The mKP equation with self-consistent sources (mKPESCS) is treated in theframework of the constrained mKP hierarchy. We introduce a new constrained mKPhierarchy which may be viewed as the stationary hierarchy of the mKP hierarchywith self-consistent sources. This offers a natural way to obtain the Laxrepresentation for the mKPESCS. Based on the conjugate Lax pairs, we constructthe generalized Darboux transformation with arbitrary functions in time $t$ forthe mKPESCS which, in contrast with the Darboux transformation for the mKPequation, provides a non-auto-B\"{a}cklund transformation between two mKPESCSswith different degrees. The formula for $n$-times repeated generalized Darbouxtransformation is proposed and enables us to find the rational solutions(including the lump solutions), soliton solutions and the solutions of breathertype of the mKPESCS.
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机译：在约束mKP层次结构的框架中处理了具有自洽源（mKPESCS）的mKP方程。我们引入了一个新的受约束的mKPhierarchy，可以将其视为具有自洽源的mKP层次的固定层次。这提供了一种自然的方式来获取mKPESCS的Lax表示。基于共轭Lax对，我们为mKPESCS构造了时间为$ t $的具有任意函数的广义Darboux变换，与mKPequation的Darboux变换相反，它提供了两个之间的非自动B \“ {a} cklund变换提出了不同程度的mKPESCSs方程，提出了$ n $倍重复广义Darboux变换的公式，使我们能够找到mKPESCS的有理解（包括块解），孤子解和呼吸型解。
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