标题： 非线性耦合的DNLS方程中的本征局域模式：分岔 显示英文标题

标题： Intrinsic localized modes for DNLS equation with competing nonlinearities: bifurcations

摘要： 我们研究由DNLS型方程描述的非线性激发，这些方程具有所谓的竞争非线性。 这些是非线性，由两个幂次项组成，系数符号不同。 这些模型的一个关键特征是存在两个主导参数：$\alpha$，它表征晶格点之间的耦合，以及$\gamma$，它量化竞争非线性之间的平衡。 我们的研究集中在本征局域模式（ILMs）上——在几个晶格点上表现出空间局域化的解。 我们研究的基本例子是立方-四次方程，该方程最近被用来在平均场近似下描述带有Lee-Huang-Yang修正的3D BEC云。 我们采用从反连续极限（ACL）开始的数值延续法，在此情况下忽略晶格点之间的耦合（情况$\alpha=0$）。 我们分析$\alpha$相关的基本ILMs分支及其在$\gamma$变化时的分岔。 我们的研究表明，除了有限数量的分支外，所有从反连续极限开始的ILMs分支都会发生分岔，并且在$\alpha$的大值下不存在。 我们给出了涉及不超过3个激发晶格点的ILMs的分岔表。 结果显示，该模型支持没有对应于ACL的非对称ILMs。 同时我们研究了当$\alpha\to \infty$时可以无限延续的ILMs分支（在此称为$\infty$-分支）。 发现对于任何$\gamma$，恰好存在两条（考虑对称性后）$\infty$-分支。 当$\gamma$增大时，这些分支经历一系列分岔。 最后，我们将结果与二次-三次方程和三次-五次方程的结果进行比较，发现（a）分岔表、（b）存在没有ACL对应解以及（c）$\infty$-分支切换的场景中没有定性差异。 显示英文摘要

摘要： We study nonlinear excitations described by DNLS-type equations with so-called competing nonlinearities. These are the nonlinearities that consist of two power terms with coefficients of different sign. A key feature of these models is the presence of two governing parameters: $\alpha$, which characterizes the coupling between lattice sites, and $\gamma$, which quantifies the balance between competing nonlinearities. Our study focuses on intrinsic localized modes (ILMs) -- solutions that exhibit spatial localization over a few lattice sites. The basic example for our study is the cubic-quartic equation that recently has been used to describe 3D BEC cloud in the mean field approximation with Lee-Huang-Yang corrections. We employ numerical continuation from the anti-continuum limit (ACL) where the coupling between the lattice sites is neglected (the case $\alpha=0$). We analyze $\alpha$-dependent branches of the basic ILMs and their bifurcations when $\gamma$ varies. Our study shows that all branches of ILMs originated at anti-continuum limit, except a finite number, bifurcate and do not exist for large values of $\alpha$. We present tables of bifurcations for the ILMs that involve not more than 3 excited lattice sites. It is shown that the model supports nonsymmetric ILMs that have no counterparts in ACL. Also we study the branches of ILMs that can be continued unlimitedly when $\alpha\to \infty$ (called here $\infty$-branches). It was found that for any $\gamma$ there are exactly two (up to symmetries) $\infty$-branches. When $\gamma$ grows these branches undergo a sequence of bifurcations. Finally, we compare our results with the results for the quadratic-cubic equation and the cubic-quintic equation and found no qualitative difference in (a) tables of bifurcations, (b) presence of solutions without ACL counterpart and (c) scenario of switching of $\infty$-branches.