On the integrability and Riemann theta functions periodic wave solutions of the Benjamin Ono equation

2018 ◽  
Vol 92 (2) ◽  
pp. 235-246
Author(s):  
Chun-Mei Fang ◽  
Shou-Fu Tian ◽  
Yang Feng ◽  
Jin-Hua Dai
Keyword(s):  
Theta Functions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Riemann Theta Functions

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.

Periodic-wave solutions and asymptotic properties for a (3+1)-dimensional generalized breaking soliton equation in fluids and plasmas

2021 ◽  
pp. 2150344
Author(s):  
Rui-Dong Chen ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Ting-Ting Jia ◽  
Gao-Fu Deng ◽  
...  
Keyword(s):  
Asymptotic Properties ◽  
Theta Functions ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Soliton Equation ◽  
One Dimensional ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Riemann Theta Functions ◽  
The One

In this paper, a (3+1)-dimensional generalized breaking soliton equation is investigated. Based on the one- and two-dimensional Riemann theta functions, one- and two-periodic-wave solutions are derived. We observe that the one-periodic wave is one-dimensional and is viewed as a superposition of the overlapping waves, placed one period apart. With certain parameters, the symmetric feature appears in the two-periodic wave, and the two-periodic wave degenerates to the one-periodic wave. With the series expansions, we explore the relations between the soliton and periodic-wave solutions. According to those relations, asymptotic properties for the periodic-wave solutions to approach to the soliton solutions under certain amplitude conditions are derived.

scholarly journals EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS

2009 ◽  
Vol 24 (21) ◽  
pp. 1677-1688 ◽  
Author(s):  
WEN-XIU MA ◽  
RUGUANG ZHOU ◽  
LIANG GAO
Keyword(s):  
Theta Functions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Riemann Matrix ◽  
Wave Solutions ◽  
Riemann Theta Functions ◽  
Bilinear Formulation ◽  
Propagation Of Waves ◽  
Hirota Bilinear ◽  
Bilinear Equations

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: ut + uxxy - 3uuy - 3uxv = 0 and ut + uxxxxy - (5uxxv + 10uxyu - 15u2v)x = 0 where vx = uy, thereby yielding their one-periodic and two-periodic wave solutions describing one-dimensional propagation of waves.

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