v 1 1 8 O ct 1 99 6 Lax Representations and Zero Curvature Representations by Kronecker Product

It is showed that Kronecker product can be applied to construct not only new Lax representations but also new zero curvature representations of integrable models. Meantime a different characteristic between continuous and discrete zero curvature equations is pointed out. Lax representation and zero curvature representation play an important role in studying nonlinear integrable models in theoretical physics. It is based on such representations that the inverse scattering transform is successfully developed (see, say, Ablowitz and Clarkson 1991). They may also provide a lot of information, such as integrals of motion, master symmetries and Hamiltonian formulation. There exist quite many integrable models to possess Lax representation or zero curvature representation (Faddeev and Takhtajan 1987, Das 1989). Two typical examples are Toda lattice (Flaschka 1974) and AKNS systems (Ablowitz, Kaup, Newell and Segur 1974) including KdV equation and nonlinear Schödinger equation. In this paper, we want to give rise to a kind of new Lax representations and new zero curvature representations by using Kronecker product of matrices, motivated by a recent progress made by Steeb and Heng (Steeb and Heng 1996). Kronecker product itself has nice mathematical properties and important applications in many fields of physics, for example, statistical physics, quantum groups, etc. (Steeb 1991). Our result for zero curvature representation also provides us with a different characteristic between continuous and discrete zero curvature equations. Institute of Mathematics, Fudan University, Shanghai 200433, P. R. of China Mathematik und Informatik, Universität-GH Paderborn, D-33098 Paderborn, Germany Dept. of Applied Mathematics, Shandong Mining Institute, Tai’an 271019, P. R. of China 1 Let IM denote the unit matrix of order M, M ∈ Z. For two matrices A = (aij)pq, B = (bkl)rs, Kronecker product A⊗ B is defined by (Steeb 1991) A⊗ B = (aijB)(pr)×(qs), (1) or equivalently by (Hoppe 1992) (A⊗ B)ij,kl = aikbjl. (2) Evidently we have a basic relation on Kronecker product (Steeb 1991, Hoppe 1992) (A⊗ B)(C ⊗D) = (AC)⊗ (BD), (3) provided that the matrices AC and BD make sense. This relation will be used to show new structure of Lax representation and zero curvature representation of integrable models. Theorem 1 (Lax representation) Assume that an integrable model (continuous or discrete) has two Lax representations L1t = [A1, L1], L2t = [A2, L2], (4) where L1, A1 and L2, A2 are M ×M and N ×N matrices, respectively. Define L3 = α1L1 ⊗ L2 + α2(L1 ⊗ IN + IM ⊗ L2), A3 = A1 ⊗ IN + IM ⊗ A2, (5) where α1, α2 are arbitrary constants. Then the same integrable model has another Lax representation L3t = [A3, L3]. Proof: First of all, we have L3t = α1(L1t ⊗ L2 + L1 ⊗ L2t) + α2(L1t ⊗ IN + IM ⊗ L2t). (6) On the other hand, using (3) we can calculate that [A3, L3] = α1([A1, L1]⊗ L2 + L1 ⊗ [A2, L2]) +α2([A1, L1]⊗ IN + IM ⊗ [A2, L2]). Now we easily find that the equalities defined by (4) implies L3t = [A3, L3]. When α2 = 0, the obtained result is exactly one in Ref. Steeb and Heng 1996. When α1 = 0, we get a new Lax representation for a given integrable model, starting 2 from two known Lax representations. Integrals of motion may also be generated from new Lax representation, because we have Fij = tr(α1L i 1 ⊗ L j 2 + α2(L i 1 ⊗ IN + IM ⊗ L j 2)) = α1tr(L i 1)tr(L j 2) + α2(Ntr(L i 1) +Mtr(L j 2)), (7) where we have used tr(A⊗B) = tr(A)tr(B) (Steeb 1991) and (L1)t = [A1, L i 1], (L i 2)t = [A2, L i 2]. Theorem 2 (Continuous zero curvature representation) Assume that a continuous integrable model has two continuous zero curvature representations U1t − V1x + [U1, V1] = 0, U2t − V2x + [U2, V2] = 0, (8) where U1, V1 and U2, V2 are M ×M and N ×N matrices, respectively. Define U3 = U1 ⊗ IN + IM ⊗ U2, V3 = V1 ⊗ IN + IM ⊗ V2. (9) Then the same integrable model has another continuous zero curvature representation U3t − V3x + [U3, V3] = 0. (10) Proof: The proof is also a direct computation. We first have U3t = U1t ⊗ IN + IM ⊗ U2t, U3x = U1x ⊗ IN + IM ⊗ U2x. Second, using (3) we can obtain that [U3, V3] = [U1, V1]⊗ IN + IM ⊗ [U2, V2]. (11) Therefore we see that (10) is true once two equalities defined by (8) hold. We remark that when we choose U3 = U1 ⊗ U2, the third zero curvature representation (10) is not certain to be true. An example will be displayed later on. 3 Theorem 3 (Discrete zero curvature representation) Assume that a discrete integrable model has two discrete zero curvature representations U1t = (EV1)U1 − U1V1, U2t = (EV2)U2 − U2V2, (12) where E is the shift operator, U1, V1 are M × M matrices, and U2, V2 are N × N matrices. Define U3 = U1 ⊗ U2, V3 = V1 ⊗ IN + IM ⊗ V2. (13) Then the same integrable model has another discrete zero curvature representation U3t = (EV3)U3 − U3V3. (14) Proof: Similarly, we first have U3t = U1t ⊗ U2 + U1 ⊗ U2t. (15) On the other hand, we may calculate that (EV3)U3 − U3V3 = ((EV1)⊗ IN + IM ⊗ (EV2))(U1 ⊗ U2) −(U1 ⊗ U2)(V1 ⊗ IN + IM ⊗ V2) = ((EV1)U1)⊗ U2 + U1 ⊗ ((EV2)U2)− (U1V1)⊗ U2 − U1 ⊗ (U2V2) = ((EV1)U1 − U1V1)⊗ U2 + U1 ⊗ ((EV2)U2 − U2V2). In the second equality above, we have used the basic relation (3). Hence we find that (14) holds if we have (12). We remark that when we choose U3 = U1 ⊗ IM + IN ⊗ U2, the third discrete zero curvature representation (14) is not certain to be true. An example will also be given later on. This is opposite to the result in the continuous case. It shows us a different characteristic between continuous and discrete zero curvature equations. In what follows, we would like to display some concrete examples to illustrate the use of the above technique of Kronecker product. Actually once we have a Lax representation or a zero curvature representation, we can obtain a new representation after choosing two required representations to be this known one. Further newer 4 representation may be constructed by use of this new representation and the process may be infinitely proceeded to. This also tells us that there exist infinitely many Lax representations or zero curvature representations once there exists one representation for a given integrable model. The concrete procedure of construction will be showed in the following examples and can be easily generalized to other integrable models, for example, in Refs. Calogero 1994, Drinfel’d and Sokolov 1984, Ma 1993, Ragnisco and Santini 1990, Tu 1990 etc. Example 1: We consider periodical Toda lattice (Flaschka 1974) ait = ai(bi+1 − bi), bit = 2(a 2 i − a 2 i−1), ai+N = ai, bi+N = bi, (16) which is a Hamiltonian system with Hamiltonian H(q1, q2, · · · , qN , p1, p2, · · · , pN) = 1 2 N ∑