Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrödinger equation
Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrödinger equation
复制标题
三次非线性薛定谔方程一维平稳解的不稳定性
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
B. Deconinck
中科院分区：
文献类型：
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作者：
R. Thelwell;J. Carter;B. Deconinck
二维三次非线性薛定谔方程允许一大族一维有界行波解。所有这些解都可以用幅度和相位来表示。具有分段恒定相位的解决方案之前已经得到了很好的研究。发现其中一些解对于一维扰动是稳定的。对于二维扰动，没有这样的解是稳定的。我们考虑较大类解的稳定性，其相位取决于一维波形的空间维度。我们使用希尔方法对此类非平凡相位解的谱稳定性进行了数值研究。我们提供的证据表明，所有此类非平凡相解对于一维和二维扰动都是不稳定的。所有情况下都会出现不稳定：对于椭圆和双曲非线性薛定谔方程，以及在聚焦和散焦情况下。
The two-dimensional cubic nonlinear Schrodinger equation admits a large family of one-dimensional bounded travelling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant phase have been well studied previously. Some of these solutions were found to be stable with respect to one-dimensional perturbations. No such solutions are stable with respect to two-dimensional perturbations. We consider stability of the larger class of solutions whose phase is dependent on the spatial dimension of the one-dimensional wave form. We study the spectral stability of such nontrivial-phase solutions numerically, using Hill's method. We present evidence which suggests that all such nontrivial-phase solutions are unstable with respect to both one- and two-dimensional perturbations. Instability occurs in all cases: for both the elliptic and hyperbolic nonlinear Schrodinger equations, and in the focusing and defocusing cases.