非齐次薛定谔方程中的局域驻波
Localized standing waves in inhomogeneous Schrodinger equations
摘要：
本文考虑光在非均匀介质中传播时所产生的非线性薛定谔方程。这种不均匀性通过方程中非线性项的非均匀系数来反映。特别是，自聚焦和自散焦非线性的组合，与自散焦区域定位在一个有限的时间间隔，进行了研究。利用数值计算，建立了相应线性化系统的线性本征模到非线性态的扩展，特别是基态和第一激发态的非线性扩展。数值计算了这些状态的（不）稳定性，得到了对称非线性解在超过临界阈值范数后变得不稳定。对称状态的不稳定性，然后研究分析通过应用的拓扑参数。正对称态不稳定性的判定归结为驻波复合相平面轨道的简单几何性质。进一步的拓扑参数适用于更高的激发态和不稳定性再次减少到简单的几何计算。对于一个相对较高的范数，它是观察到的非对称状态分叉的对称的。还考虑了非对称态的稳定性和不稳定性。
英文摘要：
A nonlinear Schrodinger equation arising from light propagation down an inhomogeneous medium is considered. The inhomogeneity is reflected through a non-uniform coefficient of the non-linear term in the equation. In particular, a combination of self-focusing and self-defocusing nonlinearity, with the self-defocusing region localized in a finite interval, is investigated. Using numerical computations, the extension of linear eigenmodes of the corresponding linearized system into nonlinear states is established, particularly nonlinear continuations of the fundamental state and the first excited state. The (in)stability of the states is also numerically calculated, from which it is obtained that symmetric nonlinear solutions become unstable beyond a critical threshold norm. Instability of the symmetric states is then investigated analytically through the application of a topological argument. Determination of instability of positive symmetric states is reduced to simple geometric properties of the composite phase plane orbit of the standing wave. Further the topological argument is applied to higher excited states and instability is again reduced to straightforward geometric calculations. For a relatively high norm, it is observed that asymmetric states bifurcate from the symmetric ones. The stability and instability of asymmetric states is also considered.
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