Almost Everything You Always Wanted to Know About the Toda Equation

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1 Jahresber. Deutsch. Math.-Verein. 103, no. 4, (2001) Mathematics Subject Classification: 35Q51, 37K10; 37K15, 39A70 Keywors an Phrases: Toa equation, Kac van Moerbeke equation, Solitons, Lax pair Preface Almost Everything You Always Wante to Know About the Toa Equation Geral Teschl, Wien Abstract The present article reviews methos from spectral theory an algebraic geometry for fining explicit solutions of the Toa equation, namely for the N-soliton solution an quasi-perioic solutions. Along they way basic concepts like Lax pairs, associate hierarchies, an Bäcklun transformations for the Toa equation are introuce. This article is suppose to give an introuction to some aspects of completely integrable nonlinear wave equations an soliton mathematics using one example, the Toa equation. Moreover, the aim is not to give a complete overview, even for this single equation. Rather I will focus on only two methos (reflecting my personal bias) an I will try to give an outline on how explicit solutions can be obtaine. More etails an many more references can be foun in the monographs by Gesztesy an Holen [19], myself [39], an Toa [40]. The contents constitutes an extene version of my talk given at the joint annual meeting of the Österreichische Mathematische Gesellschaft an the Deutsche Mathematiker-Vereinigung in September 2001, Vienna, Austria. 1 The Toa equation In 1955 Enrico Fermi, John Pasta, an Stanislaw Ulam carrie out a seemingly innocent computer experiment at Los Alamos, [15]. They consiere a simple moel for a nonlinear one-imensional crystal escribing the motion of a chain of particles with nearest neighbor interaction q(n, t) 1

2 (1) The Hamiltonian of such a system is given by H(p, q) = n Z ( p(n, t) 2 2 ) + V (q(n + 1, t) q(n, t)), where q(n, t) is the isplacement of the n-th particle from its equilibrium position, p(n, t) is its momentum (mass m = 1), an V (r) is the interaction potential. Restricting the attention to finitely many particles (e.g., by imposing perioic bounary conitions) an to the harmonic interaction V (r) = r2 2, the equations of motion form a linear system of ifferential equations with constant coefficients. The solution is then given by a superposition of the associate normal moes. It was general belief at that time that a generic nonlinear perturbation woul yiel to thermalization. That is, for any initial conition the energy shoul eventually be equally istribute over all normal moes. The aim of the experiment was to investigate the rate of approach to the equipartition of energy. However, much to everyboy s surprise, the experiment inicate, instea of the expecte thermalization, a quasi-perioic motion of the system! Many attempts were mae to explain this result but it was not until ten years later that Martin Kruskal an Norman Zabusky, [47], reveale the connections with solitons. This ha a big impact on soliton mathematics an le to an explosive growth in the last ecaes. In particular, it le to the search for a potential V (r) for which the above system has soliton solutions. By consiering aition formulas for elliptic functions, Morikazu Toa came up with the choice (2) V (r) = e r + r 1. The corresponing system is now known as the Toa equation, [41]. Figure 1: Toa potential V (r) This moel is of course only vali as long as the relative isplacement is not too large (i.e., at least smaller than the istance of the particles in the equilibrium position). For small isplacements it is approximately equal to a harmonic interaction. 2

