Abstract
From the Levi spectral problem, we obtain the (1+1)-dimensional Burgers' equation and the (2+1)-dimensional modified Kadomtsev–Petviashvili equation. Through the Miura transformation, the modified Kadomtsev–Petviashvili equation is transformed into the Kadomtsev–Petviashvili equation. Then by constructing some new Darboux transformations, we obtain new soliton solutions of Burgers' equation and find solitons fusion. By solving two (1+1)-dimensional soliton equations, new soliton solutions of the modified Kadomtsev–Petviashvili equation are obtained. And then explicit solutions of the Kadomtsev–Petviashvili equation are also obtained through the Miura transformation.
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