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Justification of the Lugiato-Lefever Model from a Damped Driven ϕ4 Equation
Akbar F.T.a, Gunara B.E.a, Susanto H.b
a Theoretical High Energy Physics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, 40132, Indonesia
b Department of Mathematics, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom
[vc_row][vc_column][vc_row_inner][vc_column_inner][vc_separator css=”.vc_custom_1624529070653{padding-top: 30px !important;padding-bottom: 30px !important;}”][/vc_column_inner][/vc_row_inner][vc_row_inner layout=”boxed”][vc_column_inner width=”3/4″ css=”.vc_custom_1624695412187{border-right-width: 1px !important;border-right-color: #dddddd !important;border-right-style: solid !important;border-radius: 1px !important;}”][vc_empty_space][megatron_heading title=”Abstract” size=”size-sm” text_align=”text-left”][vc_column_text]© 2020 by the authors.The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrodinger 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 ϕ4 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.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Author keywords” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Indexed keywords” size=”size-sm” text_align=”text-left”][vc_column_text]Lugiato-lefever equation,Nonlinear schrödinger equation,Small-amplitude approximation,ϕ4 equation[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Funding details” size=”size-sm” text_align=”text-left”][vc_column_text][{‘$’: ‘Acknowledgments: The work of FTA is partly supported by Riset ITB 2020. FTA would like to thank The Abdus Salam ICTP for associateship in 2018. The authors, BEG and HS, acknowledge Ministry of Research, Technology and Higher Education of the Republic of Indonesia for partial financial supports through the World Class Professor 2018 program. HS acknowledged Victor Brasch (Swiss Center for Electronics and Microtechnology) for his input on optical frequency combs.’}, {‘$’: ‘The contributions of the respective authors are as follows: F.T.A. conducted the analysis and prepared the article; B.E.G. provided resources and reviewed the article; H.S. conceptualised the project and reviewed the article. All authors have read and agree to the published version of the manuscript. The work of FTA is partly supported by Riset ITB 2020. FTA would like to thank The Abdus Salam ICTP for associateship in 2018. The authors, BEG and HS, acknowledge Ministry of Research, Technology and Higher Education of the Republic of Indonesia for partial financial supports through theWorld Class Professor 2018 program. HS acknowledged Victor Brasch (Swiss Center for Electronics and Microtechnology) for his input on optical frequency combs.’}][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”DOI” size=”size-sm” text_align=”text-left”][vc_column_text]https://doi.org/10.3390/MATH8050727[/vc_column_text][/vc_column_inner][vc_column_inner width=”1/4″][vc_column_text]Widget Plumx[/vc_column_text][/vc_column_inner][/vc_row_inner][/vc_column][/vc_row][vc_row][vc_column][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][/vc_column][/vc_row]