3 (3) The equation of motion in this case reas explicitly q) p(n, t) = H(p, t q(n, t) = e (q(n, t) q(n 1, t)) e (q(n + 1, t) q(n, t)), H(p, q) q(n, t) = = p(n, t). t p(n, t) The important property of the Toa equation is the existence of so calle soliton solutions, that is, pulslike waves which sprea in time without changing their size an shape. This is a surprising phenomenon, since for a generic linear equation one woul expect spreaing of waves (ispersion) an for a generic nonlinear force one woul expect that solutions only exist for a finite time (breaking of waves). Obviously our particular force is such that both phenomena cancel each other giving rise to a stable wave existing for all time! In fact, in the simplest case of one soliton you can easily verify that this solution is given by (4) q 1 (n, t) = q 0 ln 1 + γ exp( 2κn ± 2 sinh(κ)t), κ, γ > γ exp( 2κ(n 1) ± 2 sinh(κ)t) Figure 2: One soliton It escribes a single bump traveling through the crystal with spee ± sinh(κ)/κ an with proportional to 1/κ. In other wors, the smaller the soliton the faster it propagates. It results in a total isplacement 2κ of the crystal. Such solitary waves were first observe by the naval architect John Scott Russel [34], who followe the bow wave of a barge which move along a channel maintaining its spee an size (see the review article by Palais [33] for further information). Existence of soliton solutions is usually connecte to complete integrability of the system, an this is also true for the Toa equation. The motivation as a simple moel in soli state physics presente here is of course not the only application of the Toa equation. In fact, the Toa equation an relate equations are use to moel Langmuir oscillations in plasma physics, to investigate conucting polymers, in quantum cohomology, etc.. Some general books ealing with the Toa lattice are the monographs by Toa [40], [41], by Eilenberger [13], by Faeev an Takhtajan [14] an by Teschl [39]. Another goo source on soliton mathematics is the recent review article by Palais [33]. Finally, it shoul also be mentione that the Toa equation can be viewe as a iscrete version of the Korteweg-e Vries equation (see [40] or [33] for informal treatments). 3

4 2 Complete integrability an Lax pairs To see that the Toa equation is inee integrable we introuce Flaschka s variables (5) a(n, t) = 1 2 e (q(n + 1, t) q(n, t))/2, b(n, t) = 1 p(n, t) 2 an obtain the form most convenient for us ( ) ȧ(t) = a(t) b + (t) b(t), ( (6) ḃ(t) = 2 a(t) 2 a (t) 2). Here we have use the abbreviation (7) f ± (n) = f(n ± 1). To show complete integrability it suffices to fin a so-calle Lax pair, that is, two operators H(t), P 2 (t) in l 2 (Z) such that the Lax equation (8) t H(t) = P 2(t)H(t) H(t)P 2 (t) is equivalent to (6). One can easily convince oneself that the right choice is (9) H(t) = a(t)s + + a (t)s + b(t), P 2 (t) = a(t)s + a (t)s, where (S ± f)(n) = f ± (n) = f(n ± 1) are the shift operators. Now the Lax equation (8) implies that the operators H(t) for ifferent t R are unitarily equivalent: Theorem 1. Let P 2 (t) be a family of boune skew-ajoint operators, such that t P 2 (t) is ifferentiable. Then there exists a family of unitary propagators U 2 (t, s) for P 2 (t), that is, (10) t U 2(t, s) = P 2 (t)u 2 (t, s), U 2 (s, s) = 1l. Moreover, the Lax equation (8) implies (11) H(t) = U 2 (t, s)h(s)u 2 (t, s) 1. If the Lax equation (8) hols for H(t) it automatically also hols for H(t) j. Taking traces shows that (12) tr(h(t) j H j 0), j N, 4

5 are conserve quantities, where H 0 is the operator corresponing to the constant solution a 0 (n, t) = 1 2, b 0(n, t) = 0 (it is neee to make the trace converge). For example, tr(h(t) H 0) = n Z b(n, t) = 1 p(n, t) an 2 n Z (13) tr(h(t) 2 H0) 2 = b(n, t) 2 + 2(a(n, t) ) = 1 H(p, q) 2 n Z correspon to conservation of the total momentum an the total energy, respectively. The Lax pair approach was first avocate by Lax [29] in connection with the Korteweg-e Vries equation. The results presente here are ue to Flaschka [16], [17]. More informations on the trace formulas an conserve quantities can be foun in Gesztesy an Holen [18] an Teschl [37]. 3 Types of solutions The reformulation of the Toa equation as a Lax pair is the key to methos for solving the Toa equation base on spectral an inverse spectral theory for the Jacobi operator H (triiagonal infinite matrix). But before we go into further etails let me first show what kin of solutions one can obtain by these methos. The first type of solution is the general N-soliton solution (14) where (15) q N (n, t) = q 0 ln et(1l + C N (n, t)) et(1l + C N (n 1, t)), ( ) γi γ j sinh(κi)+σj sinh(κj))t C N (n, t) = e (κi+κj)n (σi 1 e (κi+κj) 1 i,j N with κ j, γ j > 0 an σ j {±1}. The case N = 1 coincies with the one soliton solution from the first section. Two examples with N = 2 are epicte below. These solutions can be obtaine by either factorizing the unerlying Figure 3: Two solitons, one overtaking the other Jacobi operator accoring to H = AA an then commuting the factors or, alternatively, by the inverse scattering transform. 5

