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We analyze the isoscalar response related to breathing modes with particularattention being paid to low-lying excitations in neutron--rich nuclei. We usethe subtracted second random--phase approximation (SSRPA) to describemicroscopically the response. By increasing the neutron excess, we study theevolution of the response in Ca isotopes going from $^{40}$Ca to $^{48}$Ca andto $^{60}$Ca as well as in $N=20$ isotones going from $^{40}$Ca to $^{36}$S andto $^{34}$Si. Finally, the case of $^{68}$Ni is investigated. We predict softmonopole modes in neutron--rich nuclei which are driven by neutron excitations.At variance with dipole pygmy modes, these neutron excitations are not onlystrongly dominant at the surface of the nucleus but over its entire volume. Theeffect of the mixing with two particle-two hole configurations induced by theSSRPA model is analyzed. The properties of such soft neutron modes areinvestigated in terms of their excitation energies, transition densities andwave--function components. Their collectivity is also discussed as a functionof the isospin asymmetry and of the mass of the nucleus. The link between suchlow--energy compression modes and a compressibility modulus introduced forneutron--rich infinite matter is finally studied.

Soft breathing modes in neutron--rich nuclei with the subtracted second random--phase approximation