expanded formula meaning in English

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And beginning with a perturbed nls equation , using a multi - scales perturbation expansion , we get the zero order and the first order equations , discuss the eigenstates of the operator in the equations , induct relevant " derivative states " , form the completeness of the bounded eigenstates of the associated operator in li space , and expand the corresponding parameters in the closure , get a series evolution equations of the coefficients in the expanded formulas , find the first order approximate solution by researching the evolution equations . this paper also gives the basis of this method - the completeness we have formed and the singular perturbation technique
) dinser方程的求解问题，讨论了自伴算子的本征函数的正交性和完备性，介绍了寻求微分方程的近似解常用的摄动方法，并从带有某种扰动项的nls方程出发，利用多重尺度的摄动方法得到了方程的零级近似方程和一级近似方程，通过对近似方程中算子的特征态的讨论，引入适当的“导出态” ，建立了算子在l _ 2空间的特征态的完备性。