The 2 + I-dimensional modified Kadomtsev-Petviashvili (mKP) equation is studied by the inverse-spectral-transform method. The initial-value problems for the mKP-I and mKP-II equations are solved by the nonlocal Riemann-Hilbert and a-problem techniques for initial data decaying sufficiently rapidly at infinity. The lump solutions for the mKP-I equation are found explicitly. Wide classes of the exact solutions for the mKP equation-namely, the rational solutions, including the plane lumps for the mKP-I equation; solutions with functional parameters; the plane solitons; and breathers-are constructed by the use of the a-dressing method based on the nonlocal a-problem. The Miura transformation between the mKP and KP equations is discussed.
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