Shape-morphing reduced-order models for nonlinear Schrödinger equations
Shape-morphing reduced-order models for nonlinear Schrödinger equations
复制标题
非线性薛定谔方程的变形降阶模型
作者：
W. Anderson;M. Farazmand
考虑一类非线性薛定谔方程描述的非线性色散波的降阶模型。我们比较两种非线性降阶建模方法：（i）减少拉格朗日方法，它依赖于变分制定的NLS和（ii）最近开发的方法降阶非线性解决方案（RONS）。首先，我们证明了令人惊讶的结果，虽然这两种方法似乎是完全不同的，他们可以得到一个单一的复值主方程的真实的和虚部。此外，对于NLS方程在一个固定的框架，我们表明，减少拉格朗日方法无法预测正确的群速度的波，而RONS预测正确的群速度。最后，对于修正后的NLS方程，其中约化拉格朗日方法是不适用的，RONS降阶模型准确地逼近真实的解决方案。
We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrödinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) the recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves, whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.