Darboux Transformations for Some Two Dimensional Affine Toda Equations

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1 ICCM 2007 Vo. III Darboux Transformations for Some Two Dimensiona Affine Toda Equations Zi-Xiang Zhou Abstract The Lax pairs for the two dimensiona A (2) 2, C(1) and D (2) +1 Toda equations have a reaity symmetry, a cycic symmetry and a unitary symmetry. The Darboux transformations for these systems are discussed. These Darboux transformations shoud be of high degree. Exact soutions are written down by computing the Darboux transformations expicity Mathematics Subject Cassification: 35Q51, 35Q58, 37K10. Keywords and Phrases: Two dimensiona affine Toda equations, Darboux transformation, Exact soutions. 1 Introduction The Toda equation is one of the most important integrabe systems. Various kinds of Toda equations have been discussed by a ot of authors after it was proposed. The two dimensiona Toda equations have been studied widey [1, 7, 18, 23] and have aso important appications in physics [3, 14, 33] as we as in differentia geometry [4, 5, 9, 10, 11, 19, 22, 26]. In 1950 s, E. Fermi, J. Pasta and S. M. Uam made a famous numerica experiment (FPU experiment), in which they anayzed the energy distribution on a noninear spring with potentia ike V (r) = 1 2 r2 + αr 3 rather than the harmonic potentia V (r) = 1 2 r2. They found out the compicated quasi-periodic phenomenon which is competey different from thermaization. This phenomenon attracted many scientists. In 1967, M. Toda introduced the potentia V (r) = e r + r 1 = 1 2 r2 1 6 r3 + o(r 3 ) to emuate the noninear spring in the FPU experiment, which ed to the ceebrated Toda equation u j,tt = e uj+1 uj e uj uj 1 (1.1) Schoo of Mathematica Sciences, Fudan University, Shanghai , China. zxzhou@fudan.edu.cn E-mai:

2 406 Zi-Xiang Zhou which is a system of integrabe ordinary differentia equations [6]. (1.1) is generaized to the two dimensiona Toda equation which is aso integrabe [20], and has the Lax pair u j,xt = e uj uj 1 e uj+1 uj, (1.2) ψ j,x = λψ j+1 + u j,x ψ j, ψ j,t = 1 λ euj uj 1 ψ j 1. (1.3) When the two dimensiona Toda equation is infinite, the Darboux transformation was presented [17, 18], which coud be naturay generaized to the two dimensiona periodic Toda equation. In [25], a kind of Bäckund transformation for the two dimensiona Toda equations corresponding to any semi-simpe Lie agebra was obtained in terms of the representation of Lie agebras. Corresponding to any affine Kac-Moody agebra, there is aso two dimensiona Toda equation [13, 15], which is aso caed the two dimensiona affine Toda equation. For any affine Kac-Moody agebra, the two dimensiona affine Toda equation is [19] ( ) ( ) w j,xt = exp c ji w i v j exp c 0i w i (j = 1,, ) (1.4) i=1 where C = (c ij ) 0 i,j is the generaized Cartan matrix of the Kac-Moody agebra and v 0, v 1,, v are Coxeter numbers, i.e. they satisfy C(v 0, v 1,, v ) T = 0. The simpest two dimensiona affine Toda equation, apart from the sinh-gordon equation, is the one corresponding to A (2) 2, which is aso caed the Tzitzeica equation or Buough-Dodd-Zhiber-Shabat equation i=1 u xt = e u e 2u. (1.5) Various Kac-Moody agebras correspond to various boundary conditions. It was shown in [15] that these equations are integrabe and the Lax pairs were presented. When g = A (1), the Toda equation is periodic. Its Lax pair has a unitary symmetry and a cycic symmetry of order. The Darboux transformation keeping these symmetries was known [17, 18]. Apart from the two dimensiona periodic Toda equation, some other integrabe systems ike Kupershmidt-Wison hierarchy have aso cycic symmetry, and their Darboux transformation can aso be constructed [27]. There are aso some work for the two dimensiona Toda equations with other Kac-Moody agebras [2, 3, 16, 21, 33], athough the symmetries are more compicated. When the Kac-Moody agebra is A (2) 2, C(1) or D (2) +1, the equations have a unitary symmetry, a reaity symmetry and a cycic symmetry of order N = (for A (2) 2 ) or N = 2 (for C(1) ) or N = (for D (2) +1 ). (In [27], these three symmetries were caed su(p, q)-reaity condition with respect to a nonstandard indefinite matric, s(n,r)-reaity condition and KW-reaity condition respectivey. In the present paper, reaity symmetry is ony restricted to the condition that

