The Alexander module of a trigonal curve. II

We study the distribution of the total and ordinary ramification points of a trigonal curve over the intersection of this curve with rational curves on a rational normal scroll. We show, among other results, that these intersections may contain all the ramification points of the trigonal curve.

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Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy

Theoretical and Mathematical Physics ◽

10.1134/s0040577917010020 ◽

2017 ◽

Vol 190 (1) ◽

pp. 18-42 ◽

Cited By ~ 2

Author(s):

Xianguo Geng ◽

Xin Zeng

Keyword(s):

Trigonal Curve

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The $$\mathrm {al}$$ al function of a cyclic trigonal curve of genus three

Collectanea mathematica ◽

10.1007/s13348-015-0138-y ◽

2015 ◽

Vol 66 (3) ◽

pp. 311-349

Author(s):

Shigeki Matsutani ◽

Emma Previato

Keyword(s):

Trigonal Curve

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

Analysis and Applications ◽

10.1142/s0219530516500214 ◽

2016 ◽

Vol 15 (05) ◽

pp. 667-697 ◽

Cited By ~ 4

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.

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Algebro-geometric integration of a modified shallow wave hierarchy

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0116 ◽

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.

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