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010 a| 2023015043;

020 a| 9781470461201q| (paperback);

020 z| 9781470474478q| (pdf);

035 a| (WaSeSS)ssj0003166192;

042 a| pcc;

049 a| EREEa| NEHH;

050 00 a| QA372b| .K547 2023;

082 00 a| 518/.642| 23/eng20230708;

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100 1 a| Kim, Kihyunc| (Mathematician);

245 10 a| On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equationh| [electronic resource] :b| existence, uniqueness, and instability /c| Kihyun Kim, Soonsik Kwon.;

260 a| Providence, RI :b| AMS, American Mathematical Society,c| 2023.;

300 a| vi, 128 pages ;c| 26 cm.;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| volume 284, number 1409;

504 a| Includes bibliographical references (pages 125-128).;

505 0 a| Notations and preliminaries -- Linearization of (CSS) under equivariance -- Profile Q(μ) -- Setup for modulation analysis -- Proof of bootstrap Lemma -- Conditional uniqueness -- Equivariant Sobolev spaces.;

506 a| Available only to authorized users.;

520 a| "We consider the self-dual Chern-Simons-Schrodinger equation (CSS), also known as a gauged nonlinear Schrodinger equation (NLS). CSS is L2-critical, admits solitons, and has the pseudoconformal symmetry. These features are similar to the L2-critical NLS. In this work, we consider pseudoconformal blow-up solutions under m-equivariance, m [greater than or equals] 1. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution u with given asymptotic profile z[asterisk]: u(t, r) [minus] 1 [absolute value of t]Q r [absolute value of t] e [minus]i r2 4[absolute value of t] eim[theta] [arrow] z[asterisk] in H1 as t [arrow] 0[minus], where Q(r)e[superscript]im[theta] is a static solution. Secondly, we show that such blowup solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of u. We construct a continuous family of solutions u(eta), 0 [less than or equal to] [eta] [much greater than] 1, such that u(0) [equals] u and for [eta] [greater than] 0, u(eta) is a global scattering solution. Moreover, we exhibit a rotational instability as [eta] [arrow] 0[plus]: u(ets) takes an abrupt spatial rotation by the angle m [plus] 1 m [pi] on the time interval [absolute value of t] [approximately less than or equal to] [eta]. We are inspired by works in the L2-critical NLS. In the seminal work of Bourgain and Wang (1997), they constructed such pseudoconformal blow-up solutions. Merle, Raphael, and Szeftel (2013) showed an instability of Bourgain-Wang solutions. Although CSS shares many features with NLS, there are essential differences and obstacles over NLS. Firstly, the soliton profile to CSS shows a slow polynomial decay r[minus](m[plus]2). This causes many technical issues for small m. Secondly, due to the nonlocal nonlinearities, there are strong long-range interactions even between functions in far different scales. This leads to a nontrivial correction of our blowup ansatz. Lastly, the instability mechanism of CSS is completely different from that of NLS. Here, the phase rotation is the main source of the instability. On the other hand, the self-dual structure of CSS is our sponsor to overcome these obstacles. We exploited the self-duality in many places such as the linearization, spectral properties, and construction of modified profiles"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Differential equations, Nonlinear.;

650 0 a| Schrödinger equation.;

650 0 a| Blowing up (Algebraic geometry);

650 7 a| Partial differential equations -- Qualitative properties of solutions -- Blow-up.2| msc;

650 7 a| Partial differential equations -- Equations of mathematical physics and other areas of application -- NLS-like equations (nonlinear Schrödinger).2| msc;

655 0 a| Electronic books.;

700 1 a| Kwŏn, Sun-sik,d| 1977-;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=30539169;

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On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equation existence, uniqueness, and instability / Kihyun Kim, Soonsik Kwon.

Author/creator	Kim, Kihyun
Other author	Kwŏn, Sun-sik, 1977-
Format	Electronic
Publication Info	Providence, RI : AMS, American Mathematical Society, 2023.
Description	vi, 128 pages ; 26 cm.
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Differential equations, Nonlinear. Schrödinger equation. Blowing up (Algebraic geometry)

Series	Memoirs of the American Mathematical Society, 0065-9266 ; volume 284, number 1409
Contents	Notations and preliminaries -- Linearization of (CSS) under equivariance -- Profile Q(μ) -- Setup for modulation analysis -- Proof of bootstrap Lemma -- Conditional uniqueness -- Equivariant Sobolev spaces.
Abstract	"We consider the self-dual Chern-Simons-Schrodinger equation (CSS), also known as a gauged nonlinear Schrodinger equation (NLS). CSS is L2-critical, admits solitons, and has the pseudoconformal symmetry. These features are similar to the L2-critical NLS. In this work, we consider pseudoconformal blow-up solutions under m-equivariance, m [greater than or equals] 1. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution u with given asymptotic profile z[asterisk]: u(t, r) [minus] 1 [absolute value of t]Q r [absolute value of t] e [minus]i r2 4[absolute value of t] eim[theta] [arrow] z[asterisk] in H1 as t [arrow] 0[minus], where Q(r)e[superscript]im[theta] is a static solution. Secondly, we show that such blowup solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of u. We construct a continuous family of solutions u(eta), 0 [less than or equal to] [eta] [much greater than] 1, such that u(0) [equals] u and for [eta] [greater than] 0, u(eta) is a global scattering solution. Moreover, we exhibit a rotational instability as [eta] [arrow] 0[plus]: u(ets) takes an abrupt spatial rotation by the angle m [plus] 1 m [pi] on the time interval [absolute value of t] [approximately less than or equal to] [eta]. We are inspired by works in the L2-critical NLS. In the seminal work of Bourgain and Wang (1997), they constructed such pseudoconformal blow-up solutions. Merle, Raphael, and Szeftel (2013) showed an instability of Bourgain-Wang solutions. Although CSS shares many features with NLS, there are essential differences and obstacles over NLS. Firstly, the soliton profile to CSS shows a slow polynomial decay r[minus](m[plus]2). This causes many technical issues for small m. Secondly, due to the nonlocal nonlinearities, there are strong long-range interactions even between functions in far different scales. This leads to a nontrivial correction of our blowup ansatz. Lastly, the instability mechanism of CSS is completely different from that of NLS. Here, the phase rotation is the main source of the instability. On the other hand, the self-dual structure of CSS is our sponsor to overcome these obstacles. We exploited the self-duality in many places such as the linearization, spectral properties, and construction of modified profiles"-- Provided by publisher.
Bibliography note	Includes bibliographical references (pages 125-128).
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2023015043
ISBN	9781470461201 (paperback)
ISBN	(pdf)

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On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equation existence, uniqueness, and instability / Kihyun Kim, Soonsik Kwon.

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