Zhang Hai-Qiang, Chen Fa
College of Science, University of Shanghai for Science and Technology, P. O. Box 253, Shanghai 200093, China.
Chaos. 2021 Feb;31(2):023129. doi: 10.1063/5.0030072.
In this paper, we construct rogue wave solutions on the periodic background for the fourth-order nonlinear Schrödinger (NLS) equation. First, we consider two types of the Jacobi elliptic function solutions, i.e., dn- and cn-function solutions. Both dn- and cn-periodic waves are modulationally unstable with respect to the long-wave perturbations. Second, on the background of both periodic waves, we derive rogue wave solutions by combining the method of nonlinearization of spectral problem with the Darboux transformation method. Furthermore, by the study of the dynamics of rogue waves, we find that they have the analogs in the standard NLS equation, and the higher-order effects do not have effect on the magnification factor of rogue waves. In addition, when the elliptic modulus approaches 1, rogue wave solutions can reduce to multi-pole soliton solutions in which the interacting solitons form weakly bound states.
在本文中，我们构造了四阶非线性薛定谔（NLS）方程在周期背景下的 rogue 波解。首先，我们考虑两种雅可比椭圆函数解，即 dn 函数解和 cn 函数解。dn 周期波和 cn 周期波对于长波扰动都是调制不稳定的。其次，在这两种周期波的背景下，我们通过将谱问题非线性化方法与达布变换方法相结合来推导 rogue 波解。此外，通过对 rogue 波动力学的研究，我们发现它们在标准 NLS 方程中有类似物，并且高阶效应不会对 rogue 波的放大因子产生影响。另外，当椭圆模量趋近于 1 时，rogue 波解可以简化为多极孤子解，其中相互作用的孤子形成弱束缚态。