6 Figure 4: Two solitons traveling in ifferent irections The secon class of solutions are (quasi-)perioic solutions which can be foun using techniques from Riemann surfaces (respectively algebraic curves). Each such solution is associate with a hyperelliptic curve of the type (16) w 2 = (z E j ), E j R, 2g+1 j=0 where E j, 0 j 2g + 1, are the ban eges of the spectrum of H (which is inepenent of t an hence etermine by the initial conitions). One obtains (17) q(n, t) = q 0 2(t b + n ln(2ã)) ln θ(z 0 2nA p0 ( + ) 2tc(g)) θ(z 0 2(n 1)A p0 ( + ) 2tc(g)), where z 0 R g, θ : R g R is the Riemann theta function associate with the hyperelliptic curve (16), an ã, b R, A p0 ( + ), c(g) R g are constants epening only on the curve (i.e., on E j, 0 j 2g + 1). If q(n, 0), p(n, 0) are (quasi-) perioic with average 0, then ã = 1 2, b = 0. Figure 5: A perioic solution associate with w 2 = (z 2 2)(z 2 1) How these solutions can be obtaine will be outline in the following sections. These methos can also be use to combine both types of solutions an put N solitons on top of a given perioic solution. 4 The Toa hierarchy The Lax approach allows for a straightforwar generalization of the Toa equation by replacing P 2 with more general operators P 2r+2 of orer 2r+2. This yiels the Toa hierarchy (18) t H(t) = P 2r+2(t)H(t) H(t)P 2r+2 (t) TL r (a, b) = 0. 6

7 To etermine the amissible operators P 2r+2 (i.e., those for which the commutator with H is of orer 2) one restricts them to the algebraic kernel of H z (19) where (20) (P 2r+2 Ker(H z) ) = 2aG r (z)s + H r+1 (z), G r (z) = g r j z j, H r+1 (z) = z r+1 + h r j z j g r+1. j=0 j=0 Inserting this into (18) shows after a long an tricky calculation that the coefficients are given by the iagonal an off-iagonal matrix elements of H j, (21) g j (n) = δ n, H j δ n, h j (n) = 2a(n) δ n+1, H j δ n. Here.,.. enotes the scalar prouct in l 2 (Z) an δ n (m) = 1 for m = n respectively δ n (m) = 0 for m n is the canonical basis. The r-th Toa equation is then explicitly given by (22) ȧ(t) = a(t)(g + r+1 (t) g r+1(t)), ḃ(t) = h r+1 (t) h r+1 (t). The coefficients g j (n) an h j (n) can be compute recursively. The Toa hierarchy was first consiere by Ueno an Takasaki [44], [45]. The recursive approach for the stanar Lax formalism, [29] was first avocate by Al ber [2]. Here I followe Bulla, Gesztesy, Holen, an Teschl [8]. 5 The Kac-van Moerbeke hierarchy (23) Consier the super-symmetric Dirac operator ( ) 0 A(t) D(t) =, A(t) 0 an choose (24) where A(t) = ρ o (t)s + + ρ e (t), A(t) = ρ o (t)s + ρ e (t), (25) ρ e (n, t) = ρ(2n, t), ρ o (n, t) = ρ(2n + 1, t) are the even an o parts of some boune sequence ρ(t). Then D(t) is associate with two Jacobi operators (26) H 1 (t) = A(t) A(t), H 2 (t) = A(t)A(t), 7

