We construct the multi-breather solutions of the focusing nonlinear Schr\"odinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.

中文翻译：

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椭圆函数背景上非线性薛定谔方程的多呼吸和高阶流氓波

通过使用它们，可以绘制出由这些调制不稳定性引起的 Akhmediev 呼吸器，并揭示它们的动态。最后，我们通过采用分支点呼吸器解决方案的特殊极限和广义达布变换，提供了流氓波和高阶流氓波解。流氓波的由此产生的动力学涉及丰富的现象：取决于背景的选择并具有相对于背景的不同速度。我们还提供了多阶和高阶流氓波解决方案的示例。我们通过采用分支点呼吸器解决方案的特殊极限和广义达布变换，提供了流氓波和高阶流氓波解。流氓波的由此产生的动力学涉及丰富的现象：取决于背景的选择并具有相对于背景的不同速度。我们还提供了多阶和高阶流氓波解决方案的示例。我们通过采用分支点呼吸器解决方案的特殊极限和广义 Darboux 变换来提供 Rogue-wave 和高阶 Rogue-wave 解决方案。流氓波的由此产生的动力学涉及丰富的现象：取决于背景的选择并具有相对于背景的不同速度。我们还提供了多阶和高阶流氓波解决方案的示例。