中文摘要：

从一个新的2×2矩阵谱问题出发导出了一族非线性孤子方程，针对前两个非平凡的孤子方程，通过谱问题的基解矩阵，利用其Lax对的规范变换，得到了孤子方程的Darboux变换。作为应用，给出了孤子方程的一些精确解，并作出了孤子图，有助于对方程所描述的自然现象进行分析和研究。

英文摘要：

Starting from a new 2 ×2 matrix spectral problem, we propose a nonlinear soliton equation. For the first two non-trivial soliton equations, gauge transformation of the Lax pair of the equation is found by using the fundamental solution matrix of the ma- trix spectral problem. Then, the Darboux transformation of the soliton equation is obtained. As an application, some exact solutions of the equation are given, and several interesting figures of the solutions are plotted, it helps to analysis and research the natural phenomena described by the equations.