Paradigm for the creation of scales and phases in nonlinear evolution equations

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dc.contributor.authorCheverry, Christophe
dc.contributor.authorFarhat, Shahnaz
dc.date.accessioned2023-05-19T20:10:30Z
dc.date.available2023-05-19T20:10:30Z
dc.date.issued2023-01-25
dc.description.abstractThe transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.
dc.description.departmentMathematics
dc.formatText
dc.format.extent59 pages
dc.format.medium1 file (.pdf)
dc.identifier.citationCheverry, C., & Farhat, S. (2023). Paradigm for the creation of scales and phases in nonlinear evolution equations. <i>Electronic Journal of Differential Equations, 2023</i>(09), pp. 1-59.
dc.identifier.issn1072-6691
dc.identifier.urihttps://hdl.handle.net/10877/16844
dc.language.isoen
dc.publisherTexas State University, Department of Mathematics
dc.rightsAttribution 4.0 International
dc.rights.holderThis work is licensed under a Creative Commons Attribution 4.0 International License.
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.sourceElectronic Journal of Differential Equations, 2022, San Marcos, Texas: Texas State University and University of North Texas.
dc.subjectNonlinear differential equations
dc.subjectNormal forms
dc.subjectIntegrability
dc.subjectBlow-up procedure
dc.subjectGeometrical optics
dc.subjectWKB analysis
dc.subjectMultiple-scale analysis
dc.subjectHamilton-Jacobi equations
dc.subjectMicrostructures
dc.subjectTurbulent flows
dc.titleParadigm for the creation of scales and phases in nonlinear evolution equations
dc.typeArticle

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