scholarly journals On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations

1987 ◽  
Vol 14 (1) ◽  
pp. 25-31 ◽  
Author(s):  
V. B. Matveev ◽  
A. O. Smirnov
Keyword(s):  
Theta Function ◽  
Trigonal Curve ◽  
Riemann Theta Function ◽  
Kp Equations

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.

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.

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