From 2D Toda hierarchy to conformal maps for domains of the Riemann sphere

2004 ◽  
pp. 207-218 ◽  
Author(s):  
Yu. Klimov ◽  
A. Korzh ◽  
S. Natanzon
Keyword(s):  
Riemann Sphere ◽  
Conformal Maps ◽  
Toda Hierarchy

scholarly journals Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

2021 ◽  
Vol 24 (3) ◽  
Author(s):  
Alexander I. Bobenko ◽  
Ulrike Bücking
Keyword(s):  
Error Estimate ◽  
Branch Point ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Convergence Result ◽  
Edge Length ◽  
Ramified Covering ◽  
Compact Riemann Surfaces ◽  
Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

Circular Arcs Fitted on a Riemann Sphere

1997 ◽  
Vol 67 (3) ◽  
pp. 311-317 ◽  
Author(s):  
Bjørn Lillekjendlie
Keyword(s):  
Riemann Sphere ◽  
Circular Arcs

Non-local integrable equations as reductions of the Toda hierarchy

1991 ◽  
Vol 160 (2) ◽  
pp. 166-172 ◽  
Author(s):  
D. Lebedev ◽  
A. Orlov ◽  
S. Pakuliak ◽  
A. Zabrodin
Keyword(s):  
Integrable Equations ◽  
Toda Hierarchy ◽  
Non Local

On the Full-Discrete Extended Generalised q-Difference Toda System

2017 ◽  
Vol 72 (8) ◽  
pp. 703-709
Author(s):  
Chuanzhong Li ◽  
Anni Meng
Keyword(s):  
Difference Equation ◽  
Hamiltonian Structure ◽  
Soliton Solutions ◽  
Toda Equation ◽  
Lax Pairs ◽  
Toda System ◽  
Toda Hierarchy

AbstractIn this paper, we construct a full-discrete integrable difference equation which is a full-discretisation of the generalised q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of an extended generalised full-discrete q-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the extended full-discrete generalised q-Toda hierarchy are given.

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