3 Darboux transformations for 2D affine Toda equations 407 a matrix takes rea vaue.) Athough the matrices in the Lax pairs are of order N N, the number of independent functions is ony. This eads to the difficuty in getting expicit soutions. In [21], the binary Darboux transformation for two dimensiona A (2) 2, C(1) and D (2) +1 Toda equations were obtained from the periodic reductions of the binary Darboux transformations for A, B and C Toda equations. In terms of the binary Darboux transformation, the soutions of the Toda equations were expressed by some integras of the soutions of the Lax pairs. In order to get the expicit soutions which are purey agebraicay expressed by the soutions of the Lax pair, usua Darboux transformations (without integras) are necessary. In [12], expicit soutions of the two dimensiona A (2) 2 Toda equation were presented for rea spectra parameters. In [30, 31, 32], the Darboux transformations for the two dimensiona A (2) 2, C(1) and D (2) +1 Toda equations were constructed and expicit soutions of those Toda equations were obtained. Usuay, the unitary symmetry is not a difficut restriction in constructing Darboux transformations. However, with the appearance of cycic symmetry, the unitary symmetry makes the probem much more compicated. In fact, when the system has ony the reaity and cycic symmetries as in the A (1) case, Theorem 1 beow can be used in constructing Darboux transformations. The owest degree of the Darboux transformations can be 1 (when a spectra parameter is rea) or 2 (when a spectra parameters are compex). However, if there is an extra unitary symmetry, the simper way is to use Theorem 2 so that the owest degree of the Darboux transformations shoud be N and N/2 (when a spectra parameter is rea) depending on whether N is odd or even, or 2N and N (when a spectra parameters are compex). Theorem 1 wi ead to the same concusions, but the procedure is more compicated. In Section 2, two genera ways of constructing Darboux transformations are reviewed. Section 3 shows the structure of the Lax pair for two dimensiona Toda equations. In Section 4, Darboux transformation for two dimensiona periodic Toda equation is reviewed. Section 5 is devoted to our work on the Darboux transformation for two dimensiona affine Toda equations mentioned above. Expicit expression of the soutions of the two dimensiona A (2) 2 Toda equation is presented as an exampe. 2 Genera constructions of Darboux transformations We review the genera construction of Darboux transformations for the Lax pair Φ x = U(x, t, λ)φ, Φ t = V (x, t, λ)φ (2.1) where U and V are N N matrix-vaued rationa functions of λ. An N N matrix G(x, t, λ) is caed a Darboux matrix if there exist Ũ(x, t, λ) and Ṽ (s, t, λ), which are aso N N matrix-vaued rationa functions of λ, such that for any soution

