Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.16492 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909603162226688 |
|---|---|
| author | Liang, Zhenguo Luo, Jiawen Zhao, Zhiyan |
| author_facet | Liang, Zhenguo Luo, Jiawen Zhao, Zhiyan |
| contents | 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 $\mathcal{H}^s-$norm as $t$ tends to infinity, such as $t^{(n-1)s}e^{λst}$ (with $λ>0$) and $t^{(2n-1)s}+ ιt^{2ns}$ (with $ι\geq 0$). 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16492 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symplectic Normal Form and Growth of Sobolev Norm Liang, Zhenguo Luo, Jiawen Zhao, Zhiyan Analysis of PDEs Dynamical Systems 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 $\mathcal{H}^s-$norm as $t$ tends to infinity, such as $t^{(n-1)s}e^{λst}$ (with $λ>0$) and $t^{(2n-1)s}+ ιt^{2ns}$ (with $ι\geq 0$). 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. |
| title | Symplectic Normal Form and Growth of Sobolev Norm |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2312.16492 |