8 whose coefficients rea (27) a 1 (t) = ρ e (t)ρ o (t), b 1 (t) = ρ e (t) 2 + ρ o (t) 2, a 2 (t) = ρ + e (t)ρ o (t), b 2 (t) = ρ e (t) 2 + ρ o (t) 2. The corresponing Lax equation (28) where (29) t D(t) = Q 2r+2(t)D(t) D(t)Q 2r+2 (t), Q 2r+2 (t) = ( P1,2r+2 (t) 0 0 P 2,2r+2 (t) gives rise to evolution equations for ρ(t) which are known as the Kac-van Moerbeke hierarchy, KM r (ρ) = 0. The first one (the Kac-van Moerbeke equation) explicitly reas ), (30) KM 0 (ρ) = ρ(t) ρ(t)(ρ + (t) 2 ρ (t) 2 ) = 0. Moreover, from the way we introuce the Kac-van Moerbeke hierarchy, it is not surprising that there is a close connection with the Toa hierarchy. To reveal this connection all one has to o is to insert ( ) D(t) 2 H1 (t) 0 (31) = 0 H 2 (t) into the Lax equation (32) t D(t)2 = Q 2r+2 (t)d(t) 2 D(t) 2 Q 2r+2 (t), which shows that the Lax equation (28) for D(t) implies the Lax equation (18) for both H 1 an H 2. This observation gives a Bäcklun transformation between the Kac-van Moerbeke an the Toa hierarchies: Theorem 2. For any given solution ρ(t) of KM r (ρ) = 0 we obtain, via (27), two solutions (a j (t), b j (t)),2 of TL r (a, b) = 0. This is the analog of the Miura transformation between the moifie an the original Korteweg-e Vries hierarchies. The Kac-van Moerbeke equation has been first introuce by Kac an van Moerbeke in [23]. The Bäcklun transformation connecting the Toa an the Kac-van Moerbeke equations has first been consiere by Toa an Waati in [43]. 8

9 6 Commutation methos Clearly, it is natural to ask whether this transformation can be inverte. In other wors, can we factor a given Jacobi operator H as A A an then compute the corresponing solution of the Kac-van Moerbeke hierarchy plus the secon solution of the Toa hierarchy? This can in fact be one. All one nees is a positive solution of the system (33) H(t)u(n, t) = 0, t u(n, t) = P 2r+2(t)u(n, t) an then one has a(t)u(t) ρ o (t) = u +, (t) (34) ρ e (t) = a(t)u + (t). u(t) In particular, starting with the trivial solution a 0 (n, t) = 1 2, b 0(n, t) = 0 an proceeing inuctively one ens up with the N-soliton solutions. The metho of factorizing H an then commuting the factors is known as Darboux transformation an is of inepenent interest since it has the property of inserting a single eigenvalue into the spectrum of H. Commutation methos for Jacobi operators in connection with the Toa an Kac-van Moerbeke equation were first consiere by Gesztesy, Holen, Simon, an Zhao [22]. For further generalizations, see Gesztesy an Teschl [20] an Teschl [38]. A secon way to obtain the N-soliton solution is via the inverse scattering transform, which was first worke out by Flaschka in [17]. 7 Stationary solutions In the remaining sections I woul like to show how two at first sight unrelate fiels of mathematics, spectral theory an algebraic geometry, can be combine to fin (quasi-)perioic solutions of the Toa equations. To reveal this connection, we first look at stationary solutions of the Toa hierarchy or, equivalently, at commuting operators (35) P 2r+2 H HP 2r+2 = 0. In this case a short calculation gives (36) (P 2r+2 Ker(H z) ) 2 = H r+1 (z) 2 4a 2 G r (z)g + r (z) =: R 2r+2 (z), where R 2r+2 (z) can be shown to be inepenent of n. That is, it is of the form (37) R 2r+2 (z) = (z E j ) 2r+1 j=0 9