4 408 Zi-Xiang Zhou Φ of (2.1), Φ = GΦ satisfies Φ x = Ũ(x, t, λ) Φ, Φt = Ṽ (x, t, λ) Φ. (2.2) The corresponding transformations of U and V are given by Ũ = GUG 1 + G x G 1, Ṽ = GV G 1 + G t G 1. (2.3) Without considering any symmetries of the Lax pair, the Darboux transformation is given by the foowing theorem. Theorem 1. [8, 24] Let λ 1,, λ N be N compex numbers such that λ j (j = 1,, N) are not a the same. Let h j be a coumn soution of the Lax pair (2.1) for λ = λ j (j = 1,, N). Let Λ = diag(λ 1,, λ N ), H = (h 1,, h N ), then when deth 0, G(x, t, λ) = λi HΛH 1 is a Darboux matrix. Now we consider the system with a unitary symmetry, i.e., there is an invertibe rea symmetric matrix K such that U and V satisfy KU(x, t, λ)k 1 = (U(x, t, λ)), KV (x, t, λ)k 1 = (V (x, t, λ)). (2.4) In this case, for any soution Φ of (2.1) with λ = µ and any soution Ψ of (2.1) with λ = µ, (Ψ KΦ) x = (Ψ KΦ) t = 0 hods. Hence Ψ KΦ = 0 identicay if it hods at one point. For the system with a unitary symmetry, the most usefu way to construct Darboux transformation is given by the foowing theorem. Theorem 2. [29] Let λ 1,, λ M be M compex numbers such that λ j, λ j (j = 1, 2,, M) are distinct. Let H j be an N r soution of the Lax pair (2.1) for λ = λ j (j = 1, 2,, M). Let Γ = (Γ ij ) 1 i,j M where are r r matrices. Then G(x, t, λ) = is a Darboux matrix. Γ ij = H i KH j λ i + λ j (i, j = 1, 2,, M) (2.5) M (λ + λ ) 1 =1 M i,j=1 H i (Γ 1 ) ij Hj K λ + λ (2.6) j In this construction, G(x, t, λ) is a poynomia of λ of degree M with matrix coefficients. Usuay we write G(x, t, λ) = M ( 1) M j G M j (x, t)λ j, G 0 (x, t) = I. (2.7) j=0

5 Darboux transformations for 2D affine Toda equations 409 Remark 1. For the simpest case M = 1, the Darboux transformation (2.7) is the same as the one given by Theorem 1 if Λ and H in Theorem 1 are chosen r N r {}}{{}}{ as Λ = diag( λ 1,, λ 1, λ 1,, λ 1 ), H = (h 1,, h r, w r+1,, w N ) where (h 1,, h r ) = H 1, w r+1,, w N are ineary independent soutions of the Lax pair (2.1) for λ = λ 1 such that wj Kh k = 0 identicay (j = r + 1,, N; k = 1,, r). For genera M, the Darboux transformation in Theorem 2 can aso be constructed by the composition of M Darboux transformations in Theorem 1, but the construction is more compicated. 3 Structure of the Lax pairs for two dimensiona Toda equations For any N N matrix A or any N dimensiona vector v, and for any integers i and j, define A ij = A i j and v i = v i where i i mod N, j j mod N and 1 i, j N. Especiay, δ ij equas 1 if i j mod N and equas 0 otherwise. Let ω = e 2πi/N, Ω = diag(1, ω 1,, ω N+1 ). Let m be a fixed integer. Let K = (K ij ) = (δ i,m j ) N N, J = (J ij ) = (δ i,j 1 ) N N be constant matrices. Then K is symmetric and Ω K = ω m 2 KΩ where Ω refers to the Hermitian conjugate of Ω. The Lax pair for the Toda equations is Φ x = U(x, t, λ)φ = (λj + P(x, t))φ, Φ t = V (x, t, λ)φ = λ 1 Q(x, t)φ (3.1) and its integrabiity conditions are Q x = [P, Q], P t + [J, Q] = 0. (3.2) There wi be different symmetries in the coefficients (P, Q) corresponding to different Kac-Moody agebras. For the Lax pair (3.1), the Darboux transformation (2.7) gives P = P + [J, G 1 ], Q = G M QG 1 M, which is derived from (2.3). For g = A (1), the corresponding N =, and the coefficients U and V satisfy a reaity symmetry and a cycic symmetry of order N as U(x, t, λ) = U(x, t, λ), V (x, t, λ) = V (x, t, λ), (3.3) ΩU(x, t, λ)ω 1 = U(x, t, ωλ), ΩV (x, t, λ)ω 1 = V (x, t, ωλ). (3.4) Under these symmetries, P = (p i δ ij ) 1 i,j N, Q = (q j δ i,j+1 ) 1 i,j N where p i (x, t) s and q i (x, t) s are rea functions. The integrabiity conditions (3.2) become q j,x = (p j+1 p j )q j, p j,t = q j 1 q j (j = 1,, N). (3.5) For g = A (2) 2 (N = 2 + 1), C (1) (N = 2) and D (2) +1 (N = 2 + 2), there is an extra unitary symmetry (2.4). Under this symmetry, we have p m j = p j, q m 1 j = q j (j = 1, 2,, N). (3.6)

