Riemann theta function solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy

2019 ◽  
Vol 140 ◽  
pp. 85-103 ◽  
Author(s):  
Xianguo Geng ◽  
Guoliang He ◽  
Lihua Wu
Keyword(s):  
Theta Function ◽  
Riemann Theta Function

On a Toda lattice hierarchy: Lax pair, integrable symplectic map and algebraic–geometric solution

2018 ◽  
Vol 32 (28) ◽  
pp. 1850344
Author(s):  
Xiao Yang ◽  
Dianlou Du
Keyword(s):  
Spectral Problem ◽  
Theta Function ◽  
Toda Lattice ◽  
Burgers Equation ◽  
Lax Pair ◽  
Discrete Variable ◽  
Riemann Theta Function ◽  
Symplectic Map ◽  
Discrete Counterpart ◽  
Toda Lattice Hierarchy

A Toda lattice hierarchy is studied by introducing a new spectral problem which is a discrete counterpart of the generalized Kaup–Newell spectral problem. Based on the Lenard recursion equation, Lax pair of the hierarchy is given. Further, the discrete spectral problem is nonlinearized into an integrable symplectic map. As a result, an algebraic–geometric solution in Riemann theta function of the hierarchy is obtained. Besides, two equations, the Volterra lattice and a (2[Formula: see text]+[Formula: see text]1)-dimensional Burgers equation with a discrete variable, yielded from the hierarchy are also solved.

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.

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