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 Schrödinger
equation via a simple transformation, our results are directly applicable also to the
linear Schrödinger equation posed on a finite interval with nonzero boundary
conditions.
Keywords
linear Lugiato–Lefever equation, linear Schrödinger
equation, finite interval, initial-boundary value problem,
periodic problem, nonzero boundary conditions, unified
transform, Fokas method
