标题： 关于广义Okamoto多项式的实根和虚根 显示英文标题

标题： On real and imaginary roots of generalised Okamoto polynomials

摘要： 最近，B. Yang 和 J. Yang 推导出 Sasa-Satsuma 方程的一族有理解，并表明只要相关的广义 Okamoto 多项式没有实根或没有虚根，其任何成员都构成一个部分异常波。 在本文中，我们推导出广义 Okamoto 多项式的实根数量和虚根数量的精确公式。 一方面，这给出了满足 Sasa-Satsuma 方程的部分异常波列表。 另一方面，它给出了第四 Painlevé 方程的有理解族，这些解在实轴或虚轴上无极点。 为了得到这些公式，我们开发了一个算法过程，从已知的种子解的奇异性分布出发，通过 Bäcklund 变换作用于形成其初始条件空间的有理曲面，来推导实解在实轴上的奇异性定性分布。 显示英文摘要

摘要： Recently, B. Yang and J. Yang derived a family of rational solutions to the Sasa-Satsuma equation, and showed that any of its members constitutes a partial-rogue wave provided that an associated generalised Okamoto polynomial has no real roots or no imaginary roots. In this paper, we derive exact formulas for the number of real and the number of imaginary roots of the generalised Okamoto polynomials. On the one hand, this yields a list of partial-rogue waves that satisfy the Sasa-Satsuma equation. On the other hand, it gives families of rational solutions of the fourth Painlev\'e equation that are pole-free on either the real line or the imaginary line. To obtain these formulas, we develop an algorithmic procedure to derive the qualitative distribution of singularities on the real line for real solutions of Painlev\'e equations, starting from the known distribution for a seed solution, through the action of B\"acklund transformations on the rational surfaces forming their spaces of initial conditions.