New Lyapunov Frameworks for Solitary Wave Stability and Effective Dynamics
报告题目(Title)：New Lyapunov Frameworks for Solitary Wave Stability and Effective Dynamics
报告人(Speaker)：Justin Holmer（美国布朗大学）
地点(Place)：后主楼1124
时间(Time)：2026年7月10日（周五）9:00-10:00
邀请人(Inviter)：许孝精
报告摘要
Classical Lyapunov stability theory for solitary waves rests on a familiar scheme: modulate along the coherent-structure manifold, remove neutral directions, and prove coercivity of an augmented energy on the transverse radiation. This talk describes several settings where that scheme remains effective only after being adapted to new geometric or dynamical features. We begin with focusing NLS in a harmonic trap, where broken translation invariance is replaced by oscillating center-of-mass dynamics and a modified Lyapunov functional. We then discuss cubic–quintic NLS kinks, where nonzero boundary states require weighted metrics and a split energy argument. The second half turns to effective dynamics: first reviewing bright NLS solitons in slowly varying potentials, then describing dark-soliton dynamics for defocusing NLS, where nonzero-background variables and local virial estimates play a central role. We close briefly with stochastic KdV solitons, where Lyapunov control is combined with stochastic modulation and localized virial estimates.
主讲人简介
Justin A. Holmer 是布朗大学数学系教授，研究方向为色散型偏微分方程、调和分析和非线性波。其工作涉及非线性薛定谔方程、孤子动力学、散射理论、爆破解及量子多体系统到非线性 PDE 的严格推导。其论文发表于 Inventiones Mathematicae、Forum of Mathematics Pi、JEMS等重要期刊。Holmer 教授于 2004 年获芝加哥大学博士学位，师从于Carlos Kenig教授，并曾获得 2009 年 Sloan Research Fellowship。
