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Online ISSN 1552-4485; Print ISSN 0033-569X
On the stability of the compacton waves for the degenerate KdV and NLS models
Authors:
Sevdzhan Hakkaev, Abba Ramadan and Atanas G. Stefanov
Journal:
Quart. Appl. Math. 80 (2022), 507-528
MSC (2020):
Primary 35Q55, 35Q53; Secondary 35Q41
DOI:
https://doi.org/10.1090/qam/1616
Published electronically:
March 28, 2022
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of $p$. Namely, we show that all waves are spectrally stable for $2<p\leq 8$, while a single mode instability occurs for $p>8$. This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range $p<8$.
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Additional Information
Sevdzhan Hakkaev
Affiliation:
Department of Mathematics, Trakya University, 22030 Edirne, Turkey; Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. bl. 8, 1113 Sofia, Bulgaria; and Faculty of Mathematics and Informatics, Shumen University, 9712 Shumen, Bulgaria
MR Author ID:
698884
Email:
s.hakkaev@shu.bg
Abba Ramadan
Affiliation:
Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487-0350
MR Author ID:
1231155
Email:
abba90ramadan@gmail.com
Atanas G. Stefanov
Affiliation:
Department of Mathematics, University of Alabama-Birmingham, University Hall, Room 4005, 1402 10th Avenue, South Birmingham, AL 35294-1241
MR Author ID:
639852
Email:
stefanov@uab.edu
Received by editor(s):
September 22, 2021
Received by editor(s) in revised form:
January 6, 2022
Published electronically:
March 28, 2022
Additional Notes:
Abba Ramadan was partially supported by a graduate research assistantship under NSF-DMS # 1614734.
Atanas Stefanov was partially supported by NSF-DMS # 1908626 and NSF-DMS # 2204788.
Article copyright:
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Brown University