The Lugiato-Lefever equation driven by a double tightly focused pump

We introduce a model of an optical cavity based on the one-dimensional Lugiato-Lefever (LL) equation, which includes the pump represented by a symmetric pair of tightly localized "hot spots" (HSs) with phase shift $\chi$ between them, and self-focusing or defocusing cubic nonlinearity. Families of bound states, pinned to the double HS, are found in the system's parameter space. They feature the effect of the symmetry breaking (SB) between peaks pinned to individual HSs, provided that the phase shift takes values $0<\chi <\pi$, and the LL equation includes the loss term. The SB, which is explained analytically, takes place in the full LL model and its linearized version alike. The same phenomenology is also explored in the framework of the LL equation with the double HS and quintic self-focusing. In that case, there are stable symmetric and asymmetric bound states, in spite of the presence of the background instability driven by the critical collapse.