Titles and Abstracts


Title: Schrödinger equation with square and inverse square potentials.


Abstract: In this talk, We discuss the transformation of solutions to Schrödinger equation of the time-dependent harmonic oscillator.

In particular, it examines the appearance of the usual Schrödinger equation through the transformation when the potential term is a singular inverse square potential.

We observe that the time-dependent coefficient arising from the transformation balances between the Laplacian and the potential, leading to the emergence of the usual Schrödinger equation.

This talk is based on joint work with professor Atsuhide Ishida (Tokyo University of Science).



Title: Blow-up dynamics of the self-dual Chern-Simons-Schrödigner equations under equivariant symmetry.


Abstract: The self-dual Chern-Simons-Schrödinger (CSS) equation is viewed as a gauged version of the 2D cubic nonlinear Schrödinger (NLS) equation. It admits static solutions and explicit blow-up solutions via the pseudoconformal transformation. I will discuss recent progress on its blow-up dynamics, from constructions, instability mechanism, and rigidity of blow-up rate. I will highlight remarkable features of (CSS) dynamics, which differ from related models like 2D cubic NLS, Schrödinger maps, wave maps, and others. This talk is based on joint works with Kihyun Kim and Sung-Jin Oh.




Title: Scattering results for the (1+4) dimensional massive Maxwell-Dirac system under Lorenz gauge condition.


Abstract:  In this talk, we consider the massive Maxwell-Dirac system under Lorenz gauge condition in (4+1) dimensional Minkowski space. We prove the global existence and scattering result for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, our main system becomes a system of quadratic Dirac equations coupled with the five wave equations. To obtain the global solution to the Maxwell-Dirac system and its behavior, we perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the free Klein-Gordon and wave propagators.




Title: Global smooth solutions to the irrotational Euler-Reisz system.


Abstract: We consider the Euler-Riesz systems which govern the motion of an inviscid compressible fluid with an interaction force satisfying the Riesz equation. In general mathematical framework, a construction of global in-time solutions is not easily understood because this system belongs to the general class of hyperbolic conservation laws with zero dissipation. Nevertheless, Guo exploited the Klein-Gordon effect to construct a global smooth irrotational solution to the Euler system involved with the Poisson interaction force. In this talk, we show the global well-posedness of the irrotational solution to the 3D Euler-Reisz system by investigating the dispersion feature of the linearized Euler-Riesz equations.




Title: Global existence and asymptotic behavior for NLS systems without coercive conservation law


Abstract: We consider a class of systems of cubic nonlinear Schrodinger equations with two components in one dimension.

It is known that cubic nonlinearity is critical in one dimension in view of the large-time behavior. (cf. modified scattering).

Compared to the single-equation case, systems exhibit a wide range of asymptotic behaviors for their solutions.

Our motivation is to comprehend the effect of general nonlinearities that are gauge-invariant and represented by cubic polynomials with real coefficients.


For systems within this class, the global existence of small solutions is not always guaranteed. One sufficient condition for global existence is the presence of a coercive conserved quantity, i.e., a conserved quantity equivalent to the (L^2)^2-norm of the solution. In previous studies of large-time behavior, the assumption of the existence of such a conserved quantity is often made, characterized as a so-called weak-null condition. Another sufficient condition for global existence is the validity of a dissipative condition, indicating the non-increasing property of a coercive quantity.


In this talk, we introduce a specific class of systems that do not satisfy the weak-null condition or the dissipative condition, and that, however, we can establish the global existence and the large-time asymptotics of small solutions. The asymptotic profile is described with a solution to the corresponding ODE system. The key is the presence of a quartic conserved quantity for the ODE system.




Title: Modified scattering operator for nonlinear Schrödinger equations with a time-decaying harmonic potential


Abstract: We consider nonlinear Schrödingerequations with a time-decaying harmonic potential.

The nonlinearity is gauge-invariant of the long-range critical order.

Kawamoto and Muramatsu show that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem.

Also, in the equation without the potential, Hayashi and Naumkin establish the existence of a modified scattering operator.

In this talk, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation.

This talk is based on a joint work with Masaki Kawamoto (Okayama University).






Title: Long time behavior of special solutions to dissipative nonlinear Schrödinger equations in the analytic class


Abstract: We consider the Cauchy problem of one dimensional dissipative nonlinear Schrödinger equations with a critical power nonlinearity.

In the previous work, the upper mass-decay estimate of solutions was shown under the analytic condition of initial data.

In this talk, we construct a special solution satisfying the lower mass-decay estimate with the same order of the previous one and it gives the optimality of mass-decay rate of solutions in the analytic class.