Ym−j共f兲兲Yj−

兺

k=1

⬁

Ym共D^{−共k−1兲}g兲 ⳵

⳵^t−k

−

兺

k=1

⬁

Ym共D^k−1f兲 ⳵

⳵^tk

. 共18兲

In particular,

关Dx,Y₀兴 = −

兺

k=1

⬁

Y₀共D^k−1f兲 ⳵

⳵^tk

, 共19兲

关Dx,Y₁兴 = − D⁻¹共Y0共f兲兲Y1−

兺

k=1

⬁

Y₁共D^{−共k−1兲}g兲 ⳵

⳵^t−k

−

兺

k=1

⬁

Y₁共D^k−1f兲 ⳵

⳵^tk

. 共20兲

Both Lemmas 2.4 and 2.5 easily can be derived from the following formula

关Dx,Y兴 = 共␣x共0兲 −␣共1兲兲⳵

⳵^t−

兺

k=1

⬁

Y共D^{−共k−1兲}g兲 ⳵

⳵^t−k

−

兺

k=1

⬁

Y共D^k−1f兲 ⳵

⳵^tk

+

兺

k=1

⬁

共␣x共k兲 −␣共k + 1兲兲 ⳵

⳵^t关k兴. 共21兲 Suppose that Eq. 共1兲admits a nontrivial n-integral. Then, by Theorem 1.2, its characteristic Lie algebra Ln is of finite dimension. Linear space of the basic vector fields 兵Yk其₁^⬁ is also finite dimensional. We have the following theorem.

Theorem 2.6: Dimension of span兵Yk其₁^⬁is finite and equal, say N if and only if the following system of equations is consistent:

Dx共␭N兲 = ␭N共AN,N− A_N+1,N+1兲 − AN+1,N,

Dx共␭N−1兲 = ␭N−1共AN−1,N−1− A_N+1,N+1兲 + ␭NA_N,N−1− A_N+1,N−1,

Dx共␭N−2兲 = ␭N−2共AN−2,N−2− A_N+1,N+1兲 + ␭N−1A_N−1,N−2+␭NA_N,N−2− A_N+1,N−2,

]

Dx共␭2兲 = ␭2共A2,2− A_N+1,N+1兲 + ␭3A_3,2+ ¯ + ␭NA_N,2− A_N+1,2,

Dx共␭1兲 = ␭1共A1,1− A_N+1,N+1兲 + ␭2A_2,1+␭3A_3,1+ ¯ + ␭NA_N,1− A_N+1,1.

0 =␭1A_1,0+␭2A_2,0+␭3A_3,0+ ¯ + ␭NA_N,0− A_N+1,0. 共22兲 Here A_k,j= D^−j共Yk−jf兲.

Proof: Suppose that the dimension of span兵Yk其₁^⬁ is finite, say N, then, by Lemma 2.1, Y₁, . . . , Y_Nform a basis in this linear space. So we can find factors␭1, . . . ,␭Nsuch that

Y_N+1=␭1Y₁+␭2Y₂+ ¯ + ␭NYN. 共23兲 Take the commutator of both sides with Dxand get by using the main commutativity relation共18兲 the following equation:

−

兺

j=0 N+1

A_N+1,jYj= Dx共␭1兲Y1+ Dx共␭2兲Y2+ ¯ + Dx共␭N兲YN

−

冉

^兺

^兺

^兺

冊

Now replace Y_N+1at the left hand side by共23兲and collect coefficients of the independent vector fields to derive the system given in the theorem.

Suppose now that the system共22兲in the theorem has a solution. Let us prove that the vector field Y_N+1is expressed in the form共23兲. Let

Z = Y_N+1−␭1Y₁−␭2Y₂− ¯ − ␭NYN. 共24兲 Let us find关Dx, Z兴.

