Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable?

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Abstract
The Kadomtsev–Petviashvili (KP) hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy. Since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the KP hierarchy provides integrable scalar equations with an arbitrary number of independent variables. It is shown, by investigating a specific equation in 3+1 dimensions, that the higher equations in the KP hierarchy are only integrable in a conditional sense. The equation under study, taken in isolation, does not pass certain well‐known and reliable integrability tests. Thus, applying Painlevé analysis, we find that solutions exist, allowing movable critical points. Furthermore, solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions. On the other hand, if the dependence of a solution on the first 2+1 variables is restricted by the fact that it should also satisfy the KP equation itself, then the integrability conditions in the other dimensions are satisfied. ‘‘Conditional integrability’’ thus means that linear techniques will provide only those solutions of equations in the hierarchy that simultaneously satisfy lower equations in the same hierarchy.

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