具有增益和损失的离散非线性薛定谔方程的坍缩动力学
Collapse dynamics for the discrete nonlinear Schrödinger equation with gain and loss
摘要：
我们讨论的有限时间崩溃，也被称为爆破，一个离散的非线性薛定谔（DNLS）方程的线性和非线性增益和损失的解决方案。这个DNLS系统出现在许多固有的离散物理背景下，作为一个更现实的推广的哈密顿DNLS晶格。通过使用有限维和无限维相空间中的能量参数（由施加的边界条件指导），我们证明了崩溃时间的分析上限和下限，对模型的散焦和聚焦情况都有效。此外，强调了线性损耗参数中临界值的存在，将有限时间崩溃与能量衰减区分开来。的数值模拟，进行了广泛的初始数据，不仅验证了我们的边界的有效性，但也表明，分析边界可以是有用的，在确定两种不同类型的崩溃动力学，即，扩展或本地化。待定的离散/振幅制度，系统表现出任何类型的崩溃和实际爆破时间的方法，在许多情况下，在很好的协议，分别与上限或下限。当这些时间位于分析界限之间时，它们与上述主要类型的崩溃动力学的非平凡混合相关联，由于散焦/聚焦效应和能量增益/损失的确证，在离散性和非线性的存在下。
英文摘要：
We discuss the finite-time collapse, also referred as blow-up, of the solutions of a discrete nonlinear Schrödinger (DNLS) equation incorporating linear and nonlinear gain and loss. This DNLS system appears in many inherently discrete physical contexts as a more realistic generalization of the Hamiltonian DNLS lattice. By using energy arguments in finite and infinite dimensional phase spaces (as guided by the boundary conditions imposed), we prove analytical upper and lower bounds for the collapse time, valid for both the defocusing and focusing cases of the model. In addition, the existence of a critical value in the linear loss parameter is underlined, separating finite time-collapse from energy decay. The numerical simulations, performed for a wide class of initial data, not only verified the validity of our bounds, but also revealed that the analytical bounds can be useful in identifying two distinct types of collapse dynamics, namely, extended or localized. Pending on the discreteness /amplitude regime, the system exhibits either type of collapse and the actual blow-up times approach, and in many cases are in excellent agreement, with the upper or the lower bound respectively. When these times lie between the analytical bounds, they are associated with a nontrivial mixing of the above major types of collapse dynamics, due to the corroboration of defocusing/focusing effects and energy gain/loss, in the presence of discreteness and nonlinearity.
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