PT-对称的非线性波方程的波结构及其稳定性分析研究

Some PT-symmetric non-Hermitian Hamiltonians have been shown to possess entirely real spectra, which broke the condition that a Hamiltonian should be Hermitian in quantum mechanics. Moreover, phase transitions in PT-symmetric systems have been observed in some experiments. In this project, on the basis of our previous research works, we focus on the symmetry breaking, integrability and stability analysis of wave structures for some PT-symmetric nonlinear wave equations using both analytic and numerical methods. Our research content mainly contains one- and higher-dimensional nonlinear Schrodinger equations with PT-symmetric potentials, PT-symmetric nonlocal continuous and discrete vector nonlinear Schrodinger equations, and PT-symemtric extensions of complex nonlinear wave equations. These result may find some mathematical properties and physical rules of PT-symmetric nonlinear waves in distinct nonlinear interactions.

PT-对称性的非厄米的Hamilton算子也拥有完全的实谱，这打破了经典量子力学中Hamilton算子是厄米的条件，在实验中已经观察到PT-对称性Hamilton系统的相位变化。本项目拟在已有相关研究工作的基础上，主要从解析方法和数值分析两个方面致力于研究PT-对称的非线性波方程的对称破缺、可积性以及波结构的演化和稳定性分析。研究的模型主要包括具有PT-对称外势的一维和高维广义非线性Schrodinger方程、PT-对称的非局部连续和离散向量非线性Schrodinger方程组、复域中非线性波方程的PT-对称延拓结构等。通过研究以期揭示PT-对称的非线性波在不同非线性作用下的数学性质和物理规律。

本项目主要研究了若干具有物理意义的PT-对称非线性波方程解的数学性质、非线性波结构产生的物理机理（包括孤子解、peakon、涡旋解、平顶有限阶光滑孤子解等）及其稳定性和相互作用。特别地，提出了两参数方法用于构造新的PT-对称的非局域可积系统，提出了一大类具有物理意义的广义PT-对称的Wadati势，使得广义非线性Schrodinger方程拥有稳定的孤子解等，研究成果发表在国际重要的学术期刊上。