6 410 Zi-Xiang Zhou The symmetries of U and V ead to the symmetries of the spectrum. Lemma 1. Suppose µ C\{0}, Φ(x, t) is a soution of (3.1) for λ = µ. If (U, V ) satisfies (3.3) and (3.4), then Φ(x, t) is a soution of (3.1) for λ = µ, and ΩΦ(x, t) is a soution of (3.1) for λ = ωµ. If, moreover, N = 2n is even and (U, V ) aso satisfies (2.4), then Ω n Φ is a soution of (3.1) for λ = µ, and Ψ = KΩ n Φ is a soution of the adjoint Lax pair Ψ x = U(µ) T Ψ, Ψ t = V (µ) T Ψ. According to Lemma 1, if µ is an eigenvaue of the Lax pair, so are ω j µ and ω j µ (j = 1,, N). Let S 1 (µ) = {ω j µ j = 1,, N}, S 2 (µ) = {ω j µ j = 1,, N}, Sj (µ) = { λ λ S k (µ)}, (k = 1, 2). (3.7) For given µ 0, if the Darboux transformation given by Theorem 2 keeps a the symmetries of the Lax pair, then λ 1,, λ M can not be arbitrary. We denote SP(µ) = {λ 1,, λ M } and SP (µ) = { λ 1,, λ M }. The reation between SP(µ), S j (µ) and SP (µ), Sj (µ) are determined by the reation among S j(µ) and Sj (µ) (j = 1, 2), which depends on the parity of N. This wi be specified in the foowing two sections. It is cear that if Γ in (2.5) exists, then SP(µ) SP (µ) =. 4 Darboux transformation for two dimensiona periodic (A (1) ) Toda equation For the two dimensiona A (1) equation, N =, p j = u j,x, q j = e uj+1 uj. The equation is u j,xt = e uj uj 1 e uj+1 uj. (4.1) The Darboux transformation can be obtained according to Theorem 1. Here we take M =, λ j = ω j µ, H j = Ω j H where µ 0 is a rea number and H is a coumn soution of the Lax pair (3.1) with λ = µ. Let H = (h 1,, h N ) T, then the Darboux matrix is given by which eads to the transformation and a new soution of the two dimensiona A (1) G ij = λδ ij µh i h i 1 δ i 1,j (4.2) φ j = λφ j µ h j h j 1 φ j 1 (4.3) Toda equation q j = h j+1h j 1 h 2 q j 1, (4.4) j

7 or equivaenty Darboux transformations for 2D affine Toda equations 411 ũ j = u j 1 + n h j h j 1 + c (4.5) where c is a constant. This construction of Darboux transformation is essentiay the same as that given in [18] The Darboux transformation with compex spectra parameters can be constructed simiary, athough its degree shoud be 2 rather than 1. 5 Darboux transformation for two dimensiona A (2) C (1), D (2) +1 Toda equations 5.1 Expicit form of the evoution equations It was known [31] that there are ony three sets of non-equivaent equations in the system (3.1) with symmetries (2.4), (3.3) and (3.4). They are (1) A (2) 2 (N = 2 + 1, m = 0): p j = p 2+1 j = u j,x (1 j ), p 2+1 = 0; q j = q 2 j = e uj+1 uj (1 j 1), q = e 2u, q 2 = q 2+1 = e u1. The evoution equations are 2, u j,xt = e uj uj 1 e uj+1 uj (2 j 1), u 1,xt = e u1 e u2 u1, u,xt = e u u 1 e 2u. (5.1) (2) C (1) (N = 2, m = 1): p j = p 2+1 j = u j,x, (1 j ); q j = q 2 j = e uj+1 uj (1 j 1), q = e 2u, q 2 = e 2u1. The evoution equations are u j,xt = e uj uj 1 e uj+1 uj (2 j 1), u 1,xt = e 2u1 e u2 u1, u,xt = e u u 1 e 2u. (5.2) (3) D (2) +1 (N = 2+2, m = 0): p j = p 2+2 j = u j,x, (1 j ), p +1 = p 2+2 = 0; q j = q 2+1 j = e uj+1 uj (1 j 1), q = q +1 = e u, q 2+1 = q 2+2 = e u1. The evoution equations are u j,xt = e uj uj 1 e uj+1 uj (2 j 1), u 1,xt = e u1 e u2 u1, u,xt = e u u 1 e u. (5.3) These equations can be changed to (1.4) by a simpe inear transformation of the unknowns (u 1,, u ). (There is a mistake of notations in [31] where the names C (1) and D (2) +1 there shoud be interchanged.) 5.2 Constructions of Darboux transformations The spectrum is different for odd and even N. Hence we wi construct the Darboux transformations for odd and even N separatey. When N = 2n + 1, a the points in S 1 (µ), S 2 (µ), S1 (µ), S 2 (µ) are distinct kπ if arg(µ) for any integer k (See Figure 1). Hence we can choose 2(2n + 1)

