Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> Equation
The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has become th...
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MDPI AG
2020-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/5/727 |
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| author | Fiki Taufik Akbar Bobby Eka Gunara Hadi Susanto |
| author_facet | Fiki Taufik Akbar Bobby Eka Gunara Hadi Susanto |
| author_sort | Fiki Taufik Akbar |
| collection | DOAJ |
| description | The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has become the basic model describing microresonator (Kerr) frequency combs. Recent works derive the Lugiato-Lefever equation from a class of damped driven <inline-formula> <math display="inline"> <semantics> <msup> <mi>ϕ</mi> <mn>4</mn> </msup> </semantics> </math> </inline-formula> equations closed to resonance. In this paper, we provide a justification of the envelope approximation. From the analysis point of view, the result is novel and non-trivial as the drive yields a perturbation term that is not square integrable. The main approach proposed in this work is to decompose the solutions into a combination of the background and the integrable component. This paper is the first part of a two-manuscript series. |
| first_indexed | 2024-03-10T20:01:44Z |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T20:01:44Z |
| publishDate | 2020-05-01 |
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| spelling | doaj.art-b1d02eae8638473ea5b81256b14a635b2023-11-19T23:33:41ZengMDPI AGMathematics2227-73902020-05-018572710.3390/math8050727Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> EquationFiki Taufik Akbar0Bobby Eka Gunara1Hadi Susanto2Theoretical High Energy Physics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 40132, IndonesiaTheoretical High Energy Physics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 40132, IndonesiaDepartment of Mathematics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UKThe Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has become the basic model describing microresonator (Kerr) frequency combs. Recent works derive the Lugiato-Lefever equation from a class of damped driven <inline-formula> <math display="inline"> <semantics> <msup> <mi>ϕ</mi> <mn>4</mn> </msup> </semantics> </math> </inline-formula> equations closed to resonance. In this paper, we provide a justification of the envelope approximation. From the analysis point of view, the result is novel and non-trivial as the drive yields a perturbation term that is not square integrable. The main approach proposed in this work is to decompose the solutions into a combination of the background and the integrable component. This paper is the first part of a two-manuscript series.https://www.mdpi.com/2227-7390/8/5/727Lugiato-Levefer equationnonlinear Schrödinger equation<i>ϕ</i><sup>4</sup> equationsmall-amplitude approximation |
| spellingShingle | Fiki Taufik Akbar Bobby Eka Gunara Hadi Susanto Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> Equation Mathematics Lugiato-Levefer equation nonlinear Schrödinger equation <i>ϕ</i><sup>4</sup> equation small-amplitude approximation |
| title | Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> Equation |
| title_full | Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> Equation |
| title_fullStr | Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> Equation |
| title_full_unstemmed | Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> Equation |
| title_short | Justification of the Lugiato-Lefever Model from a Damped Driven <i>ϕ</i><sup>4</sup> Equation |
| title_sort | justification of the lugiato lefever model from a damped driven i ϕ i sup 4 sup equation |
| topic | Lugiato-Levefer equation nonlinear Schrödinger equation <i>ϕ</i><sup>4</sup> equation small-amplitude approximation |
| url | https://www.mdpi.com/2227-7390/8/5/727 |
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