Smooth, cusped, and discontinuous traveling waves in the periodic fluid resonance equation

Persistent Link:
http://hdl.handle.net/10150/282759
Title:
Smooth, cusped, and discontinuous traveling waves in the periodic fluid resonance equation
Author:
Kruse, Matthew Thomas, 1964-
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:
The principal motivation for this dissertation is to extend the study of small amplitude high frequency wave propagation in solutions for hyperbolic conservation laws begun by A. Majda and R. Rosales in 1984. It was then that Majda and Rosales obtained equations governing the leading order wave amplitudes of resonantly interacting weakly nonlinear high frequency wave trains in the compressible Euler equations. The equations were obtained through systematic application of multiple scales and result in a pair of nonlinear acoustic wave equations coupled through a convolution operator. The extended solutions satisfy a pair of inviscid Burgers' equations coupled via a spatial convolution operator. Since then, many mathematicians have used this technique to extend the time validity of solutions to systems of equations other than the Euler equations and have arrived at similar nonlinear non-local systems. This work attempts to look at some of the basic features of the linear and nonlinear coupled and decoupled non-local equations, offering some analytic solutions and numerical insight into the phenomena associated with these equations. We do so by examining a single non-local linear equation, and then a single equation coupling a Burgers' nonlinearity with a linear convolution operator. The linear case is completely solvable. Analytic solutions are provided along with numerical results showing the fundamental properties of the linear non-local equations. In the nonlinear case some analytic solutions, including steady state profiles and traveling wave solutions, are provided along with a battery of numerical simulations. Evidence indicates the existence of attractors for solutions of the single equation with a single mode kernel. Provided resonant interaction takes place, the profile of the attractor is uniquely dependent on the kernel alone. Hamiltonian equations are obtained for both the linear and nonlinear equations with the condition that the resonant kernel must be an odd function with respect to the spacial variable. This work also offers some insight into the general understanding of nonlinear non-local systems of equations. It develops working insight for the action of the resonant mechanism between the solution and a known kernel.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.; Physics, Fluid and Plasma.; Physics, Acoustics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Brio, Moysey

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleSmooth, cusped, and discontinuous traveling waves in the periodic fluid resonance equationen_US
dc.creatorKruse, Matthew Thomas, 1964-en_US
dc.contributor.authorKruse, Matthew Thomas, 1964-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.abstractThe principal motivation for this dissertation is to extend the study of small amplitude high frequency wave propagation in solutions for hyperbolic conservation laws begun by A. Majda and R. Rosales in 1984. It was then that Majda and Rosales obtained equations governing the leading order wave amplitudes of resonantly interacting weakly nonlinear high frequency wave trains in the compressible Euler equations. The equations were obtained through systematic application of multiple scales and result in a pair of nonlinear acoustic wave equations coupled through a convolution operator. The extended solutions satisfy a pair of inviscid Burgers' equations coupled via a spatial convolution operator. Since then, many mathematicians have used this technique to extend the time validity of solutions to systems of equations other than the Euler equations and have arrived at similar nonlinear non-local systems. This work attempts to look at some of the basic features of the linear and nonlinear coupled and decoupled non-local equations, offering some analytic solutions and numerical insight into the phenomena associated with these equations. We do so by examining a single non-local linear equation, and then a single equation coupling a Burgers' nonlinearity with a linear convolution operator. The linear case is completely solvable. Analytic solutions are provided along with numerical results showing the fundamental properties of the linear non-local equations. In the nonlinear case some analytic solutions, including steady state profiles and traveling wave solutions, are provided along with a battery of numerical simulations. Evidence indicates the existence of attractors for solutions of the single equation with a single mode kernel. Provided resonant interaction takes place, the profile of the attractor is uniquely dependent on the kernel alone. Hamiltonian equations are obtained for both the linear and nonlinear equations with the condition that the resonant kernel must be an odd function with respect to the spacial variable. This work also offers some insight into the general understanding of nonlinear non-local systems of equations. It develops working insight for the action of the resonant mechanism between the solution and a known kernel.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
dc.subjectPhysics, Fluid and Plasma.en_US
dc.subjectPhysics, Acoustics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorBrio, Moyseyen_US
dc.identifier.proquest9906536en_US
dc.identifier.bibrecord.b38874660en_US
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