Quasi-periodic Jacobi operators and Riemann theta functions

1999 ◽  
pp. 153-165
Author(s):  
Gerald Teschl
Keyword(s):  
Theta Functions ◽  
Jacobi Operators ◽  
Riemann Theta Functions

scholarly journals Topological control of extreme waves

2019 ◽  
Vol 10 (1) ◽  
Author(s):  
Giulia Marcucci ◽  
Davide Pierangeli ◽  
Aharon J. Agranat ◽  
Ray-Kuang Lee ◽  
Eugenio DelRe ◽  
...  
Keyword(s):  
Nonlinear Waves ◽  
Control Strategies ◽  
Theta Functions ◽  
State Diagram ◽  
Rogue Waves ◽  
Nonlinear Phenomena ◽  
Extreme Waves ◽  
Nonlinear Wave Propagation ◽  
Riemann Theta Functions ◽  
The One

Abstract From optics to hydrodynamics, shock and rogue waves are widespread. Although they appear as distinct phenomena, transitions between extreme waves are allowed. However, these have never been experimentally observed because control strategies are still missing. We introduce the new concept of topological control based on the one-to-one correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the nonlinear Schrödinger equation solutions through Riemann theta functions. We demonstrate the concept experimentally by reporting observations of supervised transitions between waves with different genera. Considering the box problem in a focusing photorefractive medium, we tailor the time-dependent nonlinearity and dispersion to explore each region in the state diagram of the nonlinear wave propagation. Our result is the first realization of topological control of nonlinear waves. This new technique casts light on shock and rogue waves generation and can be extended to other nonlinear phenomena.

Determinantal representations of elliptic curves via Weierstrass elliptic functions

2018 ◽  
Vol 34 ◽  
pp. 125-136 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Elliptic Curves ◽  
Algebraic Curve ◽  
Theta Functions ◽  
Elliptic Functions ◽  
Determinantal Representation ◽  
Ternary Form ◽  
Riemann Theta Functions

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass $\wp$-functions in place of Riemann theta functions. An example of this approach is given.

RIEMANN THETA FUNCTIONS SOLUTIONS WITH RATIONAL CHARACTERISTICS FOR (2+1)-DIMENSIONAL SINH-GORDON EQUATION

2010 ◽  
Vol 24 (06) ◽  
pp. 575-584
Author(s):  
YANG FENG ◽  
HONG-QING ZHANG
Keyword(s):  
Gordon Equation ◽  
Theta Functions ◽  
Periodic Wave ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Wave Numbers ◽  
Riemann Theta Functions ◽  
The Waves ◽  
Hirota Bilinear

In this letter, we use the Riemann theta functions with rational characteristics and the Hirota bilinear method to construct quasi-periodic wave solutions for (2+1)-dimensional sinh-Gordon equation. This method not only conveniently obtains quasi-periodic solutions of nonlinear equations, but also directly gets the explicit expressions of frequencies, wave numbers, phase and amplitudes for the waves.

scholarly journals Computing Riemann theta functions in Sage with applications

2016 ◽  
Vol 127 ◽  
pp. 263-272 ◽  
Author(s):  
Christopher Swierczewski ◽  
Bernard Deconinck
Keyword(s):  
Theta Functions ◽  
Riemann Theta Functions
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