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The linear
Lugiato–Lefever equation with forcing and nonzero periodic or
nonperiodic boundary conditions
Joseph Wimmergren and Dionyssios Mantzavinos |
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Vol. 16 (2023), No. 5, 783–808
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Abstract |
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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. |
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Keywords
linear Lugiato–Lefever equation, linear Schrödinger
equation, finite interval, initial-boundary value problem,
periodic problem, nonzero boundary conditions, unified
transform, Fokas method
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Mathematical Subject Classification
Primary: 35G16, 35Q55
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Milestones
Received: 19 May 2022
Revised: 5 September 2022
Accepted: 12 November 2022
Published: 9 December 2023
Communicated by Kenneth S. Berenhaut
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Authors |
| Joseph Wimmergren | |
| Department of Mathematics University of Michigan Ann Arbor, MI United States |
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| Dionyssios Mantzavinos | |
| Department of Mathematics University of Kansas Lawrence, KS United States |
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| © 2023 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). |
