锯齿型Klein-Gordon模型中简并相移多呼吸子的不存在性
On the nonexistence of degenerate phase-shift multibreathers in a zigzag Klein-Gordon model
摘要：
在这项工作中，我们研究了存在低振幅四位相移多呼吸子的小值的耦合$ε$在Klein-Gordon（KG）链的相互作用长于经典的最近邻的。在适当的参数制度，所考虑的晶格承担连接到模型超出一个空间维度，即所谓的锯齿形晶格，以及二维正方形晶格。我们初步研究的持久性条件的系统，以寻求涡状波形。虽然这种方法提供了有用的见解，由于这些解决方案的退化，它不允许我们确定他们是否构成真正的解决方案，我们的系统。为了克服这一障碍，我们走了一条不同的道路。在锯齿形构型的情况下，通过Lyapunov-Schmidt分解，我们能够建立我们的模型的分支方程可以被认为是，在小能量和小耦合制度，作为一个相应的非局部离散非线性薛定谔（NL-dNLS）方程的扰动。在那里，退化相移离散孤子的不存在性的结果可以证明利用一个合适的密度电流的NL-dNLS，在最近的文献中获得的扩展。最后，简要地考虑一个一维模型轴承相似的正方形晶格，我们得出结论，上述策略是没有效率的证明存在或不存在的旋涡，由于这种配置的更高的简并性。
英文摘要：
In this work, we study the existence of low amplitude four-site phase-shift multibreathers for small values of the coupling $ε$ in Klein-Gordon (KG) chains with interactions longer than the classical nearest-neighbour ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namely the so-called zigzag lattice, as well as the two-dimensional square lattice. We examine initially the persistence conditions of the system, in order to seek for vortex-like waveforms. Although this approach provides useful insights, due to the degeneracy of these solutions, it does not allow us to determine if they constitute true solutions of our system. In order to overcome this obstacle, we follow a different route. In the case of the zigzag configuration, by means of a Lyapunov-Schmidt decomposition, we are able to establish that the bifurcation equation for our model can be considered, in the small energy and small coupling regime, as a perturbation of a corresponding non-local discrete nonlinear Schrödinger (NL-dNLS) equation. There, nonexistence results of degenerate phase-shift discrete solitons can be demonstrated by exploiting the expansion of a suitable density current of the NL-dNLS, obtained in recent literature. Finally, briefly considering a one-dimensional model bearing similarities to the square lattice, we conclude that the above strategy is not efficient for the proof of the existence or nonexistence of vortices due to the higher degeneracy of this configuration.
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