非线性Schroedinger方程在Co-Exploding坐标系下的谱分析
A Spectral Analysis of the Nonlinear Schroedinger Equation in the Co-Exploding Frame
摘要：
非线性薛定谔模型是一个典型的色散波动方程，其特征在于有限时间爆破，无论是对于超临界指数（对于固定的维度）还是对于超临界维度（对于固定的非线性指数）。在识别所谓的“共爆炸框架”中的自相似解后，动力系统对其稳定性的分析是自然的，但由于相关框架的混合Hamilton耗散特性而变得复杂。在目前的工作中，我们研究了相关的线性化问题的谱图像。我们研究了与平移、$U(1)$和共形不变性相关的3个特征值对的点谱，以及连续谱。我们发现，两个本征值成为积极的，但归因于对称性，因此不与不稳定性。除了消失的本征值，3个以上被发现是负的和真正的，而连续谱是几乎垂直的，并在左半（频谱）平面。最后，边界的微妙影响也进行了评估，并澄清了它们在所观察到的弱本征值振荡中的作用。
英文摘要：
The nonlinear Schroedinger model is a prototypical dispersive wave equation that features finite time blowup, either for supercritical exponents (for fixed dimension) or for supercritical dimensions (for fixed nonlinearity exponent). Upon identifying the self-similar solutions in the so-called "co-exploding frame", a dynamical systems analysis of their stability is natural, yet is complicated by the mixed Hamiltonian-dissipative character of the relevant frame. In the present work, we study the spectral picture of the relevant linearized problem. We examine the point spectrum of 3 eigenvalue pairs associated with translation, $U(1)$ and conformal invariances, as well as the continuous spectrum. We find that two eigenvalues become positive, yet are attributed to symmetries and are thus not associated with instabilities. In addition to a vanishing eigenvalue, 3 more are found to be negative and real, while the continuous spectrum is nearly vertical and on the left-half (spectral) plane. Finally, the subtle effects of the boundaries are also assessed and their role in the observed weak eigenvalue oscillations is clarified.
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