We consider a ground state (soliton) of a Hamiltonian PDE. We prove that if the soliton is orbitally stable, then it is also asymptotically stable. The main assumptions are transversal nondegeneracy of the manifold of the ground states, linear dispersion (in the form of Strichartz estimates) and nonlinear Fermi Golden Rule. We allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues. The theory is tailor made for the application to the translational invariant NLS in space dimension 3. The proof is based on the extension of some tools of the theory of Hamiltonian systems (reduction theory, Darboux theorem, normal form) to the case of systems invariant under a symmetry group with unbounded generators.
Lyapunov Function PDEs Method to the Stability of Some Chemical Reaction Networks
本文有助于扩展李雅普诺夫函数偏微分方程的有效性（由方和高在[SIAM Journal on Applied动力系统，18（2019），pp. 1163-1199]并且其解被证明能够表现为李雅普诺夫函数）。通过定义一类新的网络--复合平衡生产网络，证明了李雅普诺夫函数偏微分方程方法在刻画这类网络的渐近稳定性方面是有效的，在适当的条件下，也可以刻画这类网络与任意一维按物种独立网络的复合以及与任意两物种自催化非独立网络的复合.值得注意的一点是，这三类网络都是非弱可逆的、任意维数的、任意亏数的。我们将你的结果应用到一些实际的生化反应网络中，包括生-死过程，模体相关网络等，来说明有效性。
Symplectic Neural Operators for Learning Infinite Dimensional Hamiltonian Systems
Graduate School of Science（理学研究科）；Kobe University（Kobe大学）；NTT Communication Science Laboratories（NTT通信科学实验室）；Faculty of Information Science and Technology（信息科学和技术学部）；Hokkaido University（北海道大学）；Institute of Mathematics for Industry（工业数学研究所）；Kyushu University（九州大学）
