Instability in nonlinear Schrödinger breathers.
Instability in nonlinear Schrödinger breathers.
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DOI：
10.4067/s0716-09172017000400653
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发表时间：
2017-12
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影响因子：
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通讯作者：
Claudio Muñoz
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文献类型：
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作者：
Claudio Muñoz
我们考虑在一维线上提出的聚焦非线性薛定谔方程，在空间无穷远处具有非零背景条件，由均匀平面波给出。对于这个物理问题，我们研究了索博列夫空间中背景波扰动的初值问题。众所周知，该问题的相关线性动力学描述了文献中称为调制不稳定性的现象，最近也与海洋动力学中异常波浪的出现有关。从定性的角度来看，背景状态的小扰动会随着时间的推移以指数方式增加其大小。在本文中，我们表明，即使由于调制不稳定状态而导致线性动力学没有时间衰减，该方程在 Hs 中仍然是局部适定的，s > 1/2。我们应用这一结果来严格证明两种著名的 NLS 解决方案的不稳定特性：Peregrine 和 Kuznetsov-Mabreathers。
We consider the focusing Nonlinear Schrodinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study the initial value problem for perturbations of the background wave in Sobolev spaces. It is well-known that the associated linear dynamics for this problem describes a phenomenon known in the literature as modulational instability, also recently related to the emergence of rogue waves in ocean dynamics. In qualitative terms, small perturbations of the background state increase its size exponentially in time. In this paper we show that, even if there is no time decay for the linear dynamics due to the modulationally unstable regime, the equation is still locally well-posed in Hs, s > 1/2. We apply this result to give a rigorous proof of the unstable character of two well-known NLS solutions: the Peregrine and Kuznetsov-Mabreathers.