关Dx,Z兴 = 关Dx,Y_N+1兴 − Dx共␭1兲Y1− ¯ − Dx共␭N兲YN−␭1关Dx,Y₁兴 − ␭2关Dx,Y₂兴 − ¯ − ␭N关Dx,YN兴

= −

兺

j=0 N+1

A_N+1,jY_j− D_x共␭1兲Y1− ¯ − Dx共␭N兲YN

−

冉

^兺

^兺

^兺

冊

^兺

Replace now Dx共␭1兲, ... ,Dx共␭N兲 by means of the system共22兲. After some simplifications one gets

关Dx,Z兴 = − AN+1,N+1Z +

兺

k=−⬁,k⫽0

⬁

␤共k兲 ⳵

⳵^tk

. 共25兲

By Lemma 2.4 we get Z = 0. 䊐

The proof of the next three results can be found in Ref.4.

Lemma 2.7: If the operator Y₂= 0 then关X1, Y₁兴=0.

The reverse statement to Lemma 2.7 is not true as the equation t_1x= t_x+ e^tshows共see Lemma 3.4 below兲.

Lemma 2.8: The operator Y₂= 0 if and only if we have

f_t+ D⁻¹共ft₁兲ft_x= 0. 共26兲

Corollary 2.9: The dimension of the Lie algebra Lnassociated with n-integral is equal to 2 if and only if共26兲holds, or the same Y₂= 0.

Now let us introduce vector fields

C₁=关X1,Y₁兴, Ck=关X1,C_k−1兴, k ⱖ 2. 共27兲 It is easy to see that

Cm= X₁^mD⁻¹共Y0共f兲兲 ⳵

⳵^tx

+ X₁^mD⁻¹共Y0Dx共f兲兲 ⳵

⳵^txx

+ X₁^mD⁻¹共Y0D_x²共f兲兲 ⳵

⳵^txxx

+ ¯ . 共28兲

Lemma 2.10: We have

关Dx,Cm兴 = − gt_xX₁^mD⁻¹Y₀共f兲X1− X₁^mD⁻¹Y₀共f兲Y1−

兺

j=1 m

A^共m兲_j Cj, 共29兲 where

A_j^共m兲= X₁^m−j

再

gt_x

冎

k !共m − k兲!. In particular,

关Dx,C₁兴 = − gt_xX₁D⁻¹Y₀共f兲X1− X₁D⁻¹Y₀共f兲Y1−

冉

冊

Proof: We prove the lemma by induction on m. Note that for any vector field,

A =␤共0兲⳵

⳵^t+␤共1兲 ⳵

⳵^tx

+␤共2兲 ⳵

⳵^txx

+ ¯ ,

acting on the set of functions H depending on variables t₋₁, t, t_关k兴, k苸N, formula共21兲becomes

关Dx,A兴 = − 共␤共0兲gt+␤共1兲gt_x兲 ⳵

⳵^t−1

+共␤x共0兲 −␤共1兲兲⳵

⳵^t+共␤x共1兲 −␤共2兲兲 ⳵

⳵^tx

+共␤x共2兲 −␤共3兲兲 ⳵

⳵^txx

+共␤x共3兲 −␤共4兲兲 ⳵

⳵^txxx

+ ¯ .

Applying the last formula with C₁instead of A, we have

关Dx,C₁兴 = − gt_xX₁D⁻¹Y₀共f兲X1− X₁D⁻¹Y₀共f兲⳵

⳵^t+

兺

k=1

⬁

兵DxX₁D⁻¹Y₀D_x^k−1共f兲 − X1D⁻¹Y₀D_x^k共f兲其 ⳵

⳵^t_关k兴^. Since

关Y0,Dx兴G共t,t1,tx,txx,txxx, . . .兲 = ft₁Gt₁= ft₁Y₀G, i.e., Y₀Dx= DxY₀+ ft₁Y₀ and

关Dx,X₁兴H共t−1,t,t_x,t_xx,t_xxx, . . .兲 = − gt₋₁H_t

−1= − g_t

−1X₁H, then

DxX₁D⁻¹Y₀D_x^k−1共f兲 − X1D⁻¹Y₀D_x^k共f兲 = 兵DxX₁D⁻¹Y₀− X₁D⁻¹Y₀Dx其Dxk−1共f兲 = 兵DxX₁D⁻¹Y₀

