论文标题
Akhmedieev喘息的线性和非线性不稳定
The linear and nonlinear instability of the Akhmediev breather
论文作者
论文摘要
Akhmediev呼吸器（AB）及其M-Soliton概括$ ab_m $是焦点NLS方程周期性的精确解决方案，并在不稳定的背景上及时地定位于及时地定位；他们描述了$ m $不稳定的非线性模式及其相互作用的外观，并且预计它们在自然界中的周期性异常（Rogue）波（Rogue）波（Rogue）波（AWS）中起着相关作用。相当明显的是，对于NLS方程的小扰动，它们是不稳定的。关于这些解决方案在NLS动力学中的扰动，以下常见文献相信文献。对于第一个$ n $模式，让NLS背景不稳定；然后i）如果$ m $不稳定的$ ab_m $解决方案模式严格包含在此组中（$ m <n $），则$ ab_m $是不稳定的； ii）如果$ m = n $，则所谓的“不稳定性的饱和度”，则$ ab_m $解决方案是中性稳定的。我们认为，即使在饱和情况$ M = n $中，$ ab_m $ $解决方案也总是不稳定的，我们在最简单的情况下证明了这一点$ m = n = 1 $。我们首先证明了线性不稳定性，构建了两个线性化理论的$ x $周期解决方案的示例。然后，我们使用先前的结果研究了非线性不稳定性，表明i）扰动的AB初始条件演变成精确的Fermi-Pasta-Pasta-ulam-tsingou（fput）ABS的复发，它用初始数据的基本函数来描述，以领导顺序； ii）AB解决方案比背景解决方案更不稳定，其不稳定为$ t \至0 $，其中$ t $是AB外观参数。尽管AB解决方案是线性和非线性不稳定的，但它在本质上是相关的，因为它的不稳定性会产生ABS的FPUT复发。这些结果适当地概括为$ m = n> 1 $。
The Akhmediev breather (AB) and its M-soliton generalization $AB_M$ are exact solutions of the focusing NLS equation periodic in space and exponentially localized in time over the constant unstable background; they describe the appearance of $M$ unstable nonlinear modes and their interaction, and they are expected to play a relevant role in the theory of periodic anomalous (rogue) waves (AWs) in nature. It is rather well established that they are unstable with respect to small perturbations of the NLS equation. Concerning perturbations of these solutions within the NLS dynamics, there is the following common believe in the literature. Let the NLS background be unstable with respect to the first $N$ modes; then i) if the $M$ unstable modes of the $AB_M$ solution are strictly contained in this set ($M<N$), then the $AB_M$ is unstable; ii) if $M=N$, the so-called "saturation of the instability", then the $AB_M$ solution is neutrally stable. We argue instead that the $AB_M$ solution is always unstable, even in the saturation case $M=N$, and we prove it in the simplest case $M=N=1$. We first prove the linear instability, constructing two examples of $x$-periodic solutions of the linearized theory growing exponentially in time. Then we investigate the nonlinear instability using our previous results showing that i) a perturbed AB initial condition evolves into an exact Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence of ABs described in terms of elementary functions of the initial data, to leading order; ii) the AB solution is more unstable than the background solution, and its instability increases as $T\to 0$, where $T$ is the AB appearance parameter. Although the AB solution is linearly and nonlinearly unstable, it is relevant in nature, since its instability generates a FPUT recurrence of ABs. These results suitably generalize to the case $M=N>1$.
