Conservation Laws Related to the Kac-Moody-Virasoro Structure of the Potential Nizhnik-Novikov-Veselov Equation

Conservation Laws Related to the Kac-Moody-Virasoro Structure of the Potential Nizhnik-Novikov-Veselov Equation

Xi-Zhong Liu

Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China Reprint requests to X.-Z. L.; E-mail: liuxizhong123@163.com

Z. Naturforsch.66a,297 – 303 (2011); received June 22, 2010 / revised October 6, 2010

We derive the symmetry group by the standard Lie symmetry method and prove it constitutes to the Kac-Moody-Virasoro algebra. Then we construct the conservation laws corresponding to the Kac- Moody-Virasoro symmetry algebra up to second-order group invariants.

Key words:Potential Nizhnik-Novikov-Veselov Equation; Lie Point Symmetry Group;

Conservation Laws.

PACS numbers:02.30.Jr, 47.10.ab, 02.30.Ik

1. Introduction

It is of fundamental importance to obtain conserva- tion laws of a given nonlinear system. As for the soliton theory, conservation laws facilitate the study of qualita- tive properties of partial differential equations (PDEs), such as bi- or tri-Hamiltonian structures. They often guide the choice of solution methods or reveal the na- ture of special solutions. For example, the existence of a large number of conservation laws is a predictor for complete integrability of the PDE [1], i. e. solvability by the inverse scattering transform [1] and the exis- tence of solitons [2]. Conserved densities also aid in the design of numerical solvers for PDEs [3].

There are various methods to compute conservation laws of nonlinear PDEs. A common approach relies on the link between conservation laws and symmetries as stated in Noether’s theorem [4 – 6]. In the classi- cal Noether theorem [7] it is valid that if a given sys- tem of differential equations has a variational principle, then a continuous symmetry (point, contactor higher order) that leaves the action functional within a diver- gence invariant yields a conservation law [8 – 11]. In the last few years, effective methods have been devised for finding conservation laws for the very special class of so-called Lax equations. In 2003, Anco and Bluman gave the multiplier method for finding the local conser- vation laws of PDEs [12, 13]. In 2000, Kara and Ma- homed [14] presented the direct relationship between the conserved vector of a PDE and the Lie-B¨acklund symmetry generators of the PDE, from which it is pos- sible for us to obtain conservation laws from symme-

0932–0784 / 11 / 0500–0297 $ 06.00 c2011 Verlag der Zeitschrift f¨ur Naturforschung, T ¨ubingen·http://znaturforsch.com

tries (see, e. g., [15]). In the new method of Kara and Mahomed, one can use any Lie-B¨acklund operator of a PDE system to generate conservation laws without converting the operator to a canonical one, so a point symmetry generator remains of point type. Another ad- vantage of this approach is that for point symmetry the order of the generated conservation Law remains the same as for the original one remains the same. Further- more, by using this method one can prove that the Lie algebra of Lie-B¨acklund symmetry generators of the conserved form is a subalgebra of the symmetries of the system itself. In this letter, we discuss the conserva- tion laws relating to the Kac-Moody-Virasoro algebra (KMVA) of the Nizhnik-Novikov-Veselov equation (NNVE) [16 – 19] using the Lie-B¨acklund operator.

The Nizhnik-Novikov-Veselov equation

v_t+v_xxx+v_yyy+3(v∂y⁻¹v_x)x+3(v∂x⁻¹v_y)y=0, (1) whereinxandyoccur in a symmetric manner, is the famous extension of the Kortheweg-de Vries (KdV) equation

u_t+6uu_x+u_xxx=0. (2) Various integrable properties such as the inverse scat- tering transformation, B¨acklund transformation, the soliton-like solutions, and symmetry algebra of (1) are discussed by various authors [20 – 24]. We would like to derive the conservation laws related to the symme- tries and algebras for the NNVE (1) in the potential form (v=u_xy)

u_txy+u_xxxxy+u_xyyyy+3(u_xyu_xx)x+3(u_xyu_yy)y=0. (3)

