On a method for constructing algebro-geometric solutions to the zero curvature equation

1997 ◽  
Vol 110 (1) ◽  
pp. 47-56 ◽  
Author(s):  
D. P. Novikov ◽  
R. K. Romanovskii
Keyword(s):  
Curvature Equation ◽  
Zero Curvature ◽  
Geometric Solutions

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.

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.

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.

TWO FAMILIES GENERALIZATION OF AKNS HIERARCHIES AND THEIR HAMILTONIAN STRUCTURES

2010 ◽  
Vol 24 (08) ◽  
pp. 791-805 ◽  
Author(s):  
YUNHU WANG ◽  
XIANGQIAN LIANG ◽  
HUI WANG
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

By means of the Lie algebra G1, we construct an extended Lie algebra G2. Two different isospectral problems are designed by the two different Lie algebra G1 and G2. With the help of the variational identity and the zero curvature equation, two families generalization of the AKNS hierarchies and their Hamiltonian structures are obtained, respectively.

A SOLITON HIERARCHY FROM THE LEVI SPECTRAL PROBLEM AND ITS TWO INTEGRABLE COUPLINGS, HAMILTONIAN STRUCTURE

2009 ◽  
Vol 23 (05) ◽  
pp. 731-739
Author(s):  
YONGQING ZHANG ◽  
YAN LI
Keyword(s):  
Spectral Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Curvature Equation ◽  
Loop Algebras ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Time Part

A soliton-equation hierarchy from the D. Levi spectral problem is obtained under the framework of zero curvature equation. By employing two various multi-component Lie algebras and the loop algebras, we enlarge the Levi spectral problem and the corresponding time-part isospectral problems so that two different integrable couplings are produced. Using the quadratic-form identity yields the Hamiltonian structure of one of the two integrable couplings.

scholarly journals A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Spectral Problem ◽  
Lie Algebra ◽  
Difference Equations ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Zero Curvature Representation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Liouville Integrable

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

Some generalized coupled nonlinear Schrödinger equations and conservation laws

2017 ◽  
Vol 31 (32) ◽  
pp. 1750299 ◽  
Author(s):  
Wei Liu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Matrix Problem ◽  
Nonlinear Schrödinger ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Derivative Nonlinear Schrödinger Equation

Based on zero-curvature equation, a series of new four-component nonlinear Schrödinger-type equations related to a [Formula: see text] matrix problem are proposed by using the polynomial expansion of the spectral parameter. As two special reductions, a generalized coupled nonlinear Schrödinger equation and a generalized coupled derivative nonlinear Schrödinger equation are obtained. And then, the infinite conservation laws for each of these four-component nonlinear Schrödinger-type equations are constructed with the aid of the Riccati-type equations.

scholarly journals On Construction of a Super Hierarchy of the Wadati-Konno-Ichikawa Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xuemei Li ◽  
Lutong Li
Keyword(s):  
Spectral Problem ◽  
Spectral Parameter ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Matrix Spectral Problem

In this paper, a super Wadati-Konno-Ichikawa (WKI) hierarchy associated with a 3×3 matrix spectral problem is derived with the help of the zero-curvature equation. We obtain the super bi-Hamiltonian structures by using of the super trace identity. Infinitely, many conserved laws of the super WKI equation are constructed by using spectral parameter expansions.

Sign in / Sign up

Export Citation Format

Share Document