On the Normalized Ground States of Second Order PDE’s with Mixed Power Non-linearities
On the Normalized Ground States of Second Order PDE’s with Mixed Power Non-linearities
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DOI：
10.1007/s00220-019-03484-7
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发表时间：
2019-06
影响因子：
2.4
通讯作者：
A. Stefanov
中科院分区：
文献类型：
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作者：
A. Stefanov
对于每个参数，在必要条件下，我们构造具有混合幂非线性的二阶偏微分方程的归一化波，其中。我们证明这些是钟形平滑局部函数，并且计算它们的精确渐近函数。我们研究拉格朗日乘子相对于波范数的光滑度问题，即映射，这是一个与其稳定性相关的经典问题。我们表明，这与所述孤子的非简并性问题密切相关。我们提供了广泛的非线性，对于这些非线性，波是非简并的。在一些最小的额外假设下，我们表明 a.e.其中，作为 NLS 方程的解，地图是可微的，并且波在光谱上（并且在某些情况下在轨道上）是稳定的。对于相同的波，如扎哈罗夫-库兹涅佐夫系统的行波，得到了类似的结果。
For eachand under necessary conditions on the parameters, we construct normalized waves for second order PDE’s with mixed power non-linearities, with. We show that these are bell-shaped smooth and localized functions, and we compute their precise asymptotics. We study the question for the smoothness of the Lagrange multiplier with respect to thenorm of the waves, namely the map, a classical problem related to its stability. We show that this is intimately related to the question for the non-degeneracy of the said solitons. We provide a wide class of non-linearities, for which the waves are non-degenerate. Under some minimal extra assumptions, we show that a.e. in, the mapis differentiable and the wavesare spectrally (and in some cases orbitally) stable as solutions to the NLS equation. Similar results are obtained for the same waves, as traveling waves of the Zakharov–Kuznetsov system.