This paper is organized as follows. In Section 2, we derive the symmetry group to the potential NNVE (PN- NVE) by using the classical Lie method and prove the constitute infinite dimensional Kac-Moody-Virasoro symmetry algebra [25]. In Section 3, we first review some basic notions about Lie-B¨acklund operators and then use them. We finally obtain conservation laws re- lated to the infinite dimensional Kac-Moody-Virasoro symmetry algebra as PNNVE possesses up to second- order group invariants. It is emphasized that equations with the same symmetries may possess the same types of conservation laws. The last section is a short sum- mary and discussion.

2. Lie Point Symmetries and Kac-Moody-Virasoro Structure of the Potential Nizhnik-Novikov- Veselov Equation

To study the symmetry of (3), we search for the Lie point symmetry transformations in the vector form

V=X ∂

∂^x+Y

∂

∂^y+T

∂

∂^t+U

∂

∂^u,

whereX,Y,T, andUare functions with respect tox,y, t,u, which means that (3) is invariant under the point transformation

{x,y,t,u} → {x+ε^X,y+ε^Y,t+ε^T,u+ε^U} with infinitesimal parameterε^.

In other words, the symmetry of (3) can be written in the function form

σ=X u_x+Yu_y+Tu_t−U, (4) where the symmetryσ is a solution of the linearized form of (3),

σxyt+σxxxxy+σxyyyy+3uxxyσxx+3σxxyuxx

+3uxyσxxx+3σxyuxxx+3uxyyσyy+3σxyyuyy

+3u_xyσyyy+3σxyu_yyy=0,

(5)

which is obtained by substitutingu=u+εσ ^{into (3)} and dropping the nonlinear terms inσ^.

It is easy to determine X(x,y,t,u), Y(x,y,t,u), T(x,y,t,u), andU(x,y,t,u)by substituting (4) into (5) and eliminatingu_xxxxy and its higher-order derivatives by means of the PNNVE (3). The results are:

X(x,y,t,u) =1

3T_tx+X, (6)

Y(x,y,t,u) =1

3Tty+Y, (7)

T(x,y,t,u) =T, (8) U(x,y,t,u) = 1

54(x³+y³)Ttt+1

6Yty²+1 6Xtx² +Z₁x+Z₂y+Z₃,

(9)

whereX,Y,T,Z₁,Z₂, andZ₃are arbitrary functions oft.

The vector form of the Lie point symmetries reads V=

1 3T_t(t)x∂

∂^x+ 1 3T_ty∂

∂^y+T ∂

∂^t

−1

54(T_ttx³+T_tty³) ∂

∂^u

+

Y ∂

∂^y− 1 6Y_ty²∂

∂^u

+

X ∂

∂^x− 1 6X_tx² ∂

∂^u

+

−Z₁x∂

∂^u

+

−Z₂y∂

∂^u

+

−Z₃ ∂

∂^u

≡V₁(T(t)) +V₂(Y(t)) +V₃(X(t)) +V₄(Z₁(t)) +V₅(Z₂(t)) +V₆(Z₃(t)).

(10)

It is easy to verify that the symmetriesV_i,i=1, 2, 3, 4, 5, 6, constitute an infinite dimensional Kac-Moody- Virasoro [25] type symmetry algebraSSS with the fol- lowing non-zero commutation relations:

[V₁(T),V₆(Z₃)] =V₆(T Z_3t), (11) [V₂(Y), 5(Z₂)] =V₆(Y Z₂), (12) [V₁(T),V₅(Z₂)] =V₅(T Z_2t+1

3T_tZ₂), (13) [V₄(Z₁),V₁(T)] =V₄

−1

3T_tZ₁−T Z_1t

, (14)

[V₃(X),V₁(T)] =V₃

−T X_t+1 3X T_t

, (15)

