scholarly journals Algebro-Geometric Solutions for a Discrete Integrable Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Mengshuang Tao ◽  
Huanhe Dong
Keyword(s):  
Integrable Systems ◽  
Theta Functions ◽  
Discrete Systems ◽  
Integrable Equation ◽  
Integrable Hierarchy ◽  
Discrete Integrable Systems ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Riemann Theta Functions ◽  
Geometric Solutions

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

scholarly journals Direct linearization approach to discrete integrable systems associated with ℤ 𝒩 graded Lax pairs

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200036
Author(s):  
Wei Fu
Keyword(s):  
Difference Equations ◽  
Bilinear Form ◽  
Integrable Systems ◽  
Solution Structure ◽  
Discrete Systems ◽  
Lax Pairs ◽  
Discrete Integrable Systems ◽  
Direct Linearization ◽  
Linear Integral ◽  
Linear Integral Equations

Fordy and Xenitidis [ J. Phys. A: Math. Theor. 50 (2017) 165205. ( doi:10.1088/1751-8121/aa639a )] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of Z N graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel’fand–Dikii hierarchy.

Algebro-Geometric Solutions of the Harry Dym Hierarchy

2017 ◽  
Vol 18 (2) ◽  
pp. 129-136
Author(s):  
Zhu Li
Keyword(s):  
Meromorphic Function ◽  
Algebraic Curve ◽  
Asymptotic Properties ◽  
Arithmetic Genus ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Geometric Solutions

AbstractThe Harry Dym hierarchy is derived with the help of Lenard recursion equations and zero curvature equation. Based on the Lax matrix, an algebraic curve $\mathcal{K}_{n}$ of arithmetic genus $n$ is introduced, from which the corresponding meromorphic function $\phi$ and Dubrovin-type equations are given. Further, the divisor and asymptotic properties of $\phi$ are studied. Finally, algebro-geometric solutions for the entire hierarchy are obtained according to above results and the theory of algebraic curve.

A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system

2021 ◽  
pp. 2150156
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Hamiltonian System ◽  
Integrable Model ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Extended System ◽  
Akns Hierarchy ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Higher Dimensional

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

SUPER INTEGRABLE HIERARCHY AND SUPER HAMILTONIAN STRUCTURES ASSOCIATED WITH GUO HIERARCHY

2009 ◽  
Vol 23 (29) ◽  
pp. 3491-3496 ◽  
Author(s):  
NING ZHANG ◽  
HUANHE DONG
Keyword(s):  
Integrable Hierarchies ◽  
Lie Superalgebra ◽  
Integrable Hierarchy ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

A Lie superalgebra is constructed from which establishes an isospectral problems. By solving the zero curvature equation, a resulting super hierarchies of the Guo hierarchy are obtained. By making use of the super identity, the Hamiltonian structures of the above super integrable hierarchies are generated, this method can be used to other superhierarchy.

scholarly journals The generalized Giachetti-Johnson hierarchy and algebro-geometric solutions of the coupled KdV-MKdV equation

2019 ◽  
Vol 23 (3 Part A) ◽  
pp. 1697-1702
Author(s):  
Chao Yue ◽  
Tiecheng Xia ◽  
Guijuan Liu ◽  
Qiang Lu ◽  
Ning Zhang
Keyword(s):  
Lie Algebra ◽  
Theta Functions ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Riemann Theta Functions ◽  
Geometric Solutions

By using a Lie algebra A1, an isospectral Lax pair is introduced from which a generalized Giachetti-Johnson hierarchy is generated, which reduce to the coupled KdV-MKdV equation, furthermore, the algebro-geometric solutions of the coupled KdV-MKdV equation are constructed in terms of Riemann theta functions.

scholarly journals Trigonal curves and algebro-geometric solutions to soliton hierarchies II

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170233 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
General Theory ◽  
Theta Function ◽  
Theta Functions ◽  
Lax Pairs ◽  
Riemann Theta Function ◽  
Trigonal Curves ◽  
Soliton Hierarchy ◽  
Riemann Theta Functions ◽  
Geometric Solutions ◽  
Asymptotic Behaviours

This is a continuation of a study on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, we straighten out all flows in soliton hierarchies under the Abel–Jacobi coordinates associated with Lax pairs, upon determining the Riemann theta function representations of the Baker–Akhiezer functions, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviours of the Baker–Akhiezer functions. We emphasize that we analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.

Algebro-geometric integration of a modified shallow wave hierarchy

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Guoliang He ◽  
Yunyun Zhai ◽  
Zhenzhen Zheng
Keyword(s):  
Wave Equations ◽  
Meromorphic Functions ◽  
Asymptotic Properties ◽  
Trigonal Curve ◽  
Recursion Relations ◽  
Curvature Equation ◽  
The Third ◽  
Zero Curvature ◽  
Matrix Spectral Problem ◽  
Geometric Solutions

Abstract By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

VARIABLE SEPARATION AND ALGEBRAIC-GEOMETRIC SOLUTIONS OF MODIFIED TODA LATTICE EQUATION

2010 ◽  
Vol 24 (19) ◽  
pp. 2041-2055
Author(s):  
LIN LUO ◽  
ENGUI FAN
Keyword(s):  
Toda Lattice ◽  
Theta Functions ◽  
Variable Separation ◽  
Lattice Equation ◽  
Riemann Theta Functions ◽  
Jacobi Inversion ◽  
Toda Lattice Equation ◽  
Hyper Elliptic Curve ◽  
Geometric Solutions ◽  
Toda Lattice Hierarchy

In this paper, we derive a modified Toda lattice hierarchy by resorting to the Lenard operator pairs. The hyper-elliptic curve and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which some algebraic-geometric solutions of the modified Toda lattice are explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.

scholarly journals A New Super Extension of Dirac Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jiao Zhang ◽  
Fucai You ◽  
Yan Zhao
Keyword(s):  
Spectral Problem ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

We derive a new super extension of the Dirac hierarchy associated with a3×3matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy.

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