发表机构
University of Massachusetts Amherst; McMaster University（马萨诸塞大学阿默斯特分校; 麦克马斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Lyapunov--Schmidt约化等方法，研究了具有竞争非线性的NLS方程中扭结在外部势下的钉扎机制，证明了位置选择、特征值分裂及不稳定性的存在，并揭示了宇称对称性的缺失。
AI 中文摘要
外部势场打破了非线性薛定谔（NLS）方程的平移不变性，通常将连续族的驻波解转化为孤立的钉扎态。我们针对聚焦三次和散焦五次非线性竞争情形下的NLS方程中扭结这一典型情况研究该机制。利用Lyapunov--Schmidt约化方法证明了钉扎扭结的位置$s_0$在有效势$V_{\ m eff}(s)$的根附近被选择。随后，我们确定了Jacobian算子的平移零特征值在扰动下如何分裂，并证明了$V_{\ m eff}'(s_0)$的符号区分了正负能量的特征方向。我们进一步表明，如果Jacobian算子存在负特征值，则稳定性问题包含一个实部不稳定的特征值。由于零特征值嵌入在线性化算子的连续谱中，不稳定特征值的渐近展开推导是非标准的。最后，我们表明即使外部势是奇函数，扭结的持续性也缺乏宇称对称性。数值延拓、针对缓慢衰减特征函数自适应计算域的谱计算以及时间演化的直接动力学模拟，证实了不同代表性例子的解析预测。
英文摘要
External potentials break the translational invariance of nonlinear Schrödinger (NLS) equations and generally convert a continuous family of standing wave solutions into isolated pinned states. We study this mechanism for the prototypical case of a kink in the NLS equation with competing focusing cubic and defocusing quintic nonlinearities. The Lyapunov--Schmidt reduction method is used to prove that the location $s_0$ of the pinned kinks is selected near roots of an effective potential $V_{\rm eff}(s)$. We then determine how the translational zero eigenvalue of the Jacobian operator splits under the perturbation and prove that the sign of $V_{\rm eff}'(s_0)$ distinguishes eigendirections of positive and negative energy. We further show that if the Jacobian operator admits a negative eigenvalue, then the stability problem contains a real unstable eigenvalue. The derivation of the asymptotic expansion for the unstable eigenvalue is nonstandard because the zero eigenvalue is embedded in the continuous spectrum of the linearized operator. Finally, we show that the persistence of kinks lacks parity symmetry even if the external potential is odd. Numerical continuation, spectral computations with domains adapted to the slowly decaying eigenfunctions, and direct dynamical simulations of the time evolution corroborate the analytical predictions for different representative examples.
Comments20 pages, 6 figures
