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Commutativity of Pfaffianization and Bäcklund transformations: the KP equation

The Pfaffianization procedure is applied to the modified KP equation. As a result, we derive a coupled modified KP system, which is shown to constitute a bilinear Bäcklund transformation for the coupled KP (cKP) equation introduced by Hirota and Ohta. This means that commutativity of Pfaffianization and Bäcklund transformation is valid for the KP equation. As a by-product, we explicitly derive a coupled system from Bäcklund transformation formulae of the cKP equation, which solves an open problem proposed by Hirota. In addition to starting from the Bäcklund transformation, a Lax pair for the coupled KP equation is worked out. Finally, two kinds of Pfaffian solutions to the coupled modified KP equation are derived.