[V₂(Y),V₁(T)] =V₂ 1

3Y Tt−YtT

, (16)

[V₁(T₁),V₁(T₂)] =V₁(T₁T_2t−T₂T_1t), (17) [V₂(Y₁),V₂(Y₂)] =V₅

1

3(Y₁Y_2t−Y₂Y_1t)

, (18)

[V₄(Z₁),V₃(X)] =V₆(−X Z₁), (19) [V₃(X₁),V₃(X₂)] =V₄

1

3(X₁X_2t−X₂X_1t)

. (20) It should be emphasized that the algebra is infinite dimensional because the generatorsV₁,V₂,V₃,V₄,V₅, andV₆all containarbitraryfunctions. The algebra is closed because all the commutators can be expressed by the generators belonging to the generator set usually with different functions, and the generators contained differentfunctions belonging to the set. Especially, it is clear that the symmetryV₁(T)constitute an centerless Virasoro symmetry algebra.

3. Conservation Laws Related to the KMVA of PNNVE

In order to obtain conservation laws related to the symmetry (10), we need some basic notions about Lie- B¨acklund operators first.

A Lie-B¨acklund operator is given by X₀=ξⁱ ∂

∂^xi+η ∂

∂^u+ζi ∂

∂^ui+ζi1i2

∂

∂^ui₁i₂+···, (21) whereξ^i,ηand the additional coefficients are

ζi=D_i(W) +ξ^jui j,

ζi₁i₂ =D_i1i2(W) +ξ^juji₁i₂. (22) W is the Lie characteristic function defined by

W =η−ξ^juj (23)

with D_ibeing the operator of total differentiation, D_i= ∂

∂^xi+ui ∂

∂^{u+ui j} ∂

∂^uj+···, i=1,···,n,

(24)

as

ui=Di(u), ui j=DjDi(u). (25) These definitions and results related to the Lie-B¨ack- lund operator can be found in [26] and the repeated indices mean the summations according to the Einstein summation rule.

Correspondingly, the second-order Lie-B¨acklund operator of the vector fieldV defined by (10) is given

by

X₀=ξ^x ∂

∂^x+ξ^y ∂

∂^y+ξ^t ∂

∂^t+η ∂

∂^u+ζx ∂

∂^ux

+ζy ∂

∂^uy+ζt ∂

∂^ut+ζu ∂

∂^uxx+ζxy ∂

∂^uxy

+ζxt ∂

∂^uxt+ζyy ∂

∂^uyy+ζyt ∂

∂^uyt+ζtt ∂

∂^utt, (26)

where the coefficientsξ^x, ξ^y,···,ζtt can be derived straightforwardly from (21) – (25).

Theorem([14, 27]): Suppose thatX₀is a Lie-B¨ack- lund symmetry of (3) such that the conservation vector T= (T₁, T₂, T₃)is invariant underX₀. Then

X₀(T_i) +

∑

³

j=1

T_iD_jξj−

∑

³

j=1

T_jD_j(ξi) =0, i=1, 2, 3,

(27)

where D₁=D_x, D₂=D_y, D₃=D_t, andξi are deter- mined by (26).

A Lie-B¨acklund symmetryX₀ is said to be associ- ated with a conserved vectorTof (3) ifX₀andTsat- isfy relations (27).

Now, we construct the corresponding conservation laws relating to (10) in the form

D_xJ₁+D_yJ₂+D_tρ=0, (28) whereT₁=J₁,T₂=J₂,T₃=ρ^withJ1,J₂, andρ^being functions of{x,y,t,u,u_x,u_y,···,u_tt}.

