Constraints of the Kadomtsev–Petviashvili hierarchy

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Abstract
For the Kadomtsev–Petviashvili (KP) hierarchy constructed in terms of the famous Sato theory, a ‘‘k constraint’’ is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates. The eigenvalue problem of the k‐constrained system is naturally obtained from the linear system of the KP hierarchy, which takes the form of kth‐order polynomial coupled with a first‐order one, thus we are able to derive the correspondent Lax pair, recursion operator, bi‐Hamiltonian structures, and conserved quantities. The constraints for the BKP hierarchy are also sketched.