走向现代数学学术报告 - 代丹副教授（No. 736) 

题目：Asymptotics and total integrals of the $\rm{P_{I}^{2}}$ tritronqu\'{e}e solution and its Hamiltonian 

报告人：代丹 副教授（香港城市大学） 

时间：2024年9月26日 10:00 

腾讯会议ID：320-902-342 

摘要：We study the tritronqu\'{e}e solution $u(x,t)$ of the $\mathrm{P}_{\rm I}^{2}$ equation, the second member of the Painlev\'{e} I hierarchy. This particular solution is also known as the Gurevich-Pitaevskii solution of the KdV equation. It is pole-free on the real line and has various applications in mathematical physics. We obtain a full asymptotic expansion of $u(x,t)$ as $x\to\pm \infty$, uniformly for the parameter $t$ in a large interval. Based on this result, we successfully derive the total integrals of $u(x,t)$ and the associated Hamiltonian with respect to $x \in \mathbb{R}$. Surprisingly, although $u(x,t)$ exhibits significant differences between $t>0$ and $t<0$, both integrals equal zero for all $t$. 

报告人简介：代丹，香港城市大学副教授，博士生导师。主要研究方向为Riemann-Hilbert方法与渐近分析，正交多项式与特殊函数，可积系统，随机矩阵理论等。在随机矩阵特征值分布普适性猜想等领域取得了重要研究成果。相关研究成果发表在Advances in Mathematics，Communications in Mathematical Physics，SIAM Journal on Mathematical Analysis等国际重要学术期刊。主持香港研究资助局多项研究项目。