The linear Lugiato-Lefever equation with forcing and nonzero periodic or nonperiodic boundary conditions

We consider the linear Lugiato-Lefever equation formulated on a finite interval with nonzero boundary conditions. In particular, using the unified transform of Fokas, we obtain explicit solution formulae both for the general nonperiodic initial-boundary value problem and for the periodic Cauchy problem. These novel solution formulae involve integrals, as opposed to the infinite series associated with traditional solution techniques, and hence they have analytical as well as computational advantages. Importantly, as the linear Lugiato-Lefever can be related to the linear Schrodinger equation via a simple transformation, our results are directly applicable also to the linear Schrodinger equation posed on a finite interval with nonzero boundary conditions.