It is well known that the linear stability of solutions of \(1+1\) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general \(N\times N\) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for \(N=3\) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.

中文翻译：

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非线性波的可积性和线性稳定性。

众所周知，可积分的（1 + 1 \）偏微分方程解的线性稳定性可以通过谱方法非常有效地研究。我们在这里提出了线性化方程本征模式的直接构造，该方程仅使用相关的Lax对，而没有参考光谱数据和边界条件。这种局部构造是在一般的\（N \ times N \）矩阵方案中给出的，以便适用于一大类可积方程，包括多分量非线性Schrödinger系统和多波共振相互作用系统。此一般方法涉及的解析和数值计算将作为\（N = 3 \）的示例进行详细说明用于散焦，聚焦和混合状态下两个耦合的非线性Schrödinger方程的特定系统。连续波解的不稳定性在其幅度和波数的整个参数空间中得到了充分讨论。通过在频谱变量的复平面中定义和计算频谱，可以明确表示本征频率。根据它们的拓扑特性，这些光谱在参数空间中的完整分类将被显示并以图形方式显示。对于一般选择的耦合常数，连续波解是线性不稳定的。