Shijie Dong (Southern University of Science and Technology)
Title: Asymptotic behavior of the Klein-Gordon-Zakharov equations in dimension two
Abstract: The Klein-Gordon-Zakharov system is an important model in plasma physics with extensive mathematical studies. I will present some recent results regarding the global existence and the asymptotic behaviour of the Klein-Gordon-Zakharov equations in two space dimensions.
Hiroyuki Hirayama (University of Miyazaki)
Title: On traveling waves for the nonlinear Schrödinger system with quadratic three wave interaction
Abstract: We consider the system of three nonlinear Schrödinger equations with quadratic nonlinearity. In this talk, we prove the existence of traveling wave solutions to this system, which contain a translational parameter in Rd and two frequency parameters. We can see that the conditions of these parameters for the existence of traveling waves are determined by the resonance structure of the system. The traveling waves will be constructed by the minimizing problem for the action functional. For some zero mass cases, we will use the profile decomposition to obtain the compactness of the minimizing sequence. We also characterize the initial data to obtain the global solution by the traveling waves. This talk is based on joint work with Masahiro Ikeda (The University of Osaka).
Takamori Kato (Saga University)
Title: Unconditional well-posedness of periodic higher-order dispersive equations related to completely integrable systems
Abstract: In this talk, we consider the Cauchy problem of the second equations (with generalized coefficients) in the KdV hierarchy, mKdV hierarchy, NLS hierarchy and Benjamin-Ono hierarchy with periodic boundary conditions. A key is how to recover derivative losses in the nonlinear terms of these equations. The strategy to overcome this difficulty is as follows: First, we use conserved quantities, change variables and the gauge transformation to cancel out the resonant parts with derivative losses. Then, by applying the normal form reduction to the non-resonant parts, we show the wellposedness and unconditional uniqueness in low regularity setting. Part of this study is based on joint work with Kotaro Tsugawa (Chuo University).
Masaki Kawamoto (Okayama University)
Title: Modified scattering for the nonlinear Schrödinger equation with long-range potentials in 1D
Abstract: We consider the scattering problem for a nonlinear Schrödinger equation with long-range linear potentials and a gauge-invariant long-range critical power type nonlinearity.
In space dimensions two and three, the existence of modified wave operators has been established by introducing phase corrections that depends on both the linear potentials and the nonlinear term.
However, in the one-dimensional case, no such results have been obtained due to the lack of the Strichartz estimates.
In this talk, we introduce a result showing the existence of modified wave operators for small scattering data even in the one-dimensional case, similarly to the case in two and three dimensions.
In particular, by employing a proof that does not rely on Strichartz estimates, we are able to handle potentials that allow the existence of the negative eigenvalues--something that previous methods could not handle.
This work is based on joint work with Haruya Mizutani (Osaka University)
Kihyun Kim (Seoul National University)
Title: On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation
Abstract: We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices D≥3; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^1$-bounded radial solutions to energy-critical heat equations in dimensions N≥7, building upon soliton resolution for such solutions. This is a joint work with Frank Merle (IHES and CY Cergy-Paris University).
Shinya Kinoshita (Nagoya University )
Title: Small data scattering for the 2D cubic Zakharov-Kuznetsov equation
Abstract: In this talk, we consider the 2D cubic Zakharov-Kuznetsov equation. This equation can be seen as a two-dimensional generalization of the modified KdV equation. Linares–Pastor showed the global well-posedness in the energy space when mass of initial datum is below ground state. On the other hand, the scattering of the global solution is still unknown, even when the initial value is sufficiently small. In this talk, we will show the scattering by assuming a small initial value. However, this smallness requires a stronger assumption than L^2. This talk is based on joint work with Sim˜ao Correia (Instituto Superior T´ecnico).
Nobu Kishimoto (Kyoto University)
Title: On the global dynamics above the ground state threshold for NLS without radial assumption
Abstract: We consider the focusing mass-supercritical and energy-subcritical singlepower nonlinear Schrödinger (NLS) equations. We attempt to remove the radial assumption from the result of Nakanishi-Schlag (2012) on the 3D cubic NLS, which describes possible behaviors of solutions of energy at most slightly above that of the ground state, in terms of scattering, trapping by the ground state, and blowup. Although we cannot deny a certain behavior of solutions which does not appear in the radial case, we can remove the radial assumption in the case corresponding to the scattering. Moreover, even with the radial assumption, our proof extends the classification by Nakanishi-Schlag to general powers and dimensions. This talk is based on a joint work with Takafumi Akahori (Shizuoka University).