10 for some constant numbers E j R. In particular, this implies (38) (P 2r+2 ) 2 = (H E j ) 2r+1 j=0 an the polynomial w 2 = 2r+1 j=0 (z E j) is known as the Burchnall-Chauny polynomial of P 2r+2 an H. In particular, the connection between the stationary Toa hierarchy an the hyperelliptic curve (39) K = {(z, w) C 2 w 2 = (z E j )} 2r+1 j=0 is apparent. But how can it be use to solve the Toa equation? This will be shown next. We will for simplicity assume that our curve is nonsingular, that is, that E j < E j+1 for all j. The fact that two commuting ifferential or ifference operators satisfy a polynomial relation, was first shown by Burchnall an Chauny [9], [10]. The approach to stationary solutions presente here follows again Bulla, Gesztesy, Holen, an Teschl [8]. 8 Jacobi operators associate with stationary solutions Next some spectral properties of the Jacobi operators associate with stationary solutions are neee. First of all, one can show that (40) g(z, n) = G r(z, n) R 1/2 2r+2 (z) = δ n, (H z) 1 δ n, h(z, n) = H r+1(z, n) R 1/2 2r+2 (z) = δ n+1, (H z) 1 δ n. This is not too surprising, since g j an h j are by (21) just the expansion coefficients in the Neumann series of the resolvent. But once we know the iagonal of the resolvent we can easily rea off the spectrum of H. The open branch cuts of R 1/2 2r+2 (z) form an essential support of the absolutely continuous spectrum an the branch points support the singular spectrum. Since at each branch point we have a square root singularity, there can be no eigenvalues an since the singular continuous spectrum cannot be supporte on finitely many points, the spectrum is purely absolutely continuous an consists of a finite number of bans. µ 1(n) E 0 E 1 E 2 E 3 µ2(n) E 4 E 5 10

11 The points µ j (n) are the zeros of G r (z, n), (41) r G r (z, n) = (z µ j (n)), an can be interprete as the eigenvalues of the operator H n obtaine from H by imposing an aitional Dirichlet bounary conition u(n) = 0 at n. Since H n ecomposes into a irect sum H,n H +,n we can also associate a sign σ j (n) with µ j (n), inicating whether it is an eigenvalue of H,n or H +,n. Theorem 3. The ban eges {E j } 0 j 2r+1 together with the Dirichlet ata {(µ j (n), σ j (n))} 1 j r for one n uniquely etermine H. Moreover, it is even possible to write own explicit formulas for a(n + k) an b(n + k) for all k Z as functions of these ata. Explicitly one has b(n) = b (1) (n) a(n 0 1 )2 = b(2) (n) b(n) 2 ± 4 b(n ± 1) = σ j (n)r 1/2 2r+2 (µ j(n)) 2 k j (µ j(n) µ k (n)) 1 ( 2b (3) (n) 3b(n)b (2) (n) + b(n) 3 a(n 0 1 )2 12 σ j (n)r 1/2 2r+2 ± (µ j(n))µ j (n)) 2 k j (µ j(n) µ k (n)) (42) where (43) b (l) (n) = r+1 j=0 E l j µ j (n) l. These formulas alreay inicate that ˆµ j (n) = (µ j (n), σ j (n)) shoul be consiere as a point on the Riemann surface K of R 1/2 2r+2 (z), where σ j(n) inicates on which sheet µ j (n) lies. The result for perioic operators is ue to van Moerbeke [30], the general case was given by Gesztesy, Krishna, an Teschl [21]. Trace formulas for Sturm-Liouville an also for Jacobi operators have a long history. The formulas for b (l), l = 1, 2, were alreay given in [30] for the perioic case. The formulas presente here an in particular the fact that the coefficients a an b can be explicitly written own in terms of minimal spectral ata are ue to Teschl [36]. Most proofs use results on orthogonal polynomials an the moment problem. One of the classical references is [1], for a recent review article see Simon [35]. 11