In terms ofT= (J₁,J₂,ρ), (27) is equivalent to the following three equations:

1 3xT_t+X

∂^J1

∂^{x +} 1

3yT_t+Y ∂^J1

∂^{y +T}∂^J1

∂^t +

1

54(x³+y³)T_tt+1 6Y_ty²+1

6X_tx²+Z₁x+Z₂y+Z₃

·∂^J1

∂^{u +} 1

18x²T_tt+1

3X_tx+Z₁−1 3T_tu_x

∂^J1

∂^ux

+ 1

18y²Ttt+1

3Yty+Z₂−1 3Ttuy

∂^J1

∂^uy

+ 1

54(x³+y³)T_ttt+1

6Y_tty²+1

6X_ttx²+Z_1tx+Z_2ty+Z_3t

− 1

3T_ttx+X_t

u_x− 1

3T_tty+Y_t

u_y−T_tu_t ∂^J1

∂^ut

+ 1

9T_ttx+1 3X_t−2

3T_tu_xx ∂^J1

∂^uxx−2

3T_tu_xy∂^J1

∂^uxy

+ 1

18x²T_ttt+1

3X_ttx+Z_1t−1

3T_ttu_x−4 3T_tu_tx−

1

3T_ttx+X_t

u_xx− 1

3T_tty+Y_t

u_xy ∂^J1

∂^uxt (29) +

1 9Ttty+1

3Yt−2 3Ttuyy

∂^J1

∂^uyy+ 1

18y²Tttt+1

3Ytty+Z_2t−1

3Tttuy−4 3Ttuyt−

1

3Tttx+Xt

uxy

− 1

3T_tty+Y_t

u_yy ∂^J1

∂^uyt+ 1

54(x³+y³)T_tttt+1

6Y_ttty²+1

6X_tttx²+Z_1ttx+Z_2tty+Z_3tt− 1

3T_tttx+X_tt

u_x

−2 1

3T_ttx+X_t

u_xt− 1

3T_ttty+Y_tt

u_y−2 1

3T_tty+Y_t

u_yt−T_ttu_t−2T_tu_tt ∂^J1

∂^utt +4 3T_tJ₁−

1

3T_ttx+X_t

ρ=0, 1

3xT_t+X ∂^J2

∂^{x +} 1

3yT_t+Y ∂^J2

∂^{y +T}∂^J2

∂^{t +} 1

54(x³+y³)T_tt+1

6Y_ty²+1

6X_tx²+Z₁x+Z₂y+Z₃ ∂^J2

∂^u +

1

18x²T_tt+1

3X_tx+Z₁−1 3T_tu_x

∂^J2

∂^ux+ 1

18y²T_tt+1

3Y_ty+Z₂−1 3T_tu_y

∂^J2

∂^uy+ 1

54(x³+y³)T_ttt+1 6Y_tty² +1

6X_ttx²+Z_1tx+Z_2ty+Z_3t− 1

3T_ttx+X_t

u_x− 1

3T_tty+Y_t

u_y−T_tu_t ∂^J2

∂^ut + 1

9T_ttx+1 3X_t−2

3T_tu_xx ∂^J2

∂^uxx

−2

3T_tu_xy∂^J2

∂^uxy+ 1

18x²T_ttt+1

3X_ttx+Z_1t−1

3T_ttu_x−4 3T_tu_tx−

1

3T_ttx+X_t

u_xx− 1

3T_tty+Y_t

u_xy ∂^J2

∂^uxt (30) +

1 9T_tty+1

3Y_t−2 3T_tu_yy

∂^J2

∂^uyy+ 1

18y²T_ttt+1

3Y_tty+Z_2t−1

3T_ttu_y−4 3T_tu_yt−

1

3T_ttx+X_t

u_xy

− 1

3T_tty+Y_t

u_yy ∂^J2

∂^uyt+ 1

54(x³+y³)T_tttt+1

6Y_ttty²+1

6X_tttx²+Z_1ttx+Z_2tty+Z_3tt− 1

3T_tttx+X_tt

u_x