8 412 Zi-Xiang Zhou Figure 1 The spectrum S j (µ) and Sj (µ) for N = 3 (odd N). A dark dot refers to a point in SP(µ) and a circe refers to a point in SP (µ). The vertices of each triange refer to one S j (µ) or Sj (µ). SP(µ) = S 1 (µ) S 2 (µ). The Darboux transformation is constructed as foows. Let λ j = ω j 1 µ, λ 2n+1+j = λ j (j = 1, 2,, 2n+1). Let H be a coumn soution of (3.1) for λ = µ, H j = Ω j 1 H, H 2n+1+j = H j (j = 1, 2,, 2n + 1). The Darboux matrix G(x, t, λ) is constructed according to Theorem 2 with M = 2(2n + 1). Figure 2 The spectrum S j (µ) and Sj (µ) for N = 4 (even N). A dark dot refers to a point in SP(µ) and a circe refers to a point in SP (µ). The vertices of each square refer to one S j (µ) or Sj (µ). When N = 2n, S 1 (µ) = S 2(µ), S 2 (µ) = S 1(µ). Moreover, when arg(µ) kπ 2n for any integer k, S 1 (µ) S 2 (µ) = (See Figure 2). Ceary, we can not choose SP(µ) = S 1 (µ) S 2 (µ), since otherwise SP (µ) = SP(µ) woud hod, which contradicts the existence of Γ. Hence we choose SP(µ) = S 1 (µ). Now S 2 (µ) is not contained in SP(µ), but contained in SP (µ). According to Remark 1, in order to keep the symmetry between S 1 (µ) and S 2 (µ), there shoud be a symmetry between the soution h 1,, h r and w r+1, w 2n where w r+1, w 2n are defined in Remark 1. Moreover, according to (c) of Lemma 1, r = n, and w j = Ω n h j shoud hod. Therefore, the Darboux transformation is constructed as foows. Let λ j = ω j 1 µ (j = 1, 2,, 2n). Let H be an 2n n matrix soution of (3.1)

