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The Heisenberg hierarchy and its Hamiltonian structure are derived respectively by virtue of the zero curvature equation and the trace identity. With the help of the Lax matrix we introduce an algebraic curve K n of arithmetic genus n, from which we define meromorphic function φ and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel-Jacobi coordinates. Finally, we achieve the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of φ. physics, optical fibers, acoustics, mechanics, biology and mathematical finance. With the development of soliton theory, many approaches were developed from which quasiperiodic solutions for several soliton equations have been obtained in Refs. , such as the KdV, mKdV, nonlinear Schrödinger, sine-Gordon, Toda lattice and Camassa-Holm equations, etc.In this paper, we would like to construct quasi-periodic solutions of the Heisenberg hierarchy by means of the methods in Refs. [15,21]. Ferromagnetic chain equation was first proposed in 1935 by Landau and Lifshitz when studying the dispersive theory for magnetic conductivity in magnetic materials [22]. It is an important dynamical equation, which has coherent and chaotic structures depending on the nature of magnetic interactions, and frequently appears in condensation physics, quantum physics and other physics fields [23-25]. The Lax integration of the continuous Heisenberg spin chain equation are studied by Takhtajan through the inverse scattering transform method in 1977 [26]. Almost at the same time the single-soliton solution of Heisenberg spin chain in the isotropic case are obtained by Tjon and Wright [27]. The classical solutions of the continuous Heisenberg spin chain has been obtained by Jevicki and Papanicolaou using a path integral formalism that allows for semi-classical quantization of systems with spin degrees of freedom in 1979 [28]. Soon after, an explicit expression is obtained for the Miura transformation which maps the solutions of the continuous Anisotropic Heisenberg Spin Chain on solutions of the Nonlinear Schrödinger equation by Quispel and Capel in 1983 [29]. Li and Chen gave the higher order Heisenberg spin chain equations, and they proved that these evolution equations are equivalent to the evolution equation of AKNS type in 1986 [30]. Afterwards, The role of nonlocal conservation laws and the corresponding charges are analyzed in the supersymmetric Heisenberg spin chain in 1994 [31]. Cao discussed the parametric representation of the finite-band solution of the Heisenberg equation [32]. The algebraic Bethe ansatz equation has been set up for an open Heisenberg spin chain having an impurity of a different type of spin [33]. Qiao gave the involutive solutions of the higher order Heisenberg spin chain equations by virtue of the spectral problem nonlinearization method [34]. Recently, Du derived the Poisson reduction and Lie-Poisson structure for the nonlinearized spectral problem of the Heisenberg hier...

On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the relativistic Toda hierarchy are straightened out using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the relativistic Toda hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

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In this paper we obtain the discrete integrable self-dual network hierarchy associated with a discrete spectral problem. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the discrete self-dual network hierarchy are straightened using the Abel-Jacobi coordinates. The meromorphic function and the Baker-Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the discrete self-dual network hierarchy are constructed with the help of the asymptotic properties and the algebra-geometric characters of the meromorphic function, the Baker-Akhiezer function and the hyperelliptic curve.