中科院数学与系统科学研究院

数学研究所

学术报告

非线性分析讨论班

报告人：罗海军 副教授 湖南大学

题  目：Existence and orbitial stability of high-frequency standing wave solution for the fractional Schrodinger equation with potential

时  间：2024.07.23（星期二）9:00-10:00

地  点：N818

摘  要：In this talk, we will discuss the existence and orbitial stability of high-frequency standing wave solution for a class of fractional Schrodinger equation. First, we introduce a perturbation approach together with Lyapunov Schmidt reduction procedure to construct one positive standing wave with high frequency. Second, we analyze spectrum behaviors of the linearized operator to prove that the high-frequency standing wave solution is orbitally stable in the mass-subcritical case. Let us emphasize that this perturbation approach plays an important role not only in existence, and but also in orbital stability. The main difficulties are as follows: high-frequency standing waves we consider are totally different from semiclassical states, and the ground state of fractional order operator is only of algebraic decay at infinity, and potential function is of low regularity.

报告人： 李再铮 博士 河北师范大学

题  目：Rotating spirals for three-component competition systems

时  间：2024.07.23（星期二）10:00-11:00

地  点：N818

摘  要： In this talk, we discuss the existence of rotating spirals for three-component competition-diffusion systems in $B_1$. For the Neumann problem, we establish the existence of rotating spirals by applying the multi-parameter bifurcation theorem. As a byproduct, the instability of the constant positive solution is proved. In addition, for the non-homogeneous Dirichlet problem, the Rothe fixed point theorem is employed to prove the existence of rotating spirals.

报告人：张泽鑫 博士 江苏大学

题  目：Some results on the uniform bounds for nonlinear Schrodinger systems with strong competition

时  间：2024.07.23（星期二）11:00-12:00

地  点：N818

摘  要： For the positive solutions of the Gross-Pitaevskii system with strong competition, Noris et al. (Comm. Pure Appl. Math., 2010) proved that $L^{\infty}$-boundedness implies their $\alpha$-Holder boundedness for any $\alpha\in (0,1)$ uniformly as the competition parameter tends to infinity. In this talk, we mainly introduce their methods and also review some recent results.