Time: Sep 21 (Thu) 5:00-6:00 PM
Speaker: Shijie Dong (Southern University of Science and Technology)
Title: Global behavior for 2D Dirac-Klein-Gordon equations
Abstract: We consider two-dimensional Dirac-Klein-Gordon equations, which are a basic model in particle physics. We investigate the global behavior of small data solutions to the systems in the case of a massive scalar field and a massless Dirac field. More precisely, we prove sharp time decay and linear scattering for the solutions to the equations. This talk is based on joint works with Kuijie Li, Yue Ma, Zoe Wyatt, and Xu Yuan.
Time: Oct 13 (Fri) 10:00-12:00 AM
Speaker: Kawamoto Masaki (Ehime University)
Title: Modified scattering for nonlinear Schrödinger equations with long-range potentials
Abstract: We consider the scattering problem for the nonlinear Schrödinger equation with linear potentials. In particular, we mainly focus on the case where both of nonlinearities and potentials are of long-range type in the sense of scattering. In this talk, we introduce a concrete form of the asymptotic behavior of solutions of this equation and show the modified scattering for the associated final state problem, as well as the existence of modified wave operators. This talk is based on the joint work with Haruya Mizutani (Osaka University).
Time: Oct 25 (Wed) 5:00-6:00 PM
Speaker: Ikkei Shimizu (Osaka University)
Title: Phase transition threshold and stability of magnetic skyrmions
Abstract: We consider variational problem for the Landau-Lifshitz energy with the Dzyaloshinskii-Moriya interaction. This is one of the mathematical model describing the formation of magnetization in chiral ferromagets, where critical points of the energy correspond to equilibrium state in reality. This energy admits an explicit critical point with vortex-like configuration, called magnetic skyrmion. In this talk, we consider the stability of magnetic skyrmions, showing the following result: There is an explicit threshold value of the parameter in the energy below which the skyrmion is stable, while it becomes unstable above the threshold. This result is consistent with the established phase diagram in physics. This is joint work with S. Ibrahim (UVic).
Time: Nov 8 (Wed) 5:00-6:00 PM
Speaker: Charles Collot (CY Cergy Paris Université)
Title: Asymptotic stability of traveling waves for one-dimensional nonlinear Schrodinger equations.
Abstract: We consider one dimensional nonlinear Schrodinger equations around a traveling wave. We prove its asymptotic stability for general nonlinearities, under the hypotheses that the orbital stability condition of Grillakis-Shatah-Strauss is satisfied and that the linearized operator does not have a resonance and only has 0 as an eigenvalue. As a by-product of our approach, we show long-range scattering for the radiation remainder. Our proof combines for the first time modulation techniques and the study of space-time resonances. We rely on the use of the distorted Fourier transform, akin to the work of Buslaev and Perelman and, of Krieger and Schlag, and on precise computations and estimates of space-time resonances to handle its interaction with the soliton. This is joint work with Pierre Germain.
Time: Nov 22 (Wed) 5:00-6:00 PM
Speaker: Irfan Glogić (University of Vienna)
Title: On generic blowup for the supercritical wave maps equation.
Abstract: We consider wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere. Numerical simulations of this model indicate that in the energy supercritical case, $d \geq 3$, generic large data lead to finite time blowup via an explicitly known self-similar solution. In the effort of rigorously proving these observations, many works have been produced over the last decade, starting with the pioneering work of Aichelburg-Donninger-Schörkhuber. In this talk, we outline a novel general framework for the analysis of spatially global stability of self-similar solutions to semilinear wave equations. We then implement this scheme in the aforementioned context of wave maps, thereby obtaining the first nonlinear stability result that is global-in-space. At the end, we discuss further open problems as well as the new mathematical challenges that our approach generates.
Time: Dec 6 (Wed) 10:00-11:00 AM
Speaker: Zaher Hani (University of Michigan)
Title: Long time justification of wave turbulence theory.
Abstract: Wave turbulence theory is the theory of statistical physics for nonlinear waves. Over the past few years, the mathematical understanding of this theory started to emerge following the maturation of various techniques in dispersive PDE, probability, and math physics. This included the rigorous derivation of the central foundation of wave turbulence theory, namely the wave kinetic equation, in a series of works jointly with Yu Deng (USC). We shall review this progress in this talk, and focus on a most recent result in which we extend this derivation to arbitrarily long time intervals. The latter is the first result of its kind of any nonlinear kinetic limit (be it particle or wave).