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For the superconducting phase with a d-wave order parameter and zerotemperature the magnetic susceptibility of the t-J model is calculated usingthe Mori projection operator technique. Conditions for the appearance of anincommensurate magnetic response below the resonance frequency are identified.A fast decay of the tails of the hole coherent peaks and a weak intensity ofthe hole incoherent continuum near the Fermi level are enough to produce anincommensurate response using different hole dispersions established for$p$-type cuprates, in which such response was observed. In this case, thenesting of the itinerant-electron theory or the charge modulation of the stripetheory is unnecessary for the incommensurability. The theory reproduces thehourglass dispersion of the susceptibility maxima with their location in themomentum space similar to that observed experimentally. The upper branch of thedispersion stems from the excitations of localized spins, while the lower oneis due to the incommensurate maxima of their damping. The narrow and intensiveresonance peak arises if the frequency of these excitations at theantiferromagnetic momentum lies below the edge of the two-fermion continuum;otherwise the maximum is broad and less intensive.

Magnetic incommensurability in $p$-type cuprate perovskites