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

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170233 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
General Theory ◽  
Theta Function ◽  
Theta Functions ◽  
Lax Pairs ◽  
Riemann Theta Function ◽  
Trigonal Curves ◽  
Soliton Hierarchy ◽  
Riemann Theta Functions ◽  
Geometric Solutions ◽  
Asymptotic Behaviours

This is a continuation of a study on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, we straighten out all flows in soliton hierarchies under the Abel–Jacobi coordinates associated with Lax pairs, upon determining the Riemann theta function representations of the Baker–Akhiezer functions, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviours of the Baker–Akhiezer functions. We emphasize that 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 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 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.

scholarly journals The generalized Giachetti-Johnson hierarchy and algebro-geometric solutions of the coupled KdV-MKdV equation

2019 ◽  
Vol 23 (3 Part A) ◽  
pp. 1697-1702
Author(s):  
Chao Yue ◽  
Tiecheng Xia ◽  
Guijuan Liu ◽  
Qiang Lu ◽  
Ning Zhang
Keyword(s):  
Lie Algebra ◽  
Theta Functions ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Riemann Theta Functions ◽  
Geometric Solutions

By using a Lie algebra A1, an isospectral Lax pair is introduced from which a generalized Giachetti-Johnson hierarchy is generated, which reduce to the coupled KdV-MKdV equation, furthermore, the algebro-geometric solutions of the coupled KdV-MKdV equation are constructed in terms of Riemann theta functions.

scholarly journals An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

1998 ◽  
Vol 10 (03) ◽  
pp. 345-391 ◽  
Author(s):  
F. Gesztesy ◽  
R. Ratnaseelan
Keyword(s):  
Theta Function ◽  
Systematic Approach ◽  
Algebraic Methods ◽  
Trace Formulas ◽  
Lax Pairs ◽  
Akns Hierarchy ◽  
Alternative Approach ◽  
Geometric Setting ◽  
Fundamental Polynomial ◽  
Geometric Solutions

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall–Chaundy curves, Baker–Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.

VARIABLE SEPARATION AND ALGEBRAIC-GEOMETRIC SOLUTIONS OF MODIFIED TODA LATTICE EQUATION

2010 ◽  
Vol 24 (19) ◽  
pp. 2041-2055
Author(s):  
LIN LUO ◽  
ENGUI FAN
Keyword(s):  
Toda Lattice ◽  
Theta Functions ◽  
Variable Separation ◽  
Lattice Equation ◽  
Riemann Theta Functions ◽  
Jacobi Inversion ◽  
Toda Lattice Equation ◽  
Hyper Elliptic Curve ◽  
Geometric Solutions ◽  
Toda Lattice Hierarchy

In this paper, we derive a modified Toda lattice hierarchy by resorting to the Lenard operator pairs. The hyper-elliptic curve and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which some algebraic-geometric solutions of the modified Toda lattice are explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.

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 Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

INTEGRABLE COUPLINGS FOR THE ABLOWITZ–LADIK LATTICE THROUGH ENLARGING ASSOCIATED LAX PAIRS

2006 ◽  
Vol 20 (05) ◽  
pp. 253-259
Author(s):  
NING ZHANG ◽  
XI-XIANG XU ◽  
HONG-XIANG YANG
Keyword(s):  
Discrete Systems ◽  
Lax Pairs ◽  
Spectral Problems ◽  
Soliton Hierarchy ◽  
Integrable Couplings

A direct way to construct integrable couplings for discrete systems is introduced through enlarging associated spectral problems. As an application, the procedure for the Ablowitz–Ladik lattice soliton hierarchy is employed.

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