A Super Extension of Kaup Newell Hierarchy

Transcript

1 Commun. Theor. Phys. Beijing China pp c Chinese Physical Society and IOP Publishing Ltd Vol. 54 No. 4 October 5 00 A Super Extension of Kaup Newell Hierarchy GENG Xian-Guo ÞÁ and WU Li-Hua ÛÙ Department of Mathematics Zhengzhou University 00 Kexue Road Zhengzhou China Received December Abact With the help of the zero-curvature equation and the super trace identity we derive a super extension of the Kaup Newell hierarchy associated with a 3 3 matrix spectral problem and establish its super bi-hamiltonian uctures. Furthermore infinite conservation laws of the super Kaup Newell equation are obtained by using spectral parameter expansions. PACS numbers: 0.30.Jr Key words: super nonlinear evolution equations super bi-hamiltonian uctures conservation laws Introduction During the last forty years or so soliton theory has achieved great success and its development is still quite exciting. Meanwhile theoretical physics has developed a new fruitful conception of super model theories where the anticommuting odd variables of the Grassmann algebra are equally treated with the usual commuting even variables. So far many classical integrable equation have been extended to be the super completely integrable equations such as super KdV [ 3] super AKNS [4 7] super KP [8] and so on. [9 6] Various systematic methods in the classical integrable systems have been developed to obtain exact solutions of the super integrable systems such as the inverse scattering transformation [7] the Bäcklund and Darboux transformations [8 0] the bilinear transformation of Hirota [ ] and others [3 4] by which some explicit solutions are found. The aim of the present paper is to propose a super extension of the Kaup Newell KN hierarchy and establish its super bi-hamiltonian uctures. The first nontrivial equation in the hierarchy is u t = u x + u v uαβ x + u x αβ + 4uαβ x 4αα x v t = v x + uv vαβ ββ x x α t = α x + uvα u x β uβ x x β t = β x + uvβ x + u x vβ + uvβ x vα x v x α αββ x which is just the well-known KN equation [5] as α = 0 and β = 0. Infinite conservation laws of the super KN equation are derived as a direct consequence of the spectral parameter expansions. This paper is arranged as follows. In Sec. we introduce a 3 3 matrix spectral problem with two commuting potentials and two anti-commuting potentials which is an extension of the spectral problem associated with the famous KN equation. Using zero-curvature equation we derive a hierarchy of super KN equations. In Sec. 3 we establish the super bi-hamiltonian uctures of the super KN hierarchy by using super trace identity. [6 7] In Sec. 4 we conuct the infinite conservation laws of the super KN equation with the aid of the spectral parameter expansions. Super KN Hierarchy In this section we shall derive the super KN hierarchy. To this end we introduce a 3 3 matrix spectral problem: φ λ λu λα φ x = Uφ φ = φ U = v λ λβ φ 3 β α 0 where u v λ φ φ are the commuting variables which can be indicated by the the degree mod p as pu = pv = pλ = pφ = pφ = 0; α β and φ 3 are the anticommuitng variables which can be indicated by p as pα = pβ = pφ 3 =. Here λ is assumed to be a constant spectral parameter. We first solve the stationary zero-curvature equation where V x + [U V ] = 0 V = V ij pv = pv = pv = pv = pv 33 = 0 pv 3 = pv 3 = pv 3 = pv 3 =. Equation 3 is equivalent to V x + λuv + λαv 3 vv βv 3 = 0 V x λv + λuv V + λαv 3 + αv 3 = 0 V 3x λv 3 + λuv 3 + λαv 33 V λβv = 0 V x + vv V + λv + λβv 3 βv 3 = 0 V x + vv + λβv 3 λuv + αv 3 = 0 Supported by National Natural Science Foundation of China under Grant No Innovation Scientists and Technicians Troop Conuction Projects of Henan Province and SRFDP Corresponding author wulihuaxyz3@63.com