− X₁D⁻¹兵DxY₀+ ft₁Y₀其其Dx k−1共f兲

=关Dx,X₁兴D⁻¹Y₀D_x^k−1共f兲X1共D⁻¹Y₀共f兲兲D⁻¹Y₀D_x^k−1共f兲

− D⁻¹共Y0共f兲兲X1D⁻¹Y₀D_x^k−1共f兲 = − gt₋₁X₁D⁻¹Y₀D_x^k−1共f兲

− X₁D⁻¹共Y0共f兲兲D⁻¹Y₀D_x^k−1共f兲 − D⁻¹共Y0共f兲兲X1D⁻¹Y₀D_x^k−1共f兲.

Therefore,

关Dx,C₁兴 = − gt_xX₁D⁻¹Y₀共f兲X1− X₁D⁻¹Y₀共f兲⳵

⳵^t−

兺

k=1

⬁

X₁共D⁻¹Y₀共f兲兲D⁻¹Y₀D_x^k−1共f兲 ⳵

⳵^t_关k兴

− gt₋₁

兺

k=1

⬁

X₁D⁻¹Y₀D_x^k−1共f兲 ⳵

⳵^t关k兴− D⁻¹共Y0共f兲兲

兺

k=1

⬁

X₁D⁻¹Y₀D_x^k−1共f兲 ⳵

⳵^t关k兴= − gt_xX₁D⁻¹Y₀共f兲X1

− X₁D⁻¹Y₀共f兲Y1−共gt₋₁+ D⁻¹共ft₁兲兲C1

that proves the base of mathematical induction. Assuming Eq.共29兲is true for m − 1, we have 关Dx,Cm兴 = 关Dx,关X1,C_m−1兴兴 = − 关X1,关Cm−1,Dx兴兴 − 关Cm−1,关Dx,X₁兴兴 = 关X1,关Dx,C_m−1兴兴 + 关Cm−1,gt₋₁X₁兴

=关X1,关Dx,C_m−1兴兴 + Cm−1共gt₋₁兲X1− gt₋₁Cm=

冋

−

兺

j=1 m−1

A^共m−1兲_j Cj

册

− gt_xX₁^mD⁻¹Y₀共f兲X1− X₁^mD⁻¹Y₀共f兲Y1− X₁^m−1D⁻¹Y₀共f兲C1−

兺

j=1 m−1

X₁共Aj共m−1兲兲Cj

−

兺

j=1 m−1

A^共m−1兲_j C_j+1+ g_t

−1t_xX₁^m−1D⁻¹Y₀共f兲X1− g_t

−1C_m= − g_t

xX₁^mD⁻¹Y₀共f兲X1− X₁^mD⁻¹Y₀共f兲Y1

−兵A_m−1^共m−1兲+ gt₋₁其Cm−兵X₁^m−1D⁻¹Y₀共f兲 + X1共A₁^共m−1兲兲其C1−

兺

j=2 m−1

兵X1共A共m−1兲j 兲 + A^共m−1兲_j−1 其Cj

= − gt_xX₁^mD⁻¹Y₀共f兲X1− X₁^mD⁻¹Y₀共f兲Y1−

兺

j=1 m

A^共m兲_j Cj, where

A₁^共m兲= X₁^m−1D⁻¹Y₀共f兲 + X1共A₁^共m−1兲兲 = X₁^m−1

再

冎

再

gt_x

冎

= X₁^m−1

再

冎

A^共m兲_j = X₁共Aj共m−1兲兲 + A共m−1兲j−1 = X₁X₁^m−1−j

再

g_t

x

冎

+ X₁^m−j

再

gt_x

冎

再

gt_x

冎

A_m^m= A_m−1^共m−1兲+ gt₋₁=共m − 1兲gt₋₁− gt

gt_x

+ gt₋₁= mgt₋₁− gt

gt_x

that finishes the proof of the lemma. 䊏

Assume equation t_1x= f共t,t1, tx兲 admits a nontrivial n-integral. Then we know that the dimen- sion of Lie algebra L_nis at least 2 by Corollary 2.9.