In this paper, we consider applications of the connection between the soliton theory and the commuting nonselfadjoint operator theory, established by Livšic and Avishai. An approach to the inverse scattering problem and to the wave equations is presented, based on the Livšic operator colligation theory (or vessel theory) in the case of commuting bounded nonselfadjoint operators in a Hilbert space, when one of the operators belongs to a larger class of nondissipative operators with asymptotics of the corresponding nondissipative curves. The generalized Gelfand–Levitan–Marchenko equation of the cases of different differential equations (the Korteweg–de Vries equation, the Schrödinger equation, the Sine–Gordon equation, the Davey–Stewartson equation) are derived. Relations between the wave equations of the input and the output of the generalized open systems, corresponding to the Schrödinger equation and the Korteweg–de Vries equation, are obtained. In these two cases, differential equations (the Sturm–Liouville equation and the 3-dimensional differential equation), satisfied by the components of the input and the output of the corresponding generalized open systems, are derived.

在本文中，我们考虑孤子理论与通勤的非自伴算子理论之间的联系的应用，这是由利夫西奇和阿维沙伊建立的。当换位有界非自伴算子在希尔伯特空间中交换时（其中一个算子属于较大类），基于Livšic算子积算理论（或器皿理论），提出了一种求解逆散射问题和波动方程的方法。非耗散算子与相应非耗散曲线的渐近性。推导了不同微分方程（Korteweg – de Vries方程，Schrödinger方程，Sine – Gordon方程，Davey – Stewartson方程）情况下的广义Gelfand – Levitan – Marchenko方程。得到了广义开放系统输入和输出的波动方程之间的关系，对应于Schrödinger方程和Korteweg – de Vries方程。在这两种情况下，可以推导相应的广义开放系统的输入和输出分量所满足的微分方程（Sturm – Liouville方程和3维微分方程）。