12 9 Algebro-geometric solutions of the Toa equations The iea now is to choose a stationary solution of TL r (a, b) = 0 as the initial conition for TL s (a, b) an to consier the time evolution in our new coorinates {E j } 0 j 2r+1 an {(µ j (n), σ j (n))} 1 j r. From unitary equivalence of the family of operators H(t) we know that the ban eges E j o not epen on t. Moreover, the time evolution of the Dirichlet ata follows from the Lax equation (44) t (H(t) z) 1 = [P 2s+2 (t), (H(t) z) 1 ]. Inserting (40) an (41) yiels (45) t µ j(n, t) = 2G s (µ j (n, t), n, t) σ j(n, t)r 1/2 2r+2 (µ j(n, t)) k j (µ k(n, t) µ j (n, t)), where G s (z) has to be expresse in terms of µ j using (42). Again, this equation shoul be viewe as a ifferential equation on K rather than R. A closer investigation shows that each Dirichlet eigenvalue µ j (n, t) rotates in its spectral gap. At first sight it looks like we have not gaine too much since this flow is still highly nonlinear, but it can be straightene out using Abel s map from algebraic geometry. So let us review some basic facts first. Our hyperelliptic curve K is in particular a compact Riemann surface of genus r an hence it has a basis of r holomorphic ifferentials which are explicitly given by (46) ζ j = k=1 c j (k) zk 1 z R 1/2 2r+2 (z). (At first sight these ifferentials seem to have poles at each ban ege, but near such a ban ege we nee to use a chart z E j = w 2 an z = 2ww shows that each zero in the enominator cancels with a zero in the numerator). Given a homology basis a j, b j for K they are usually normalize such that (47) ζ k = δ j,k a j an one sets ζ k =: τ jk. b j Now the Jacobi variety associate with K is the r-imensional torus C r mo L, where L = Z r + τz r an the Abel map is given by (48) A p0 (p) = p p 0 ζ mo L, p, p 0 K. 12

13 Theorem 4. The Abel map straightens out the ynamical system ˆµ j (0, 0) ˆµ j (n, t) both with respect to n an t (49) A p0 (ˆµ j (n, t)) = A p0 (ˆµ j (0, 0)) 2nA p0 ( + ) tu s, where U s can be compute explicitly in terms of the ban eges E j. (50) Sketch of proof. Consier the function (compare (19)) φ(p, n, t) = H r+1(p, n, t) + R 1/2 2r+2 (p) = 2a(n, t)g r(p, n + 1, t) p K, 2a(n, t)g r (p, n, t) H r+1 (p, n, t) R 1/2 2r+2 (p), whose zeros are ˆµ j (n + 1, t), an whose poles are ˆµ j (n, t), +. Abel s theorem implies (51) A p0 ( + ) + A p0 (ˆµ j (n, t)) = A p0 ( ) + A p0 (ˆµ j (n + 1, t)), which settles the first claim. To show the secon claim we compute (52) t A p0 (ˆµ j ) = µ j c(k) = 2 k=1 µ k 1 j σ j R 1/2 2r+2 (µ j) G s (µ j ) c(k) l j (µ j µ l ) µk 1 The key iea is now to rewrite this as an integral G s (z) (53) G r (z) zk 1 z, Γ j,k j. where Γ is a close path encircling all points µ j. By (41) this is equal to the above expression by the resiue theorem. Moreover, since the integran is rational we can also compute this integral by evaluating the resiue at, which is given by (54) G s (z) G r (z) = G s(z) g(z) = 2 1 2r+2 (z) = zs+1 (1 + O(z s )) R 1/2 2r+2 (z) c(l) s r+l (E) =: U s, R 1/2 l=max{1,r s} since the coefficients of G s coincie with the first s coefficients in the Neumann series of g(z) by (21). Here j (E) are just the coefficients in the asymptotic expansion of 1/R 1/2 2r+2 (z). 13

