[01229] Cauchy problem for Deterministic and Stochastic nonlinear dispaersive equations
Session Time & Room : 5C (Aug.25, 13:20-15:00) @G709
Type : Proposal of Minisymposium
Abstract : We consider Cauchy problem for Deterministic and Stochastic nonlinear dispersive equations. For the Deterministic nonlinear Schrödinger equation (NLS), some global dynamics will be discussed, that is scattering, blowing up, growing up, or uniform bound for the solutions of NLS. For the Stochastic NLS, we will deal with nonlinear equation with a multiplicable noise. We also intoroduce and discuss paracontrolled calculus to prove local well-posedness of a renormalized version of the stochastic nonlinear wave equation.
[03890] The well-posedness of the stochastic nonlinear Schrödinger equations in H^2
Format : Talk at Waseda University
Author(s) :
Shunya Hashimoto (Saitama university)
Abstract : We consider the well-posedness of $H^2$-solutions in initial value problems for the stochastic nonlinear Schrödinger equations with power-type nonlinear terms with multiplicative noise. For the proof, we use the rescaling approach, which transforms the stochastic equation into a random equation in which no white noise appears. Unlike $L^2$- and $H^1$-solution, there are two difficulties in the $H^2$-solution: first, the lack of the smoothness of nonlinear functions, and second, the treatment of white noise that reappears.
[04966] Time beharior of solutions to nonlinear Schr\"odinger equation with a potential
Format : Talk at Waseda University
Author(s) :
Masaru Hamano (Waseda University)
Abstract : In this talk, we deal with the Cauchy problem of a nonlinear Schr\"odinger equation with a potential. In particular, we consider time behavior of solutions to the equation with initial data, whose energy is equal to that of the Talenti function.
[03584] On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction
Format : Talk at Waseda University
Author(s) :
Noriyoshi Fukaya (Tokyo University of Science)
Abstract : We consider existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction.
The Schrödinger operator with a point interaction describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation.
In this talk we consider the stability/instability of standing waves with small and large frequencies.
This talk is based on a joint work with Vladimir Georgiev (University of Pisa) and Masahiro Ikeda (RIKEN).
[05020] Convergence of the intermediate long wave equation from a statistical perspective
Format : Talk at Waseda University
Author(s) :
Guopeng Li (The Maxwell Institute for Mathematical Sciences)
Abstract : The intermediate long wave equation (ILW) models water waves of finite depth, connecting the Benjamin-Ono equation (deep-water limit) and the KdV equation (shallow-water limit). Convergence problems of ILW (in both the deep-water and shallow-water limits) have attracted attention from both the applied and theoretical points of view. In this talk, I will discuss convergence problems from a statistical viewpoint. I first consider the convergence problem of the Gibbsian ensembles. In this case, I establish convergence of the Gibbs measures and then also show convergence of invariant Gibbs dynamics to that of the Benjamin-Ono and KdV equations (without
uniqueness).
ILW is known to be completely integrable and thus possesses infinitely many conservation laws. In the second part of the talk, I consider invariant dynamics for ILW associated with higher order conservation laws. Due to a complicated nature of the dispersion, even the construction of measures associated with higher order conservation laws turns out to be highly non-trivial. By considering a suitable combination of higher order conservation laws, I overcome this issue and construct invariant dynamics for ILW with a fixed depth parameter. In the final part, I will discuss convergence of the invariant dynamics associated with higher order conservation laws.
This talk is based on a joint work with Tadahiro Oh (Edinburgh), Guangqu Zheng (Liverpool), and Andreia Chapouto (UCLA).
