可积耦合AKNS方程的达布变换及其精确显式解

Darboux Transformation of the Integrable Coupling AKNS Equations and Its Exact Explicit Solutions

由于许多物理现象需要建立有两个或多个分量的波动模型用以说明不同的模式、频率和极化现象。此外,只有多分量系统才能从理论和实践上解释一些多个物理场能量的交换。因此,给定一个可积系统,我们如何构造一个非平凡的微分方程系统,使它是可积的并且包含原系统为一个子系统,是可积耦合研究的重要问题之一。利用一个稳定方程推导可积耦合AKNS方程,然后给出一次达布变换,其中的元素可以用两个行列式的商来表示。通过比较一次达布变换的形式和特点,推导出用行列式表示的N次达布变换公式。进而利用种子解,通过N次达布变换进行迭代,可以得到任意阶孤子解。作为达布变换的应用,我们求出了精确显式单孤子解。

For many physical phenomena,it is necessary to establish wave models with two or more components to explain different patterns,frequencies and polarization phenomenna.In addition,only multi-component systems can explain the energy exchange of multiple physical fields theoretically and practically.Therefore,for a integrable system,how to construct a non-trivial differential equation system that is integrable and contains the original system as a subsystem is one of the important problems in the study of integrable coupling.In this work,the integrable couplings of the Ablowitz-Kaup-Newell-Segur equation are constructed based on a stationary equation.Then a Darboux transformation in which the elements can be expressed by the quotient of two determinants is obtained,and the production process is proved strictly.By comparing the forms and characteristics of the one-fold Darboux transformation,N-fold Darboux transformation formula which can be demonstrated as determinants is derived.Therefore,by means of seed solutions and N-fold Darboux transformation,any-order soliton solutions can be derived.As the application of Darboux tarnsformation,we solve the exact explicit one-soliton solutions.