We numerically study solitary waves in the coupled nonlinear Schr\"odinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change its stability type at the saddle-node bifurcation point.

中文翻译：

---

耦合非线性薛定谔方程中孤立波分岔和谱稳定性的数值计算

我们数值研究耦合非线性 Schr\"odinger 方程中的孤立波。我们检测基本孤立波的干草叉分岔并计算相应特征值问题的特征值和特征函数，以确定在干草叉分岔处产生的孤立波的光谱稳定性。我们的数值结果证明了作者最近获得的理论结果。我们还计算了与呈现鞍节点分叉的分叉孤立波的零特征值相关的广义特征函数，并表明它在鞍节点分叉点处不会改变其稳定性类型.