arXiv subjects

Huajie Song

Publications and source records attributed to Huajie Song.

3 recordsLinked to original sources

arXiv2026

Generalization of a localized-state formation mechanism in finite lattices with interaction nonlinearity

We study how time-periodic, spatially localized states are born from the linear spectrum of a \emph{finite} lattice as the nonlinearity is switched on. In earlier work we treated this question for a diatomic chain with on-site nonlinearity and developed a framework that continues a near-edge linear mode in amplitude and controls the resulting perturbation series uniformly in the chain length. The present paper shows that the same framework applies to the more difficult case of Fermi--Pasta--Ulam--Tsingou (FPUT) interaction nonlinearity. The key is a structural relation between the FPUT and on-site nonlinearities, which allows the estimates obtained in the on-site setting to be transferred to the FPUT setting. As before, the analysis yields a quantitative radius of convergence, $\eps=\Theta(1/\sqrt{n})$ for a chain of length $2n$, below which the near-edge mode stays extended and above which the orbit localizes and its frequency leaves the band. The diatomic chain is used only as a test case; both the formation mechanism and the method are model-independent and are expected to extend to other short-range nonlinearities and to higher dimensions.

nlin.PS↗

arXiv2025

Analytical estimations of edge states and extended states in large finite-size lattices

The bulk boundary correspondence, one of the most significant features of topological matter, theoretically connects the existence of edge modes at the boundary with topological invariants of the bulk spectral bands. However, it remains unspecified in realistic examples how large the size of a lattice should be for the correspondence to take effect. In this work, we employ the diatomic chain model to introduce an analytical framework to characterize the dependence of edge states on the lattice size and boundary conditions. In particular, we apply asymptotic estimates to examine the bulk boundary correspondence in long diatomic chains as well as precisely quantify the deviations from the bulk boundary correspondence in finite lattices due to symmetry breaking and finite size effects. Moreover, under our framework the eigenfrequencies near the band edges can be well approximated where two special patterns are detected. These estimates on edge states and eigenfrequencies in linear diatomic chains can be further extended to nonlinear chains to investigate the emergence of new nonlinear edge states and other nonlinear localized states. In addition to one-dimensional diatomic chains, examples of more complicated and higher dimensional lattices are provided to show the universality of our analytical framework.

math-ph↗

arXiv2023

Emergence of purely nonlinear localized states with frequencies exited from spectral bands

In this work, we revisit the classic model of diatomic chain with cubic nonlinearity and investigate the formation mechanism of nonlinear localized time-periodic solutions (breathers) with frequencies exited the spectral bands. First we employ the long-chain limit to obtain estimates of linear eigenstates, especially those with frequencies near the band edges. As the strength of nonlinearity grows, some frequencies can cross the band edges to turn isolated while the corresponding states gradually change from non-localized to localized. Based on the estimates of linear eigenstates, we derive analytical approximations of these nonlinear states and prove their validity for frequencies within the bands. Moreover, the process of states growing localized are illustrated in both analytical and numerical approaches. Although here we place emphasis on nonlinear middle-localized states with the most generic boundary conditions, the results can also be extended to a wider range of localized states including edge states.

nlin.PS↗