Modified constrained KP hierarchy and bi-Hamiltonian structures (Q2221308)

The author presents a new class of KP-like integrable hierarchy, called {modified constrained KP hierarchy}. It differs from the standard constrained KP hierarchy in [\textit{W. Oevel} and \textit{W. Strampp}, Commun. Math. Phys. 157, No. 1, 51--81 (1993; Zbl 0793.35095)] by the addition of a spectral parameter. The constraint is given by \[ \hat{L}=\partial^n-u_{n-2}\partial^{n-2}-\cdots-u_1\partial-u_0-\lambda\,q\partial^{-1}r, \] where \(q\) and \(r\) are, the eigenfunctions of the scattering problem for a KP hierarchy. The standard constrained KP hierarchy is recovered with \(\lambda=1\). The author presents the hierarchies obtained in the case \(n=1,2,3\) and their bi-Hamiltonian structures. Despite the identification of a gauge transform linking the \(n=2\) and \(n=3\) cases, the Yajima-Oikawa and Melnikov hierarchies (and their bi-Hamiltonian structures as well) are not obtained by Miura transforms, which would have been very cumbersome, but using a trace identity.