9 Darboux transformations for 2D affine Toda equations 413 for λ = µ such that H T KΩ n H = (Ω n H) KH = 0 at certain point (x 0, t 0 ). Then (c) of Lemma 1 impies that H T KΩ n H = 0 hods identicay. Let H j = Ω j 1 H (j = 1, 2,, 2n). The Darboux matrix G(x, t, λ) is constructed according to Theorem 2 with M = 2n. After the Darboux transformation, the coefficients of the Lax pair are changed to Ũ = λj + P, Ṽ = 1 Q λ where P = P + [J, G 1 ], Q = G M QG 1 M. Moreover, we have Theorem 3. [30, 31] The Darboux transformation satisfies the foowing symmetries: Unitary: G(x, t, λ) KG(x, t, λ) = M ( λ + λ)(λ λ)k, (5.4) Cycic: ΩG(x, t, ω 1 λ)ω 1 = G(x, t, λ), (5.5) Reaity: G(x, t, λ) = G(x, t, λ) (5.6) where M = 2(2n + 1) for N = 2n + 1 and M = 2n for N = 2n. (5.4) is a direct concusion of the construction of the Darboux transformation (2.6). (5.5) can be verified directy. (5.6) is aso obtained directy when N is odd. However, when N is even, since the spectrum does not have the reaity symmetry expicity, direct verification of (5.6) is very difficut. Instead, et (x, t, λ) = G(x, t, λ) G(x, t, λ), then it can be proved that (x, t, λ i )H i = 0, (x, t, λ i )Ω n Hi = 0 hod. Then we can get (x, t, λ) 0 after proving the invertibiity of a matrix composing λ i s and H i s. Theorem 3 impies that the Darboux transformation keeps a the symmetries (2.4), (3.3) and (3.4). =1 5.3 Expicit expressions of the soutions In order to get expicit expressions of the soutions we shoud derive the expicit expression of G 2(2n+1). (2.6) is too compicated to be computed directy even by computer. Therefore, we need to represent the matrix Γ, regarded as a inear transformation, in another basis, so that the computation can be simpified. For exampe, when N is odd, the soution of the two dimensiona A (2) 2 Toda equation can be obtained expicity as foows. Theorem 4. [30] Suppose (u 1,, u ) be a soution of the two dimensiona A (2) 2 kπ Toda equation (5.1), µ is a compex number with arg(µ) for any integer 2(2 + 1) k, H = (h 1,, h 2+1 ) T is a coumn soution of the Lax pair for λ = µ, then ũ k = u k + n ζ k+1 ζ k (k = 1, 2,, ) (5.7)

10 414 Zi-Xiang Zhou is a new soution of the two dimensiona A (2) 2 ζ k = 1 µ µ s=2+2 k 2+1 k 1 4 µ k s=1 Toda equation (5.1) where k+s 1 h s h s ( µ) 2+1 k s µ h s h s ( µ) 4+2 k s µ k+s s=1 h s h s ( 1) k+s 2+1 s=2+2 k h s h s ( 1) k+s 2. (5.8) When N = 2n is even, the soutions for C (1) (n = ) and D (2) +1 (n = + 1) cannot be written down in such a simpe form without matrix operation. However, they can be simpified so that they ony depends on the the inverse of an n n matrix rather than the origina 2n 2n matrices. kπ Ti now, we need the condition arg(µ) when N = 2n + 1 and 2(2n + 1) arg(µ) kπ 2n when N = 2n for any integer k. However, if arg(µ) = kπ 2(2n + 1) when N = 2n + 1 or arg(µ) = kπ when N = 2n for certain integer k, the 2n reations among the sets S 1 (µ), S 2 (µ), S1 (µ) and S 2 (µ) are changed. In fact, when N is odd, we have either S 2 (µ) = S 1 (µ), S2(µ) = S1(µ) or S1(µ) = S 1 (µ), S2 (µ) = S 2(µ). When N is even, we have S1 (µ) = S 2 (µ) = S 1(µ) = S 2 (µ). The Darboux transformations can be constructed differenty [30, 32]. This work was supported by Nationa Basic Research Program of China (973 Program) (2007CB814800) and STCSM (06JC14005). References [1] M. J. Abowitz & P. A. Carkson, Soitons, Noninear Evoution Equations and Inverse Scattering, Cambridge University Press, [2] M. Ader & P. Van Moerbeke, Toda versus Pfaff attice and reated poynomias, Duke Math. J. 112 (2002), [3] H. Aratyn, C. P. Constantinidis, L. A. Ferreira, J. F. Gomes & A. H. Zimerman, Hirota soitons in the affine and the conforma affine Toda modes, Nucear Physics B406 (1993) [4] A. I. Bobenko, A constant mean curvature tori in R 3, S 3 and H 3 in terms of theta-functions, Math. Ann. 290 (1991), [5] J. Boton & L. M. Woodward, The affine Toda equations and minima surfaces, in: A. P. Fordy & J. C. Wood (Eds.), Harmonic Maps and Integrabe System, Vieweg (1994) [6] H. Faschka, On the Toda attice II. Inverse Scattering soution, Progr. Theor. Phys. 51 (1974), [7] A. P. Fordy & J. Gibbons, Integrabe noninear Kein-Gordon equations and Toda attices, Commun. Math. Phys. 77 (1980),

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