Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies.

*(English)*Zbl 0906.35099
Mem. Am. Math. Soc. 641, 79 p. (1998).

The monograph represents a deep study of interplay between the Toda and Kac-van Moerbeke hierarchies of integrable nonlinear difference equations and a thorough construction of their algebro-geometric quasi-periodic solutions. The reader will find a substantial account which provides more essential details of this technique than one can usually find in the standard literature. The authors have obtained a number of new results, among which there is a complete derivation of all real-valued algebro-geometric quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy. Many of these results are obtained by original and non-standard methods which are often simpler and more streamlined than the existing ones, and this circumstance makes all exposition fresh and instructive.

In the introduction the authors summarize briefly main results of the work and explain its construction. In Chapter 2 the authors build the Toda hierarchy of integrable difference equations by means of a recursive approach first advocated by Al’ber. This approach, though equivalent to the conventional one, has preference to yield naturally the Burchnal -Chaundy polynomials and hence the underlying hyperelliptic curves, creating in such a way a basis for subsequent algebro-geometric construction.

The stationary Baker-Akhiezer function of the Toda hierarchy is constructed in Chapter 3 by the original method which goes back to a classic representation of positive divisors of an algebraic curve by Jacobi and which has been applied to algebraic integrable equations by Mumford and McKean. The authors express the Baker-Akhiezer function in terms of the Riemann theta-functions, write down accurately the expressions for the normalizing constants, and discuss numerous properties of these functions. Chapter 4 represents a digression into the spectral properties of self-adjoint Jacobi operators in \(l^2(\mathbb{Z})\) for the limit point case. The authors build the Green functions and the Weyl matrices of the Jacobi operators and describe their properties in details. Along with it a construction of appropriate spectra and trace formulas are derived also.

In Chapter 5 the authors construct the algebro-geometric finite-gap solutions for the stationary Toda hierarchy. The major result of this chapter is expression of the Toda variables \(a(n), b(n)\) in terms of the theta-function associated with the appropriate hyperelliptic curve. Some of these formulas are new. In conclusion of the chapter a criterion for the Toda variables to be periodic is presented. In Chapter 6 the authors build the algebro-geometric finite-gap solutions for the non-stationary Toda hierarchy with the stationary solution as initial condition. The authors obtain the equations for the time evolution of the Dirichlet eigenvalues and study properties of their solutions. Then the authors derive for the case of Toda flow the expressions for time-dependent Baker-Akhiezer function and Toda variables in terms of theta-functions.

A connection between the Toda hierarchy and the Kac-van Moerbeke hierarchy is considered in Chapter 7. It appears that these hierarchies are related in such a way as the Gel’fand-Dickey hierarchy and the Drinfeld-Sokolov hierarchy. In all these cases the hierarchies are counterparts of Miura-type transformation which is based on a factorization method. The discrete analog of the properly generalized Miura transformation in connection with the factorization method was employed systematically for the first time by Adler (1981) and developed further by Gesztesy, Holden, Simon and Zhao (1993).

In Chapter 8 the authors summarize basic facts on the finite-gap Dirac-type difference operators and study briefly their spectral properties. At last in chapter 9 the authors complete the main object of the work and construct all real-valued algebro-geometric quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy. Along with it they consider also appropriate isospectral manifolds. In conclusion of the chapter a number of possible applications of these results to different completely integrable lattice models is discussed shortly.

In order to make the exposition self-contained the authors include three Appendices on hyperelliptic curves and theta-functions, on periodic Jacobi operators and on the simplest explicit examples of algebro-geometric solutions for the genus 0 and 1.

In the introduction the authors summarize briefly main results of the work and explain its construction. In Chapter 2 the authors build the Toda hierarchy of integrable difference equations by means of a recursive approach first advocated by Al’ber. This approach, though equivalent to the conventional one, has preference to yield naturally the Burchnal -Chaundy polynomials and hence the underlying hyperelliptic curves, creating in such a way a basis for subsequent algebro-geometric construction.

The stationary Baker-Akhiezer function of the Toda hierarchy is constructed in Chapter 3 by the original method which goes back to a classic representation of positive divisors of an algebraic curve by Jacobi and which has been applied to algebraic integrable equations by Mumford and McKean. The authors express the Baker-Akhiezer function in terms of the Riemann theta-functions, write down accurately the expressions for the normalizing constants, and discuss numerous properties of these functions. Chapter 4 represents a digression into the spectral properties of self-adjoint Jacobi operators in \(l^2(\mathbb{Z})\) for the limit point case. The authors build the Green functions and the Weyl matrices of the Jacobi operators and describe their properties in details. Along with it a construction of appropriate spectra and trace formulas are derived also.

In Chapter 5 the authors construct the algebro-geometric finite-gap solutions for the stationary Toda hierarchy. The major result of this chapter is expression of the Toda variables \(a(n), b(n)\) in terms of the theta-function associated with the appropriate hyperelliptic curve. Some of these formulas are new. In conclusion of the chapter a criterion for the Toda variables to be periodic is presented. In Chapter 6 the authors build the algebro-geometric finite-gap solutions for the non-stationary Toda hierarchy with the stationary solution as initial condition. The authors obtain the equations for the time evolution of the Dirichlet eigenvalues and study properties of their solutions. Then the authors derive for the case of Toda flow the expressions for time-dependent Baker-Akhiezer function and Toda variables in terms of theta-functions.

A connection between the Toda hierarchy and the Kac-van Moerbeke hierarchy is considered in Chapter 7. It appears that these hierarchies are related in such a way as the Gel’fand-Dickey hierarchy and the Drinfeld-Sokolov hierarchy. In all these cases the hierarchies are counterparts of Miura-type transformation which is based on a factorization method. The discrete analog of the properly generalized Miura transformation in connection with the factorization method was employed systematically for the first time by Adler (1981) and developed further by Gesztesy, Holden, Simon and Zhao (1993).

In Chapter 8 the authors summarize basic facts on the finite-gap Dirac-type difference operators and study briefly their spectral properties. At last in chapter 9 the authors complete the main object of the work and construct all real-valued algebro-geometric quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy. Along with it they consider also appropriate isospectral manifolds. In conclusion of the chapter a number of possible applications of these results to different completely integrable lattice models is discussed shortly.

In order to make the exposition self-contained the authors include three Appendices on hyperelliptic curves and theta-functions, on periodic Jacobi operators and on the simplest explicit examples of algebro-geometric solutions for the genus 0 and 1.

Reviewer: E.D.Belokolos (Kyïv)

##### MSC:

35Q58 | Other completely integrable PDE (MSC2000) |

39A70 | Difference operators |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

39A12 | Discrete version of topics in analysis |

35Q51 | Soliton equations |

14H52 | Elliptic curves |