2023/12/27 by Liang, Zhenguo, Luo, Jiawen, Zhao, Zhiyan · 1 citation
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.16492
For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term solution behaviors within Sobolev space. This classification is achieved through the utilization of Metaplectic and Schrödinger representations. Each pattern of Sobolev norm behavior corresponds to a specific n-dimensional symplectic normal form, as detailed in Theorems 1.1 and 1.2. When applied to periodically or quasi-periodically forced n-dimensional quantum harmonic oscillators, we identify novel growth rates for the Hs-norm as t tends to infinity, such as t(n-1)seλst (with λ>0) and t(2n-1)s+ ιt2ns (with ι≥ 0). Notably, we demonstrate that stability in Sobolev space, defined as the boundedness of the Sobolev norm, is essentially a unique characteristic of one-dimensional scenarios, as outlined in Theorem 1.3. As a byproduct, we discover that the growth rate of the Sobolev norm for the quantum Hamiltonian can be directly described by that of the solution to the classical Hamiltonian which exhibits the ``fastest" growth, as articulated in Theorem 1.4.