The decomposition of the time reversal operator (DORT) method is a selective detection and focusing technique widely used in acoustics. The background theory is based on the time reversal (TR) where a signal collected from an array of transducers is time reversed and then back-propagated into the medium to focus on selected targets. The DORT method was developed to detect damages and inhomogeneities which induce linear changes of the elastic moduli.However, material may experience some type of damage (cracks, voids) which may affect the nonlinear elastic wave propagation behaviour with a small undetectable changes of the linear elastic moduli.This paper presents an extension of this method for the detection of multiple linear and nonlinear scatterers. In the presence of nonlinear elastic scatterers, forcing the solid with a harmonic excitation, the time reversal operator can be obtained not only at the fundamental excitation frequency but also at the odd harmonics. At the fundamental harmonic, inhomogeneities or linear damages can be individually selected, but using the odd harmonics nonlinear elastic damages can be detected. Thus, by decomposing the operator at the 3rd harmonic is possible to focus on nonlinear scatterers and to distinguish them from the linear inhomogeneities. The Preisach-Mayergoz (PM) material constitutive model was used for modeling the presence of nonlinear elastic hysteretic scatterers.This paper presents a complete mathematical analysis of this method for 1 and 2D structures. The paper includes numerical simulations applied to 1 and 2 dimensional structures showing the capability of the method to focus selectively on linear and nonlinear inhomogeneous medium.