Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations
Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations
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DOI：
10.1016/j.jmaa.2010.05.070
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发表时间：
2010-11
影响因子：
1.3
通讯作者：
Shou‐Fu Tian;Hong-qing Zhang
中科院分区：
文献类型：
--
作者：
Shou‐Fu Tian;Hong-qing Zhang
本文基于多维黎曼 theta 函数，提出了 Hirota-Riemann 方法的清晰而直接的推广，以显式构造非线性方程（例如 Caudrey-Dodd-Gibbon-Sawada-Kotera 方程和（2+1）维破缺孤子方程）的多周期黎曼 theta 函数周期波解。在这些周期波中，一周期波是众所周知的椭圆曲线波，其表面图案是一维的，通常将其用作周期波的一维模型。二周期波是一周期波的直接推广，其表面图案是二维的，因此在两个独立的水平方向上具有两个独立的空间周期。提出了一个限制程序来详细分析多周期波的渐近行为，并严格建立了周期波解和孤子解之间的关系。这种广义的 Hirota-Riemann 方法也可以在一类非线性差分方程（例如 Toeplitz 晶格方程）上得到证明。
In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota–Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota–Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.