The Focusing Nonlinear Schrödinger Equation in the semiclassical regime: Peregrine breathers near the gradient catastrophe
数学专题报告
报告题目(Title)：The Focusing Nonlinear Schrödinger Equation in the semiclassical regime: Peregrine breathers near the gradient catastrophe
报告人(Speaker)：Marco Bertola（Concordia University，加拿大）
地点(Place)：Zoom会议ID：962 703 61978 密码：324797
时间(Time)：3月27日(周三)，上午9:00-10:00
邀请人(Inviter)：王灯山
报告摘要
I will review the main message in the work [Comm. Pure Appl. Math. (2013) 66, no.5, 678-752] in collaboration with Alex Tovbis. In the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schr\"odinger equation (NLS) with decaying potentials there is a first time where the genus-zero dispersionless approximation fails due to a gradient catastrophe. I will address the (scaling) behaviour in a full neighbourhood of this point of gradient catastrophe. This allows us to address and partially verify a conjecture of Dubrovin-Grava-Klein (since then proved by O. Costin et al) about the role of the integrale tritronqu\'ee in the description of the asymptotic behaviour, establishing a universality with respect to a large class of initial datum. The main feature of the analysis are that the maximum asymptotic amplitude of each spike in the genus two region (near the gradient catastrophe) is exactly three times the one at the gradient catastrophe and the shape of each spike is universally the one of the rational breather solution (aka rogue wave) to the NLS (due to Peregrine). The method of proof is based upon the nonlinear steepest descent method and allows us to conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. I will try to describe the phenomenon qualitatively in the first half and possibly show some key ingredient of the proof in the second half; a technical but quite crucial and -to my knowledge- new ingredient is the fact that the local parametrix (constructed from the $\psi$ function of Painlev\'e\ I) needs to be studied in a neighborhood of the pole of the Painlev\'e transcendent.
主讲人简介
Marco Bertola, Full Professor, Head of Department of Mathematics and Statistics, Concordia University. His research interests are Mathematical Physics, Integrable systems, Inverse problems, Asymptotics in nonlinear integrable equations and Moduli spaces. He has published more than 100 papers on Invent. Math., Comm. Pure Appl. Math., Comm. Math. Phys., IMRN, Adv. Math.
