Department of Physics, Shaanxi University of Science & Technology, Xi'an, 710021, China.
Sci Rep. 2018 Jan 22;8(1):1374. doi: 10.1038/s41598-018-19756-6.
The tunable band-gap structure is fundamentally important in the dynamics of both linear and nonlinear modes trapped in a lattice because Bloch modes can only exist in the bands of the periodic system and nonlinear modes associating with them are usually confined to the gaps. We reveal that when a momentum operator is introduced into the Gross-Pitaevskii equation (GPE), the bandgap spectra of the periodic system can be shifted upward parabolically by the growth of the constant momentum coefficient. During this process, the band edges become asymmetric, in sharp contrast to the standard GPE with an external periodic potential. Extended complex Bloch modes with asymmetric profiles can be derived by applying a phase transformation to the symmetric profiles. We find that the inherent parity-time symmetry of the complex system is never broken with increasing momentum coefficient. Under repulsive interactions, solitons with different numbers of peaks bifurcating from the band edges are found in finite gaps. We also address the existence of embedded solitons in the generalized two-dimensional GPE. Linear stability analysis corroborated by direct evolution simulations demonstrates that multi-peaked solitons are almost completely stable in their entire existence domains.
可调谐带隙结构在晶格中捕获的线性和非线性模式的动力学中至关重要，因为布洛赫模式只能存在于周期性系统的能带中，而与之相关的非线性模式通常局限于间隙中。我们揭示了当动量算子被引入 Gross-Pitaevskii 方程（GPE）时，通过增加常数动量系数，周期性系统的带隙谱可以抛物线式地向上移动。在这个过程中，带边缘变得不对称，与具有外部周期性势的标准 GPE 形成鲜明对比。通过对对称轮廓进行相位变换，可以得出具有不对称轮廓的扩展复布洛赫模式。我们发现，随着动量系数的增加，复杂系统固有的宇称时间对称性从未被打破。在排斥相互作用下，在有限的间隙中发现了从带边缘分叉的具有不同峰值数的孤子。我们还解决了广义二维 GPE 中嵌入孤子的存在问题。线性稳定性分析通过直接演化模拟得到了证实，表明多峰孤子在其整个存在域内几乎完全稳定。
