The coupled Sasa–Satsuma hierarchy: Trigonal curve and finite genus solutions

2016 ◽  
Vol 15 (05) ◽  
pp. 667-697 ◽  
Author(s):  
Yunyun Zhai ◽  
Xianguo Geng
Keyword(s):  
Theta Function ◽  
Asymptotic Expansions ◽  
Meromorphic Functions ◽  
Trigonal Curve ◽  
Riemann Theta Function ◽  
Function Representation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Finite Genus

Based on the Lenard recursion equations and the stationary zero-curvature equation, we derive the coupled Sasa–Satsuma hierarchy, in which a typical number is the coupled Sasa–Satsuma equation. The properties of the associated trigonal curve and the meromorphic functions are studied, which naturally give the essential singularities and divisors of the meromorphic functions. By comparing the asymptotic expansions for the Baker–Akhiezer function and its Riemann theta function representation, we arrive at the finite genus solutions of the whole coupled Sasa–Satsuma hierarchy in terms of the Riemann theta function.

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.

Quasi-periodic Solutions to the K(−2, −2) Hierarchy

2016 ◽  
Vol 71 (7) ◽  
pp. 639-645
Author(s):  
Lihua Wu ◽  
Xianguo Geng
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Characteristic Polynomial ◽  
Theta Function ◽  
Hyperelliptic Curve ◽  
Arithmetic Genus ◽  
Riemann Theta Function ◽  
Function Representation

AbstractWith the help of the characteristic polynomial of Lax matrix for the K(−2, −2) hierarchy, we define a hyperelliptic curve 𝒦n+1 of arithmetic genus n+1. By introducing the Baker–Akhiezer function and meromorphic function, the K(−2, −2) hierarchy is decomposed into Dubrovin-type differential equations. Based on the theory of hyperelliptic curve, the explicit Riemann theta function representation of meromorphic function is given, and from which the quasi-periodic solutions to the K(−2, −2) hierarchy are obtained.

Algebro-geometric constructions of the Heisenberg hierarchy

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhu Li
Keyword(s):  
Meromorphic Function ◽  
Theta Function ◽  
Algebraic Curve ◽  
Hamiltonian Structure ◽  
Asymptotic Properties ◽  
Trace Identity ◽  
Arithmetic Genus ◽  
Geometric Constructions ◽  
Curvature Equation ◽  
Zero Curvature

AbstractThe 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 ${\mathcal{K}}_{n}$ 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.

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

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170232 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
General Theory ◽  
Theta Function ◽  
Meromorphic Functions ◽  
Linear Combinations ◽  
Riemann Theta Function ◽  
Characteristic Equations ◽  
General Properties ◽  
Trigonal Curves ◽  
Soliton Hierarchy ◽  
Geometric Solutions

This is the first part of a study, consisting of two parts, on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, explore general properties of meromorphic functions defined as ratios of the Baker–Akhiezer functions, and determine zeros and poles of the Baker–Akhiezer functions and their Dubrovin-type equations. 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.

scholarly journals Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Chao Yue ◽  
Tiecheng Xia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Theta Function ◽  
Meromorphic Functions ◽  
Reaction Diffusion ◽  
Arithmetic Genus ◽  
Trigonal Curve ◽  
Matrix Spectral Problem ◽  
Geometric Solutions

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

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.

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