QUASI-PERIODIC SOLUTIONS OF THE DISCRETE mKdV HIERARCHY

Abstract

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...

The discrete mKdV equation revisited: a Riemann-Hilbert approach

“…Here we confine ourselves to the system (1.4). In [6], the minus type dmKdV equation is the compatibility condition of the Lax pair…”

“…we note that there are many other differential-difference equations which can be transformed into the dmKdV equation [8][9][10][11][12][13][14][15]. The dmKdV equation has widely applications in the fields as plasma physics, electromagnetic waves in ferromagnetic, antiferromagnetic or dielectric systems, and can be solved by the method of inverse scattering transform, Hirota bilinear, Algebro-geometric approach and others [3,6,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Rational solutions for three semi-discrete modified Korteweg–de Vries-type equations

“…Recently, soliton solutions and Jordan-block solutions for the equation (1.2) [21] was derived through the generalized Cauchy matrix approach [22]. For the sd-mKdV equation (1.3), many approaches, such as inverse scattering transform [23,24], Darboux transformation [25,26], bilinear approach [27], discrete Jacobi sub-equation method [28], algebro-geometric approach [29], Riemann-Hilbert approach [30] and Deift-Zhou nonlinear steepest descent method [31], have been developed to construct its exact solutions.…”

Algebro‐geometric constructions of the Wadati‐Konno‐Ichikawa flows and applications

“…Seeking algebro‐geometric solutions of soliton equations is an important and interesting subject in recent years, because algebro‐geometric solutions describe the quasi‐periodic behavior of nonlinear phenomenon and reveal inherent structure mechanism of solutions. In a series of papers , algebro‐geometric solutions for many soliton equations associated with 2 × 2 or 3 × 3 matrix spectral problems have been obtained, such as the Korteweg‐de Vries KdV, nonlinear Schrödinger, modified KdV, discrete modified KdV, Boussinesq, sine‐Gordon, Toda lattice, and Camassa‐Holm equations. The main aim of this paper is to construct the explicit theta function representations of solutions for the entire Wadati–Konno–Ichikawa (WKI) hierarchy related to WKI equations which were first discovered by Wadati, Konno, and Ichikawa in 1979 .…”