Abstract
We give a method for finding explicit solutions of the Kadomtsev-Petviashvili (KP) equation which are associated to rank-three commuting ordinary differential operators. Explicit formulae for the coefficients of rank-three commuting differential operators are used to reduce the KP equation to a nonlinear partial differential equation in only two variables. This equation, which is obtained in a similar way to the rank-two Krichever-Novikov (KN) equation, is easier to solve than KP and some solutions are given.
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