Jinyeop Lee(The University of British Columbia)
Title: Derivation of the Chern–Simons–Schrödinger equation from the dynamics of an almost-bosonic-anyon gas
Abstract: We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern–Simons–Schrödinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern–Simons–Schrödinger equation. This talk is based on a work with Th´eotime Girardot, (https://arxiv.org/abs/2412.13080).
Hayato Miyazaki (Kagawa University)
Title: $L^{2}$-decay of solutions to nonlinear Schrödinger equations with attractive-dissipative nonlinearity
Abstract: We discuss $L^{2}$-decay estimates for solutions to nonlinear Schrödinger equations with a power-type dissipative nonlinearity.
The equation does not possess an $L^{2}$-conservation law, and the $L^{2}$-norm of solutions decreases monotonically over time.In particular, it is known that the $L^{2}$-norm decays over time if the power of the nonlinearity is less than or equal to the Barab-Ozawa exponent; otherwise, it does not.
In this talk, we focus on the case where the power is below the Barab-Ozawa exponent, and
present recent results on the decay rate of the $L^{2}$-norm under the attractive-dissipative condition, in which solutions may exhibit local concentration.
This talk is based on joint work with Naoyasu Kita (Kumamoto University) and Takuya Sato (Ehime University).
Jia Shen (Nankai University)
Title: Global theory of nonlinear Schrödinger equations in the weighted space
Abstract: I will report some recent progress of the global well-posedness and scattering for the defocusing nonlinear Schrödinger equations (NLS) in the weighted space, which is based on the joint work with Prof. Yifei Wu. We will first show the global well-posedness of 3D quadratic NLS with radial data in the critical weighted space. Previously, Killip, Masaki, Murphy, and Visan proved its conditional global well-posedness and scattering in such space. Our result removes the a priori assumption for the global well-posedness part. Next, we will consider the scattering of mass subcritical NLS. Previously, it is shown by Tsutsumi-Ogawa that the scattering holds in the first-order weighted space, and by Lee that the continuity of the scattering operator breaks down in L^2. We extend the scattering result below the first-order weight, and give the scattering with a large class of L^2-data based on probabilistic method.
Shun Tsuhara (Hokkaido University)
Title: Existence of solutions for the nonlinear Schrödinger equation with a nonlinear boundary condition in the half-space
Abstract: We study the initial-boundary value problem for the nonlinear Schrödinger equation with a nonlinear Neumann boundary condition in the half-space. In the one-dimensional case, Batal--Özsarı(2016) established the well-posedness of the problem in $H^1$, and more recently, Hayashi--Ogawa--Sato(2025) proved well-posedness in $L^2$. However, higher-dimensional cases have not been extensively investigated. As far as the speaker is aware, the existence of $L^2$-solutions in higher dimensions has been partially studied by Ogawa--Sato--T.(2024), while other cases remain unclear. In this talk, we consider the existence of $H^1$-solutions for the problem in the two-dimensional half-plane. Our approach is based on a new representation formula for solutions to the linear problem introduced by Audiard(2019). Using this representation, we establish boundary Strichartz estimates for the linear solution, including estimates with respect to both time and spatial derivatives.
Kota Uriya (Okayama University of Science)
Title: Asymptotic behavior of the solution to systems of cubic nonlinear Schrödinger equations in one dimension
Abstract: We consider the asymptotic behavior of a general system of cubic nonlinear Schrödinger equations in one dimension. We identify a new class of cubic NLS systems for which the global boundedness and asymptotic behavior of small solutions can be established, even in the absence of any effective conserved quantity. The key to this analysis lies in utilizing conserved quantities for the reduced ordinary differential system derived from the original NLS systems. This is joint work with Professor S. Masaki (Hokkaido University) and Professor J. Segata (Kyushu University).
Chanjin You (Penn State University)
Title: Stability of translation-invariant equilibria for the Hartree equation
Abstract: The Hartree equation is a mean-field model describing many-body quantum systems. In this talk, I will focus on the stability of translation-invariant equilibria for the Hartree equation. In the short-range regime, I will present a criterion for phase mixing estimates for the density associated with the perturbation from the Penrose stable equilibrium. In the long-range regime, we establish that the Penrose type stability fails for every equilibrium. Furthermore, in the Coulomb potential case, we show that the leading behavior of the density is dominated by the Klein-Gordon type dispersion relation.