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.

scholarly journals QUASI-PERIODIC SOLUTIONS OF THE RELATIVISTIC TODA HIERARCHY

2012 ◽  
Vol 19 (04) ◽  
pp. 1250030 ◽  
Author(s):  
DONG GONG ◽  
XIANGUO GENG
Keyword(s):  
Meromorphic Function ◽  
Periodic Solutions ◽  
Continuous Flow ◽  
Hyperelliptic Curve ◽  
Algebraic Curves ◽  
Asymptotic Properties ◽  
Toda Hierarchy ◽  
Discrete Flow

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.

A hierarchy of generalized Jaulent–Miodek equations and their explicit solutions

2017 ◽  
Vol 15 (01) ◽  
pp. 1850002
Author(s):  
Xianguo Geng ◽  
Liang Guan ◽  
Bo Xue
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Theta Function ◽  
Algebraic Curves ◽  
Explicit Solutions ◽  
Two Systems ◽  
Energy Dependent

A hierarchy of generalized Jaulent–Miodek (JM) equations related to a new spectral problem with energy-dependent potentials is proposed. Depending on the Lax matrix and elliptic variables, the generalized JM hierarchy is decomposed into two systems of solvable ordinary differential equations. Explicit theta function representations of the meromorphic function and the Baker–Akhiezer function are constructed, the solutions of the hierarchy are obtained based on the theory of algebraic curves.

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.

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.

scholarly journals ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY

1999 ◽  
Vol 11 (07) ◽  
pp. 823-879 ◽  
Author(s):  
R. DICKSON ◽  
F. GESZTESY ◽  
K. UNTERKOFLER
Keyword(s):  
Periodic Solutions ◽  
Theta Function ◽  
Systematic Approach ◽  
Recursive Construction ◽  
Lax Pairs ◽  
Function Representation ◽  
Geometric Solutions

We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes associated Burchnall–Chaundy curves, Baker–Akhiezer functions and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. The principal aim of this paper is a detailed theta function representation of all algebro-geometric quasi-periodic solutions and related quantities of the Bsq 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.

QUASI-PERIODIC SOLUTIONS OF THE DISCRETE mKdV HIERARCHY

2013 ◽  
Vol 10 (03) ◽  
pp. 1250094 ◽  
Author(s):  
XIANGUO GENG ◽  
DONG GONG
Keyword(s):  
Meromorphic Function ◽  
Periodic Solutions ◽  
Continuous Flow ◽  
Hyperelliptic Curve ◽  
Algebraic Curves ◽  
Asymptotic Properties ◽  
Mkdv Hierarchy

On the basis of the theory of algebraic curves, the continuous flow related to the discrete mKdV hierarchy is 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 mKdV hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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