Using Wigner–Moyal formulation for the pair-ion plasma, with the residual electrons present we have derived the nonlinear Schrödinger equation for the kinetic envelope soliton with the help of Krylov–Bogoliubov–Mitropolsky approach. This is a full-quantum mechanical treatment of the problem where all the species (both ion and electrons) are treated on the same footing. The stability property of the nonlinear Schrödinger equation is then analyzed with the help of Wigner-transform in phase space. Our approach has the advantage that one can study the stability for various form of initial spectral distribution. The standard modulational stability is a special case of this approach. The experimental observation of the temperature difference between the species in pair ion is taken care of by the Cairns non-thermal distribution. It is observed that non-thermality has a good influence on the propagation characteristics and on the stability properties.

对于配对离子等离子体，使用Wigner-Moyal公式，在存在剩余电子的情况下，借助Krylov-Bogoliubov-Mitropolsky方法，得出了动力学包络孤子的非线性Schrödinger方程。这是对所有物种（离子和电子）都在同一基础上进行处理的问题的全量子力学处理。然后借助Wigner变换在相空间中分析非线性Schrödinger方程的稳定性。我们的方法的优点是可以研究各种形式的初始光谱分布的稳定性。标准调制稳定性是这种方法的特例。凯恩斯的非热分布照顾了成对离子中物种间温度差的实验观察。