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Scattering for a Generalized Benjami...

Morgan, Adam George,

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Scattering for a Generalized Benjamin-Bona-Mahony Equation /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Scattering for a Generalized Benjamin-Bona-Mahony Equation // Adam George Morgan.
作者:	Morgan, Adam George,
面頁冊數:	1 electronic resource (206 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-05, Section: B.
Contained By:	Dissertations Abstracts International86-05B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30988522
ISBN:	9798342745192

Scattering for a Generalized Benjamin-Bona-Mahony Equation /
Morgan, Adam George,

Scattering for a Generalized Benjamin-Bona-Mahony Equation /Adam George Morgan. - 1 electronic resource (206 pages)

Source: Dissertations Abstracts International, Volume: 86-05, Section: B.

The Benjamin-Bona-Mahony equation (BBM) is a nonlinear dispersive PDE that models surface gravity waves moving through a shallow layer of water. In the long-wave limit, BBM is expected to produce the same physical predictions as the more well-known Korteweg-de Vries equation (KdV). When describing intermediate or short wavelengths, however, KdV and BBM differ markedly. For example, the linearized BBM features finite speed of propagation and a dispersion relation with a nonzero inflection point while the linearized KdV does not. Now, global-in-time behaviour of small solutions of KdV-type equations with a nonlinear term of the form upux has been well-understood since the turn of the century: provided p > 2, small solutions tend to solutions of the linearized PDE as t → ∞. In the jargon of nonlinear dispersive waves, this asymptotic "forgetting" of the nonlinearity is called scattering. For BBM-type equations with the same nonlinearity, however, scattering has only been established for p > 4 and progress on the problem has almost ceased since the mid-1990s. In light of the similarities and differences between KdV and BBM, it is natural to ask if one can shrink the scattering exponent for BBM-type equations down to the KdV threshold p > 2, and in this thesis I discuss some recent progress I have made towards resolving this question. Specifically, I establish linear dispersive decay and scattering for a BBM-type equation with p = 3. In particular, the presence of a nonzero inflection point in the dispersion relation does not obstruct dispersive decay for this choice of nonlinear term. To deal with the difficulties that appear for p = 3 < 4, I apply the method of space-time resonances, which reduces closing bootstrap estimates to bounding multidimensional oscillatory integrals.

English

ISBN: 9798342745192Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Benjamin-Bona-Mahony equation

Scattering for a Generalized Benjamin-Bona-Mahony Equation /
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245 1 0 $a Scattering for a Generalized Benjamin-Bona-Mahony Equation / $c Adam George Morgan.
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520 $a The Benjamin-Bona-Mahony equation (BBM) is a nonlinear dispersive PDE that models surface gravity waves moving through a shallow layer of water. In the long-wave limit, BBM is expected to produce the same physical predictions as the more well-known Korteweg-de Vries equation (KdV). When describing intermediate or short wavelengths, however, KdV and BBM differ markedly. For example, the linearized BBM features finite speed of propagation and a dispersion relation with a nonzero inflection point while the linearized KdV does not. Now, global-in-time behaviour of small solutions of KdV-type equations with a nonlinear term of the form upux has been well-understood since the turn of the century: provided p > 2, small solutions tend to solutions of the linearized PDE as t → ∞. In the jargon of nonlinear dispersive waves, this asymptotic "forgetting" of the nonlinearity is called scattering. For BBM-type equations with the same nonlinearity, however, scattering has only been established for p > 4 and progress on the problem has almost ceased since the mid-1990s. In light of the similarities and differences between KdV and BBM, it is natural to ask if one can shrink the scattering exponent for BBM-type equations down to the KdV threshold p > 2, and in this thesis I discuss some recent progress I have made towards resolving this question. Specifically, I establish linear dispersive decay and scattering for a BBM-type equation with p = 3. In particular, the presence of a nonzero inflection point in the dispersion relation does not obstruct dispersive decay for this choice of nonlinear term. To deal with the difficulties that appear for p = 3 < 4, I apply the method of space-time resonances, which reduces closing bootstrap estimates to bounding multidimensional oscillatory integrals.
546 $a English
590 $a School code: 0779
650 4 $a Mathematics. $3 515831
650 4 $a Applied mathematics. $3 2122814
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Theoretical physics. $3 2144760
653 $a Benjamin-Bona-Mahony equation
653 $a Harmonic analysis
653 $a Nonlinear dispersive equations
653 $a Partial differential equations
653 $a Scattering theory
690 $a 0405
690 $a 0364
690 $a 0642
690 $a 0753
710 2 $a University of Toronto (Canada). $b Mathematics. $e degree granting institution. $3 3818021
720 1 $a Pusateri, Fabio G. $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-05B.
790 $a 0779
791 $a Ph.D.
792 $a 2024
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30988522