−2 1

3T_ttx+X_t

u_xt− 1

3T_ttty+Y_tt

u_y−2 1

3T_tty+Y_t

u_yt−T_ttu_t−2T_tu_tt ∂^J2

∂^utt

+4 3T_tJ₂−

1 3T_tty+Y_t

ρ=0, 1

3xT_t+X ∂ρ

∂^x+ 1

3yT_t+Y ∂ρ

∂^{y +T}

∂ρ

∂^{t +} 1

54(x³+y³)T_tt+1

6Y_ty²+1

6X_tx²+Z₁x+Z₂y+Z₃ ∂ρ

∂^u +

1

18x²T_tt+1

3X_tx+Z₁−1 3T_tu_x

∂ρ

∂^ux+ 1

18y²T_tt+1

3Y_ty+Z₂−1 3T_tu_y

∂ρ

∂^uy+ 1

54(x³+y³)T_ttt+1 6Y_tty² +1

6X_ttx²+Z_1tx+Z_2ty+Z_3t− 1

3T_ttx+X_t

u_x− 1

3T_tty+Y_t

u_y−T_tu_t ∂ρ

∂^ut+ 1

9T_ttx+1 3X_t−2

3T_tu_xx ∂ρ

∂^uxx

−2

3T_tu_xy ∂ρ

∂^uxy+ 1

18x²T_ttt+1

3X_ttx+Z_1t−1

3T_ttu_x−4 3T_tu_tx−

1

3T_ttx+X_t

u_xx− 1

3T_tty+Y_t

u_xy ∂ρ

∂^uxt (31) +

1 9T_tty+1

3Y_t−2 3T_tu_yy

∂ρ

∂^uyy+ 1

18y²T_ttt+1

3Y_tty+Z_2t−1

3T_ttu_y−4 3T_tu_yt−

1

3T_ttx+X_t

u_xy

− 1

3T_tty+Y_t

u_yy ∂ρ

∂^uyt+ 1

54(x³+y³)T_tttt+1

6Y_ttty²+1

6X_tttx²+Z_1ttx+Z_2tty+Z_3tt− 1

3T_tttx+X_tt

u_x

−2 1

3T_ttx+X_t

u_xt− 1

3T_ttty+Y_tt

u_y−2 1

3T_tty+Y_t

u_yt−T_ttu_t−2T_tu_tt ∂ρ

∂^utt +2

3T_tρ=0. The solutionsJ₁,J₂, andρof (29) – (31) can be directly

solved:

ρ=f₀(t)K₁(t₁,t₂,t₃,···,t₁₂), (32)

J₂= [f₁(t) +f₂(t)y]K₁(t₁,t₂,t₃,···,t₁₂) + f₃(t)K₂(t₁,t₂,t₃,···,t₁₂), (33)

J₁= [f₄(t) +f₅(t)x]K₁(t₁,t₂, t₃,···,t₁₂) + f₆(t)K₃(t₁, t₂,t₃, ···,t₁₂), (34) where K₁, K₂, and K₃ are arbitrary functions of {t₁,t₂,t₃, ···,t₁₂}, and f_i,i=0,1,···, 10 are func- tions fixed by:

f₀=T⁻²³, (35)

f₁=Y T⁻⁵³, f₂=1

3T_tT⁻⁵³, f₃=T⁻⁴³, (36) f₄=X_1tT⁻¹³, f₅=1

3T_tT⁻⁵³, f₆=T⁻⁴³, (37) with the invariants being

t₁= (x−T¹³X₁)T⁻¹³, (38) t₂= (y−Y₁T¹³)T⁻¹³, (39) t₃= 1

54T⁻⁴³

54uT⁴³−T_tT¹³y³−T_tx³T¹³

−9X T¹³x²−9Y y²T¹³ −54yY₂T+18yY₃T

−54Y₄T⁴³+54Y₅T⁴³−18Y₆T⁴³

+54Y₇T⁴³−18Y₈T⁴³−54xY₉T+18xY₁₀T ,

(40)

t₄= 1

18T⁻²³ 18Tu_x−6X x−T_tx²

−18Y₁₀T²³+6Y₁₁T²³

, (41)