2 No. 4 A Super Extension of Kaup Newell Hierarchy 595 V 3x + λv 3 + vv 3 + λβv 33 V λαv = 0 V 3x βv V 33 + αv + λv 3 vv 3 = 0 V 3x βv + αv V 33 λuv 3 λv 3 = 0 V 33x βv 3 + αv 3 + λαv 3 + λβv 3 = 0 4 where each entry V ij = V ij A B C ρ is the function of A B C ρ with pa = pb = pc = 0 pρ = p = : V = λa V = λb V 3 = λρ V = C V = λa V 3 = λ V 3 = V 3 = ρ V 33 = 0. 5 Substituting Eq. 5 into Eq. 4 yields B x + λua + λb αρ = 0 C x λva λc + λβ = 0 ρ x λu + λβb + λαa + λρ = 0 x λβa vρ λ + αc = 0 A x = uc α vb βρ. 6 Functions A B C ρ are expanded into Laurent series in λ: A = j 0 ρ = j 0 A j λ j B = j 0 B j λ j C = j 0 C j λ j ρ j λ j = j 0 j λ j. 7 Substituting Eq. 7 into Eq. 6 we obtain the Lenard recursion equation KG j = JG j+ JG 0 = 0 8 where G j = C j B j j ρ j A j K and J are two operators defined by 0 0 α K = / 0 α 0 / v/ 0 u v α/ β/ u 0 β 0 v J = 0 β u/ / α / 0 β u v α/ β/ It is easy to see that kerj = {c 0 g 0 c 0 R} with g 0 = v u β α T. In order to find a general representation of solution of Eq. 8 we introduce a Lenard recursion equation Kg j = Jg j+ j 0 0 with condition to identify constants of integration as zero when acting with operator J upon Kg j. This means that g j is uniquely determined by the recursion equation 0. It is easy to see that v x + uv vαβ ββ x g = u x + u v uαβ β x uvβ α x + uvα u x β uβ x uv + αβ 4 v xx uvv x 3 8 u v vα xβ 3 vαβ x 3 ββ xx + 3 uv αβ + 3uvββ x 4 u xx 3 4 uu xv 3 8 u3 v + αα x + 3 uα xβ 3 uαβ x + 3 u vαβ + 3 u ββ x g = β xx 3 u xvβ 3 uv xβ 3uvβ x u v β + vα x + v x α. α xx 3 u xvα 3 uv xα 3uvα x + 3u x β x + 3 u xxβ 3 4 u v α + 3uu x vβ + 3 u v x β + 3u vβ x 4 u xv 4 uv x + αβ x α x β 3 8 u v 3 uββ x 3 uvαβ Since the equation JG 0 = 0 has a general solution G 0 = c 0 g 0 then we obtain by operating with J K j upon Eq. and using Eq. 8 that G j = c 0 g j + c g j + + c j g 0 j 0 where c 0 c... c j are arbitrary constants of integration and g j = g j g j g 3 j g 4 j g 5 T j. Let φ satisfies the spectral problem and an auxiliary problem φ tm = V m φ 3 where each entry V m ij = V ij A m B m C m ρ m m in the matrix V m is a polynomial of eigenparameter λ with A m = A j λ m j B m = B j λ m j

3 596 GENG Xian-Guo and WU Li-Hua Vol. 54 C m = C j λ m j ρ m = ρ j λ m j m = j λ m j. 4 Then the compatibility condition of and 3 yields the zero-curvature equation U tm Vx m + [U V m ] = 0 which is equivalent to a hierarchy of super KN equations u tm = B mx αρ m v tm = C mx α tm = ρ mx β tm = mx + αc m vρ m. 5 This can be written as u tm v tm α tm β tm T = c 0 X m + c X m + + c m X 0 m 0 6 with X j = PKG j = PJG j+ where P is the projective map: a b c d e T a b c d T. The first nontrivial member in the hierarchy 6 is as follows u t = c 0[u x + u v uαβ x + u x αβ + 4uαβ x 4αα x ] c u x v t = c 0[ v x + uv vαβ ββ x x ] c v x α t = c 0[α x + uvα u x β uβ x x ] c α x β t = c 0[ β x + uvβ x + u x vβ + uvβ x vα x v x α αββ x ] c β x 7 which is respectively reduced to the famous KN equation or the super KN equation when α = 0 β = 0 c 0 = c = 0 or c 0 = c = 0. 3 Super Bi-Hamiltonian Structures In this section we shall establish the super bi- Hamiltonian uctures of Eq. 6 with the help of super trace identity [6 7] U 0 V U λ dx = λ γ [ U ] λ γ V 8 λ U 0 where γ is a constant to be fixed and U 0 = u v α β T. Through direct calculations we have V U = λa + uc α + βρ λ U u V = λc U U v V = λb α V = λ U β V = λρ. 9 Substituting Eqs. 7 and 9 into Eq. 8 we get u v α λa + uc α + βρdx = λ γ β λ λγ λc λb λ λρ 0 u v α A j+ + uc j α j + βρ j dx = γ j + C j B j j ρ j. β A direct calculation gives that when c k 0 and k=0 c 0 = c = = c k = c k+ = = c j+ = 0 0 k j + then γ k = + k. Using Eqs. and we arrive at u v α H j = C j B j j ρ j j H 0 = c 0 uv + αβdx β j 5 g j+ k H j = c + ug j k + /αg3 j k + /βg4 j k k dx + c j uv + αβdx c j+ dx j. k j On the other hand we obtain from Eqs. 9 and 5 that u tm v tm α tm β tm C m = K B m m = J ρ m C m+ B m+ m+ ρ m+ m 0 3 where K and J are two super skew-symmetric operators defined by 0 0 α u u + u v u α u β v u v v β + v α v β K = J = α u β + α v α 0 u α α +. 4 α β v β u β v + β α β β

