We report here an exact analytical travelling wave solution for a non-linear electron plasma wave excited by an intense ultra-relativistic proton beam. It demonstrates the underlying physics of longitudinal electric field characteristics of the excited wake wave formed behind the drive beam. The results are further supplemented by a fully relativistic particle-in-cell code OSIRIS in two-dimensional geometry. The investigation is further extended by providing an analytical description of the wake wave excited by an equi-spaced train of small proton bunches with the inclusion of the non-relativistic plasma ion dynamics. Our results show that the amplitude of the wake field does not grow indefinitely with the increase in the number of proton bunches. On the contrary, it saturates to a definitive limit.

中文翻译：

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强超相对论质子束激发等离子体尾场

我们在此报告了由强超相对论质子束激发的非线性电子等离子体波的精确解析行波解。它展示了在驱动光束后面形成的激发尾波的纵向电场特性的基本物理特性。结果得到了二维几何中完全相对论的细胞内粒子代码 OSIRIS 的进一步补充。通过提供由等距小质子束串激发的尾波的分析描述，包括非相对论等离子体离子动力学，进一步扩展了研究。我们的结果表明，尾流场的幅度不会随着质子束数量的增加而无限增长。相反，它饱和到一个确定的限制。