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Texas A&M University

Nonlinear Partial Differential Equations

Spring 2022


Date:March 8, 2022
Speaker:Tristan Léger, Princeton University
Title:Long-time behavior of quadratic Klein-Gordon with internal mode in 3d
Abstract:In this talk I will present results on the small data regime of a quadratic Klein-Gordon equation with potential in 3d. This is motivated by the question of asymptotic stability of coherent structures in dispersive equations such as solitons.

The main novelty is the presence of an internal mode. We construct global solutions in this setting, and find that they have new features compared to the case with no internal mode. In particular they exhibit growth near certain frequencies, and have anomalously slow time decay. This is joint work with Fabio Pusateri.

Date:March 29, 2022
Speaker:Benjamin Harrop-Griffiths, UCLA
Title:The derivative NLS on the line
Abstract:The derivative nonlinear Schrödinger equation arises as an effective equation for the dynamics of Alfvén waves in plasmas. While this model has been analyzed in depth since the 1970s, the last two years have seen a significant amount of progress on the behavior of solutions in Sobolev spaces. In this talk we discuss some of these recent results and present a proof of well-posedness for all initial data on the line with finite mass. This is joint work with Rowan Killip, Maria Ntekoume, and Monica Vişan.

Date:April 12, 2022
Speaker:Gong Chen, University of Kentucky
Title:Long-time dynamics of 1d cubic nonlinear Schrödinger equations with a trapping potential
Abstract:We will consider the long-time dynamics of small solutions to the 1d cubic nonlinear Schrödinger equation (NLS) with a trapping potential. I will illustrate that every small solution will decompose into a small solitary wave and a radiation term which exhibits the modified scattering. The analysis also establishes the long-time behavior of solutions to a perturbation of the integrable cubic NLS with the appearance of solitons.

Date:April 19, 2022
Speaker:Kihyun Kim, IHES
Title:On long-term dynamics for the equivariant self-dual Chern-Simons-Schrödinger equation
Abstract:In this talk, I will discuss some recent developments on the long-term dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. CSS is a gauge-covariant 2D cubic nonlinear Schrödinger equation. I will first discuss a strong rigidity of the dynamics (soliton resolution) for this model, which is a consequence of the self-duality and non-local nonlinearity that are very specific features of CSS. Next, I will discuss the blow-up dynamics and introduce an interesting instability mechanism (rotational instability) of some finite-time blow-up solutions.