14 Since the poles an zeros of the function φ(z), which appeare in the proof of the last theorem, as well as their image uner the Abel map are known, a function having the same zeros an poles can be written own using Riemann theta functions (Jacobi s inversion problem an Riemann s vanishing theorem). The Riemann Roch theorem implies that both functions coincie. Finally, the function φ(z) has also a spectral interpretation as Weyl m-function, an thus explicit formulas for the coefficients a an b can be obtaine from the asymptotic expansion for z. This prouces the formula in equation (17). The first results on algebro-geometric solutions of the Toa equation were given by Date an Tanaka [11]. Further important contributions were mae by Krichever, [24] [28], van Moerbeke an Mumfor [31], [32]. The presentation here follows Bulla, Gesztesy, Holen, an Teschl [8] respectively Teschl [39]. Another possible approach is to irectly use the spectral function of H an to consier its t epenence, see Berezanskiĭ an coworkers [3] [7]. For some recent evelopments base on Lie theoretic methos an loop groups I again recommen the review by Palais [33] as starting point. Acknowlegments I thank Fritz Gesztesy for his careful scrutiny of this article leaing to several improvements, as well as Wolfgang Bulla an Karl Unterkofler for many valuable suggestions. References [1] N. Akhiezer, The Classical Moment Problem, Oliver an Boy, Lonon, [2] S. J. Al ber, Associate integrable systems, J. Math. Phys. 32, (1991). [3] Yu. M. Berezanskiĭ, Integration of nonlinear ifference equations by the inverse spectral problem metho, Soviet Math. Dokl., 31 No. 2, (1985). [4] Yu. M. Berezanski, The integration of semi-infinite Toa chain by means of inverse spectral problem, Rep. Math. Phys., 24 No. 1, (1985). [5] Yu. M. Berezansky, Integration of nonlinear nonisospectral ifferenceifferential equations by means of the inverse spectral problem, in Nonlinear Physics. Theory an experiment, (es E. Alfinito, M. Boiti, L. Martina, F. Pempinelli), Worl Scientific, (1996). 14

15 [6] Yu. M. Berezansky an M. I. Gekhtman, Inverse problem of the spectral analysis an non-abelian chains of nonlinear equations, Ukrain. Math. J., 42, (1990). [7] Yu. Berezansky an M. Shmoish, Nonisospectral flows on semi-infinite Jacobi matrices, Nonl. Math. Phys., 1 No. 2, (1994). [8] W. Bulla, F. Gesztesy, H. Holen, an G. Teschl Algebro-Geometric Quasi-Perioic Finite-Gap Solutions of the Toa an Kac-van Moerbeke Hierarchies, Mem. Amer. Math. Soc , [9] J. L. Burchnall an T. W. Chauny, Commutative orinary ifferential operators, Proc. Lonon Math. Soc. Ser. 2, 21, (1923). [10] J. L. Burchnall an T. W. Chauny, Commutative orinary ifferential operators, Proc. Roy. Soc. Lonon A118, (1928). [11] E. Date an S. Tanaka Analogue of inverse scattering theory for the iscrete Hill s equation an exact solutions for the perioic Toa lattice, Prog. Theoret. Phys. 56, (1976). [12] P. Deift, L.C. Li, an C. Tomei, Toa flows with infinitely many variables, J. Func. Anal. 64, (1985). [13] S. N. Eilenberger, An Introuction to Difference Equations, Springer, New York, [14] L. Faeev an L. Takhtajan, Hamiltonian Methos in the Theory of Solitons, Springer, Berlin, [15] E. Fermi, J. Pasta, S. Ulam, Stuies of Nonlinear Problems, Collecte Works of Enrico Fermi, University of Chicago Press, Vol.II, ,1965. Theory, Methos, an Applications, 2n e., Marcel Dekker, New York, [16] H. Flaschka, The Toa lattice. I. Existence of integrals, Phys. Rev. B 9, (1974). [17] H. Flaschka, On the Toa lattice. II. Inverse-scattering solution, Progr. Theoret. Phys. 51, (1974). [18] F. Gesztesy an H. Holen, Trace formulas an conservation laws for nonlinear evolution equations, Rev. Math. Phys. 6, (1994). [19] F. Gesztesy an H. Holen, Soliton Equations an their Algebro- Geometric Solutions I III, Cambrige Series in Avance Mathematics, in preparation. 15

