Ma Li-Yuan, Zhu Zuo-Nong
Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, P. R. China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Sep;90(3):033202. doi: 10.1103/PhysRevE.90.033202. Epub 2014 Sep 9.
In this paper, we investigate nonintegrable semidiscrete Hirota equations, including the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation. We focus on the topics on gauge-equivalent structures and dynamical behaviors for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that, under the discrete gauge transformations, the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation are, respectively, gauge equivalent to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We prove that the two discrete gauge transformations are reversible. We study the dynamical properties for the two nonintegrable semidiscrete Hirota equations. The exact spatial period solutions of the two nonintegrable semidiscrete Hirota equations are obtained through the constructions of period orbits of the stationary discrete Hirota equations. We discuss the topic regarding whether the spatial period property of the solution to the nonintegrable semidiscrete Hirota equation is preserved to that of the corresponding gauge-equivalent nonintegrable semidiscrete equations under the action of discrete gauge transformation. By using the gauge equivalent, we obtain the exact solutions to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We also give the numerical simulations for the stationary discrete Hirota equations. We find that their dynamics are much richer than the ones of stationary discrete nonlinear Schrödinger equations.
在本文中，我们研究了不可积半离散广田方程，包括不可积半离散广田（-）方程和不可积半离散广田（+）方程。我们关注这两个不可积半离散方程的规范等价结构和动力学行为的相关主题。通过使用规定离散曲率的概念，我们表明，在离散规范变换下，不可积半离散广田（-）方程和不可积半离散广田（+）方程分别规范等价于不可积广义半离散修正海森堡铁磁体方程和不可积广义半离散海森堡铁磁体方程。我们证明这两个离散规范变换是可逆的。我们研究了这两个不可积半离散广田方程的动力学性质。通过构建定常离散广田方程的周期轨道，得到了这两个不可积半离散广田方程的精确空间周期解。我们讨论了在离散规范变换作用下，不可积半离散广田方程解的空间周期性质是否保留到相应规范等价的不可积半离散方程这一主题。通过使用规范等价，我们得到了不可积广义半离散修正海森堡铁磁体方程和不可积广义半离散海森堡铁磁体方程的精确解。我们还给出了定常离散广田方程的数值模拟。我们发现它们的动力学比定常离散非线性薛定谔方程的动力学丰富得多。
