Essays on nonlinear waves: Patterns under water; pulse propagation through random media

Persistent Link:
http://hdl.handle.net/10150/282787
Title:
Essays on nonlinear waves: Patterns under water; pulse propagation through random media
Author:
Komarova, Natalia, 1971-
Issue Date:
1998
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
This is a collection of essays on weakly and strongly nonlinear systems and possible ways of solving/interpreting them. Firstly, we study sand patterns which are often observed on sea (river) beds. One of the most common features looks like straight rolls perpendicular to the water motion. In many cases, the straight rolls are superimposed on a much longer wave so that two vastly different length scales coexist. In general, there are at least two mechanisms responsible for the growth of periodic sand waves. One is linear instability, and the other is nonlinear coupling between long waves and short waves. One novel feature of this work is to suggest that the latter can be much more important than the former one for the generation of long waves. A weakly nonlinear analysis of the corresponding physical system suggests that the nonlinear coupling leads to the growth of the longer features if the amplitude of the shorter waves has a non-zero curvature. For the case of a straight channel and a tidal shallow sea, we derive nonlinear amplitude equations governing the dynamics of the main features. Estimates based on these equations are consistent with measurements. Secondly, we consider strongly nonlinear systems with randomness. The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion. The problem is then shown to be isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random (in space) potential. It is proven that the SIT result continues to hold when the uniform medium of inhomogeneously broadened two-level atoms is replaced by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem suggests that nonlinearity can improve the transparency of the medium. Also, the small amplitude, almost monochromatic limit of SIT is taken and results in an envelope equation which is an exactly integrable combination of NLS and a modified SIT equation. Some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.; Engineering, Marine and Ocean.; Physics, Fluid and Plasma.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Interdisciplinary Program inApplied Mathematics
Degree Grantor:
University of Arizona
Advisor:
Newell, Alan C.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleEssays on nonlinear waves: Patterns under water; pulse propagation through random mediaen_US
dc.creatorKomarova, Natalia, 1971-en_US
dc.contributor.authorKomarova, Natalia, 1971-en_US
dc.date.issued1998en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThis is a collection of essays on weakly and strongly nonlinear systems and possible ways of solving/interpreting them. Firstly, we study sand patterns which are often observed on sea (river) beds. One of the most common features looks like straight rolls perpendicular to the water motion. In many cases, the straight rolls are superimposed on a much longer wave so that two vastly different length scales coexist. In general, there are at least two mechanisms responsible for the growth of periodic sand waves. One is linear instability, and the other is nonlinear coupling between long waves and short waves. One novel feature of this work is to suggest that the latter can be much more important than the former one for the generation of long waves. A weakly nonlinear analysis of the corresponding physical system suggests that the nonlinear coupling leads to the growth of the longer features if the amplitude of the shorter waves has a non-zero curvature. For the case of a straight channel and a tidal shallow sea, we derive nonlinear amplitude equations governing the dynamics of the main features. Estimates based on these equations are consistent with measurements. Secondly, we consider strongly nonlinear systems with randomness. The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion. The problem is then shown to be isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random (in space) potential. It is proven that the SIT result continues to hold when the uniform medium of inhomogeneously broadened two-level atoms is replaced by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem suggests that nonlinearity can improve the transparency of the medium. Also, the small amplitude, almost monochromatic limit of SIT is taken and results in an envelope equation which is an exactly integrable combination of NLS and a modified SIT equation. Some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
dc.subjectEngineering, Marine and Ocean.en_US
dc.subjectPhysics, Fluid and Plasma.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineInterdisciplinary Program inApplied Mathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorNewell, Alan C.en_US
dc.identifier.proquest9912090en_US
dc.identifier.bibrecord.b39118459en_US
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