16 [20] F. Gesztesy an G. Teschl, Commutation methos for Jacobi operators, J. Diff. Eqs. 128, (1996). [21] F. Gesztesy, M. Krishna, an G. Teschl, On isospectral sets of Jacobi operators, Com. Math. Phys. 181, (1996). [22] F. Gesztesy, H. Holen, B. Simon, an Z. Zhao, On the Toa an Kac-van Moerbeke systems, Trans. Amer. Math. Soc. 339, (1993). [23] M. Kac an P. van Moerbeke, On an explicitly soluble system of nonlinear ifferential equations, relate to certain Toa lattices, Av. Math. 16, (1975). [24] I. M. Krichever, Algebraic curves an nonlinear ifference equations, Russian Math. Surveys. 334, (1978). [25] I. M. Krichever, Nonlinear equations an elliptic curves, Rev. of Science an Technology 23, (1983). [26] I. M. Krichever, Algebro-geometric spectral theory of the Schröinger ifference operator an the Peierls moel, Soviet Math. Dokl. 26, (1982). [27] I. M. Krichever, The Peierls moel, Funct. Anal. Appl. 16, (1982). [28] I. Krichever, Algebraic-geometrical methos in the theory of integrable equations an their perturbations, Acta Appl. Math. 39, (1995). [29] P. D. Lax Integrals of nonlinear equations of evolution an solitary waves, Comm. Pure an Appl. Math. 21, (1968). [30] P. van Moerbeke, The spectrum of Jacobi Matrices, Inv. Math. 37, (1976). [31] P. van Moerbeke an D. Mumfor The spectrum of ifference operators an algebraic curves, Acta Math. 143, (1979). [32] D. Mumfor, An algebro-geometric construction of commuting operators an of solutions to the Toa lattice equation, Korteweg e Vries equation an relate non-linear equations, Intl. Symp. Algebraic Geometry, , Kyoto, [33] R. S. Palais, The symmetries of solitons, Bull. Amer. Math. Soc., 34, (1997). 16

17 [34] J. S. Russel, Report on waves, 14th Mtg. of the British Assoc. for the Avance of Science, John Murray, Lonon, pp plates, [35] B. Simon, The classical moment problem as a self-ajoint finite ifference operator, Avances in Math. 137, (1998). [36] G. Teschl, Trace Formulas an Inverse Spectral Theory for Jacobi Operators, Comm. Math. Phys. 196, (1998). [37] G. Teschl, Inverse scattering transform for the Toa hierarchy, Math. Nach. 202, (1999). [38] G. Teschl, On the Toa an Kac-van Moerbeke hierarchies, Math. Z. 231, (1999). [39] G. Teschl, Jacobi Operators an Completely Integrable Nonlinear Lattices, Math. Surv. an Monographs 72, Amer. Math. Soc., Rhoe Islan, 2000 [40] M. Toa, Theory of Nonlinear Lattices, 2n enl. e., Springer, Berlin, [41] M. Toa, Theory of Nonlinear Waves an Solitons, Kluwer, Dorrecht, [42] M. Toa, Selecte Papers of Morikazu Toa, e. by M. Waati, Worl Scientific, Singapore, [43] M. Toa an M. Waati, A canonical transformation for the exponential lattice, J. Phys. Soc. Jpn. 39, (1975). [44] K. Ueno an K. Takasaki, Toa lattice hierarchy, in Avance Stuies in Pure Mathematics 4, (e. K. Okamoto), North-Hollan, Amsteram, 1 95 (1984). [45] K. Ueno an K. Takasaki, Toa lattice hierarchy. I, Proc. Japan Aca., Ser. A 59, (1983). [46] K. Ueno an K. Takasaki, Toa lattice hierarchy. II, Proc. Japan Aca., Ser. A 59, (1983). [47] N. J. Zabusky an M. D. Kruskal, Interaction of solitons in a collisionless plasma an the recurrence of initial states, Phys. Rev. Lett. 15, (1965). 17

18 Geral Teschl Institut für Mathematik Strulhofgasse 4 A-1090 Wien URL: geral/ Eingegangen

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