t₅=−1

18T⁻²³ −18u_yT+T_ty²+6Y y +18Y₂T²³−6Y₃T²³

,

(42)

t₆=u_tT+u_yY+1

3u_yT_ty+X u_x+1 3T_txu_x

− 1

54T_tty³− 1

54T_ttx³−1

6X_tx²−1 6Y_ty²

−Z₂y−Z₁x−Z₃,

(43)

t₇=−1

9T⁻¹³(−9u_xxT+3X+T_tx), (44)

t₈=u_xyT²³, (45)

t₉=−1

18T¹³(−18u_xtT−18u_xyY−6u_xyyT_t

−18u_xxX−6u_xxxT_t+6X_tx+18Z₁

−6T_tu_x+x²T_tt),

(46)

t₁₀=−1

9T⁻¹³(−9Tu_yy+3Y+T_ty), (47)

t₁₁=−1

18T¹³(−18Tuyt−6xuxyTt−18Yuyy

−6T_tu_yyy−18X u_xy+18Z₂−6T_tu_y +y²T_tt+6Y_ty),

(48)

t₁₂=1

3T_tX u_x−1

9T_tx²X_t−1

3X xX_t+T X_tu_x−Z₂Y

−Z_3tT+1

3T T_ttxu_x+2

3T xT_tu_xt+2

3Tyu_ytT_t+2 3X T_tyu_xy +2

3u_xxxX T_t+2

3u_yyyY T_t+2

9xyT_t²u_xy+2 3u_xyxY T_t +1

3Tu_yyT_tt+Tu_tT_t+Tu_yY_t−TyZ_2t−T xZ_1t

−1

18y²Y T_tt− 1

54x³T_tT_tt− 1

54y³T_tT_tt+1 9u_yyy²T_t² +1

9u_xxx²T_t²−X Z₁+u_yyY²+T²u_tt+u_xxX² (49)

−1

18x²X T_tt+2u_xyY X−1

6x²X_ttT+2Tu_ytY−1 6Ty²Y_tt

−1

54Ty³T_ttt− 1

54T x³T_ttt+2Tu_xtX−1

3yZ₂T_t−1 3yY_tY

−1

9y²Y_tT_t+1

3u_yY T_t−1

3xT_tZ₁+1

9u_yyT_t²+1 9u_xxT_t², where

X_1t=X T⁻⁴³, Y_1t=−Y T⁻⁴³,

Y_2t=Z₂T⁻²³, Y_3t=Y²T⁻⁵³, (50) Y_4t=Z₃T⁻¹, Y_5tt=T⁻²³Z₂,

Y_6t=T⁻⁴³YY₃, Y_7t=X T⁻⁴³Y₉, (51) Y_8t=X T⁻⁴³Y₁₀, Y_9t=T⁻²³Z₁,

Y_10t=T⁻⁵³X². (52) To determine the functions ofK₁,K₂, and K₃, we substitute (32), (33), and (34) into (28) which yields the complicated equation

J_1,x+J_1,uu_x+J_1,uxu_xx+J_1,uyu_xy+J_1,utu_xt+J_1,uxxu_xxx +J_1,uxyu_xxy+J_1,uxtu_xxt+J_1,uyyu_xyy+J_1,uytu_xyt

+J_1,uttuxtt+J_2,y+J_2,uuy+J_2,uxuxy+J_2,uyuyy+J_2,utuyt

+J_2,uxxu_xxy+J_2,uxyu_xyy+J_2,uxtu_xyt+J_2,uyyu_yyy +J_2,uytu_yyt+J_2,uttu_ytt+ρt+ρuu_t+ρuxu_xt+ρuyu_yt +ρutu_tt+ρuxxu_xxt+ρuxyu_xyt+ρuxtu_xtt+ρuyyu_yyt +ρuytu_ytt+ρuttu_ttt=0. (53)