Various plasma phenomena will be discussed using a fundamentalfluid model for plasmas, called the Euler-Poisson system. These include plasma sheaths and plasma soliton. First we will briefly introduce recent results on the stability of plasma sheath solutions, and the quasi-neutral limit of the Euler-Poisson system in the presence of plasma sheaths. Another example of ourinterest is plasma solitary waves, for which we discuss existence, stability, and the time-asymptotic behavior. To study the nonlinear stability of solitary waves, the global existence of smooth solutions must be established, which is completelyopen. As a negative answer for global existence, we look into the finite-time blow-up results for the Euler-Poisson system, and discuss the related open questions.

In this talk, we propose the Landau-Lifshitz type system augmented with Chern-Simons gauge terms, which can be considered as the geometric analog of so-called the Chern-Simons-Schrodinger equations. We first derive its self-dual equations through the energy minimization so that we can provide $N$-equivariant solitons. We next deliver basic ideas of constructing $N$-equivariant solitary waves for non-self-dual cases and investigating their qualitative properties.

In this talk, I will introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimeswith odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-timetailson stationary spacetimes. Moreover, the method also applies to dynamical spacetimes. In this case, I will explain how the late-timetailsare in general different(!) from the stationary case in the presence of dynamical and/or nonlinear perturbations of the problem. This is joint work with Jonathan Luk(Stanford).

In this talk, I will discuss some recent developments on the long-term dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariantsymmetry. CSS is a gauge-covariant 2D cubic nonlinear Schrödingerequation, which admits the L2-scaling/pseudoconformalinvariance and soliton solutions.I will first discuss soliton resolution for this model, which is a remarkable consequence of the self-duality and non-local nonlinearity that are distinguished features of CSS. Next, I will discuss the blow-up dynamics (singularity formation) for CSS and introduce an interesting instability mechanism (rotational instability) of finite-time blow-up solutions. This talk is based on joint works with SoonsikKwon and Sung-JinOh.