4 No. 4 A Super Extension of Kaup Newell Hierarchy 597 Therefore we obtain the super bi-hamiltonian uctures of Eq. 6 u tm u u v tm α tm = K v H m = J v H m+ m 0. 5 β tm α α β β Notice that /U 0 dx = 0. For brevity we take c j+ = 0 in Eq.. Especially for m = the super KN equation 7 can be written as u ut = K v α t H α β with the Hamiltonian function H = c 0 u xv + 4 u v uvαβ uββ x + αβ x α x β dx + c uv + αβdx. 4 Infinite Conservation Laws In what follows we will conuct conservation laws of super KN equation 7. We introduce the variables F = φ φ G = φ 3 φ 6 where pf = 0 pg =. From Eq. we have F x = v + λf λuf λαfg + λβg G x = β + αf + λg λufg. 7 We expand F G in powers of λ as follows F = f j λ j G = g j λ j 8 where pf j = 0 pg j =. Substituting Eq. 8 into Eq. 7 and comparing the coefficients of the same power of λ we obtain f = v g = β + vα f = 4 v x + 8 uv ββ x g = β x + vα uvβ f 3 = 8 v xx + 6 u xv 6 u v uvv x 3 4 ββ xx + 4 v xαβ + 4 vα xβ 4 vαβ x uvββ x g 3 = β xx v xα + vα x u xvβ 3 4 uv xβ uvβ x 3 8 uv α + αββ x u v β 9 and a recursion formula for f n and g n g n+ = g nx αg n + u f l g n+ l n f n+ = f nx βg nx αβf n + u f l f n+ l + α f l g n+ l Because of uβ f l g n+ l n. 30 φ x = φ t t φ x φ we derive the conservation laws of Eq. 7 t where + uf + αg = A + BF + ρg 3 x A = c 0 λ + c 0 uv + αβ + c B = c 0 λu + c 0 u x + u v uαβ c u ρ = c 0 λα + c 0 α x + uvα u xβ uβ x c α. 3 Assume that σ = +uf +αg θ = A+BF +ρg. Then Eq. 3 can be written as σ t = θ x which is the right form of conservation laws. We expand σ and θ as series in powers of λ with the coefficients which are called conserved densities and currents respectively σ = + σ j λ j θ = c 0 λ + c + θ j λ j 33 where c 0 c are constants of integration for Eq.. Then the first two conserved densities and currents read σ = uv + αβ σ = 4 uv x + 8 u v + αβ x uββ x uvαβ θ = c 0 4 uv x 4 u xv 3 8 u v + α x β

5 598 GENG Xian-Guo and WU Li-Hua Vol. 54 αβ x + 3 uvαβ + 3 uββ x c uv + αβ θ = c 0 8 uv xx + 8 u3 v 3 8 u xv x 3 8 u vv x uββ xx uv xαβ 3 4 uvα xβ 3 u vββ x uvαβ x αβ xx vαα x u xvαβ 3 4 u v αβ 3 4 u xββ x + α x β x c 4 uv x + 8 u v + αβ x + uββ x uvαβ. 34 The recursion relations for σ n and θ n n are σ n = uf n + αg n [ θ n = c 0 uf n+ αg n+ + u x + u v uαβ + α x + uvα ] u xβ uβ x g n c uf n + αg n 35 where f n and g n can be calculated from Eq. 30. f n References [] B.A. Kupershmidt Phys. Lett. A [] P. Mathieu J. Math. Phys [3] B.A. Kupershmidt J. Phys. A: Math. Gen L869. [4] M. Gürses and Ö. Oğuz Phys. Lett. A [5] M. Gürses and Ö. Oğuz Lett. Math. Phys [6] Y.S. Li and L.N. Zhang Nuovo Cimento A [7] Z. Popowicz J. Phys. A: Math. Gen [8] Yu.I. Manin and A.O. Radul Commun. Math. Phys [9] Q.P. Liu J. Phys. A: Math. Gen L45. [0] S.Q. LÜ X.B. Hu and Q.P. Liu J. Phys. Soc. Jpn [] I. Yamanaka and R. Sasaki Prog. Theor. Phys [] G.H.M. Roelofs and P.H.M. Kersten J. Math. Phys [3] J.C. Beuneli A. Das and Z. Popowicz J. Math. Phys [4] C. Devchanda and J. Schiffb J. Math. Phys [5] C.M. Yung Phys. Lett. B [6] K. Ikeda Lett. Math. Phys [7] M. Chaichian and P.P. Kulish Phys. Lett. 78B [8] Q.P. Liu Lett. Math. Phys [9] Q.P. Liu and Mañas M Phys. Lett. B [0] M. Siddiq M. Hassan and U. Saleemon J. Phys. A: Math. Gen [] A.S. Carstea Nonlinearity [] A.S. Carstea J. Nonlinear Math. Phys. 8 Suppl [3] A.R. Chowdlhury and S. Roy J. Math. Phys [4] P. Mathieu Lett. Math. Phys [5] D.J. Kaup and A.C. Newell J. Math. Phys [6] X.B. Hu J. Phys. A: Math. Gen [7] W.X. Ma J.S. He and Z.Y. Qin J. Math. Phys