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

Joseph Wimmergren and Dionyssios Mantzavinos

Vol. 16 (2023), No. 5, 783–808
Abstract

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
Mathematical Subject Classification
Primary: 35G16, 35Q55
Milestones
Received: 19 May 2022
Revised: 5 September 2022
Accepted: 12 November 2022
Published: 9 December 2023

Communicated by Kenneth S. Berenhaut
Authors
Joseph Wimmergren
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Dionyssios Mantzavinos
Department of Mathematics
University of Kansas
Lawrence, KS
United States