Lugiato–Lefever equation  The model usually designated as Lugiato–Lefever equation (LLE) was formulated in 1987 by Luigi Lugiato and Renè Lefever   The same equation governs two kinds of patterns: stationary patterns that arise in the planes orthogonal with respect to the direction of propagation of light ("transverse patterns") and patterns that form in the longitudinal direction ("longitudinal" "patterns"), travel along the cavity with the velocity of light in the medium and give rise to a sequence of pulses in the output of the cavity. The case of longitudinal patterns is intrinsically linked to the phenomenon of “Kerr frequency combs” in microresonators, discovered in 2007 by Tobias Kippenberg and collaborators, that has raised a very lively interest, especially because of the applicative avenue it has opened. Figure 1 shows a light beam that propagates in the formula_1 direction, while formula_2 and formula_3 are the transverse directions. If we assume that the electric field as formula_4 , where formula_5 denotes time, is linearly polarized and therefore can be treated as a scalar, we can express it in terms of the slowly varying normalized complex envelope formula_6 in this way  where formula_8 is the frequency of the light beam that is injected into the cavity and formula_9 of the light velocity in the Kerr medium that fills the cavity. For definiteness, consider a ring cavity (Fig. 2) of very high quality (High-Q cavity). In the original LLE, one assumes conditions such that the envelope formula_10 is independent of the longitudinal variable formula_1 ( i.e. uniform along the cavity), so that formula_12. The equation reads  where formula_14 and formula_15, formula_16 are normalized temporal and spatial variables, i.e. formula_17, formula_18, formula_19, with formula_20 being the cavity decay rate or cavity linewidth, formula_21 the diffraction length in the cavity. formula_22 is the cavity detuning parameter, with formula_23 being the cavity frequency nearest to formula_8. In the righthand side of Eq.(), formula_25 is the normalized amplitude of the input field that is injected into the cavity,  the second is the decay term, the third is the detuning term, the fourth is the cubic nonlinear term that takes into account the Kerr medium, the last term with the transverse Laplacian formula_26 describes diffraction in the paraxial approximation.