國立高雄大學圖資館 |

Language: English

Back

Asymptotics for Some Dispersive Equations with Slow Dispersion.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Asymptotics for Some Dispersive Equations with Slow Dispersion.
Author:	Stewart, Gavin Scot.
Published:	Ann Arbor : ProQuest Dissertations & Theses, 2022
Description:	254 p.
Notes:	Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
Notes:	Advisor: Germain, Pierre.
Contained By:	Dissertations Abstracts International84-01B.
Subject:	Mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29061952
ISBN:	9798837544361

Asymptotics for Some Dispersive Equations with Slow Dispersion.
Stewart, Gavin Scot.

Asymptotics for Some Dispersive Equations with Slow Dispersion. - Ann Arbor : ProQuest Dissertations & Theses, 2022 - 254 p.

Source: Dissertations Abstracts International, Volume: 84-01, Section: B.

Thesis (Ph.D.)--New York University, 2022.

This item must not be sold to any third party vendors.

We consider two nonlinear dispersive equations with degenerate (linear) dispersion relations. This degeneracy in the linear dispersion leads to slow decay for solutions, and produces self-similar modifications to the linear scattering behavior for small data.We first study the complex modified Korteweg-de Vries (complex mKdV) equation. This equation simplifies to the usual mKdV equation when the initial data is real, and like mKdV it is completely integrable. However, for general initial data the nonlinear term is not a derivative, which prevents us from using the scaling vector field method developed in previous works for mKdV. Instead, we write the solution as a sum of a (time varying) self-similar profile and a remainder term. The modulation of the self-similar term is controlled by the mean of the solution, and the remainder term obeys stronger decay bounds. We can then obtain the desired asymptotics by (i) controlling the remainder in a weighted L2 space, (ii) bounding the rate of change of the mean, and (iii) performing a stationary phase estimate to identify an additional logarithmic correction to the linear scattering behavior.Next, we consider the Ablowitz-Ladik equation. This equation was first introduced by Ablowitz and Ladik as a completely integrable discretization of the one dimensional cubic nonlinear Schrodinger equation. However, near the frequencies where the dispersion relation is degenerate, we can obtain complex mKdV as a (formal) continuum limit. Based on this, we define approximately self-similarly solutions to Ablowitz-Ladik to be a time dependent truncation of the mKdV self-similar solutions. We then decompose the solution to Ablowitz-Ladik as the sum of a self-similar piece and a remainder term and show that the argument for complex mKdV is robust enough to be adapted for this equation. Since we only have access to approximately self-similar solutions, we obtain new error terms which we must carefully control.

ISBN: 9798837544361Subjects--Topical Terms:

184409
Mathematics.
Subjects--Index Terms:

Ablowitz-Ladik equation

Asymptotics for Some Dispersive Equations with Slow Dispersion.
LDR:03220nmm a2200397 4500 001 636087
005 20230501063856.5
006 m o d
007 cr#unu||||||||
008 230724s2022 ||||||||||||||||| ||eng d
020 $a 9798837544361
035 $a (MiAaPQ)AAI29061952
035 $a AAI29061952
040 $a MiAaPQ $c MiAaPQ
100 1 $a Stewart, Gavin Scot. $3 942376
245 1 0 $a Asymptotics for Some Dispersive Equations with Slow Dispersion.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2022
300 $a 254 p.
500 $a Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
500 $a Advisor: Germain, Pierre.
502 $a Thesis (Ph.D.)--New York University, 2022.
506 $a This item must not be sold to any third party vendors.
520 $a We consider two nonlinear dispersive equations with degenerate (linear) dispersion relations. This degeneracy in the linear dispersion leads to slow decay for solutions, and produces self-similar modifications to the linear scattering behavior for small data.We first study the complex modified Korteweg-de Vries (complex mKdV) equation. This equation simplifies to the usual mKdV equation when the initial data is real, and like mKdV it is completely integrable. However, for general initial data the nonlinear term is not a derivative, which prevents us from using the scaling vector field method developed in previous works for mKdV. Instead, we write the solution as a sum of a (time varying) self-similar profile and a remainder term. The modulation of the self-similar term is controlled by the mean of the solution, and the remainder term obeys stronger decay bounds. We can then obtain the desired asymptotics by (i) controlling the remainder in a weighted L2 space, (ii) bounding the rate of change of the mean, and (iii) performing a stationary phase estimate to identify an additional logarithmic correction to the linear scattering behavior.Next, we consider the Ablowitz-Ladik equation. This equation was first introduced by Ablowitz and Ladik as a completely integrable discretization of the one dimensional cubic nonlinear Schrodinger equation. However, near the frequencies where the dispersion relation is degenerate, we can obtain complex mKdV as a (formal) continuum limit. Based on this, we define approximately self-similarly solutions to Ablowitz-Ladik to be a time dependent truncation of the mKdV self-similar solutions. We then decompose the solution to Ablowitz-Ladik as the sum of a self-similar piece and a remainder term and show that the argument for complex mKdV is robust enough to be adapted for this equation. Since we only have access to approximately self-similar solutions, we obtain new error terms which we must carefully control.
590 $a School code: 0146.
650 4 $a Mathematics. $3 184409
650 4 $a Applied mathematics. $3 377601
653 $a Ablowitz-Ladik equation
653 $a Asymptotics
653 $a Modified Korteweg-de Vries
653 $a Slow dispersion
653 $a Self-similar solutions
653 $a Space-time resonances
690 $a 0405
690 $a 0364
710 2 $a New York University. $b Mathematics. $3 603182
773 0 $t Dissertations Abstracts International $g 84-01B.
790 $a 0146
791 $a Ph.D.
792 $a 2022
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29061952