关于非线性薛定谔方程的变形
On a deformation of the nonlinear Schrödinger equation
摘要：
我们研究了最近在可积系统族变形的背景下导出的非线性薛定谔方程的变形。这种系统的方法也导致了已知的可积方程，如Camassa-Holm方程。虽然这个新方程还没有被证明是完全可积的，它的孤立波解表现出典型的孤子行为，包括近弹性碰撞。我们将首先集中在驻波的解决方案，这可以是平滑的或峰值，然后，在数值模拟的帮助下，我们将研究孤立波，它们的相互作用，最后流氓波调制不稳定制度。有趣的是，在孤立波碰撞过程中或在流氓波事件的解决方案的结构是尖锐的，并有更大的振幅比在经典的NLS方程。
英文摘要：
We study a deformation of the nonlinear Schrödinger equation recently derived in the context of deformation of hierarchies of integrable systems. This systematic method also led to known integrable equations such as the Camassa-Holm equation. Although this new equation has not been shown to be completely integrable, its solitary wave solutions exhibit typical soliton behaviour, including near elastic collisions. We will first focus on standing wave solutions, which can be smooth or peaked, then, with the help of numerical simulations, we will study solitary waves, their interactions and finally rogue waves in the modulational instability regime. Interestingly the structure of the solution during the collision of solitary waves or during the rogue wave events are sharper and have larger amplitudes than in the classical NLS equation.
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