Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Compactons and their variational properties for degenerate KdV and NLS in dimension 1


Authors: Pierre Germain, Benjamin Harrop-Griffiths and Jeremy L. Marzuola
Journal: Quart. Appl. Math. 78 (2020), 1-32
MSC (2010): Primary 35Q53, 35Q55
DOI: https://doi.org/10.1090/qam/1538
Published electronically: April 17, 2019
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Abstract: We analyze the stationary and traveling wave solutions to a family of degenerate dispersive equations of KdV- and NLS-type. In stark contrast to the standard soliton solutions for nondegenerate KdV and NLS equations, the degeneracy of the elliptic operators studied here allows for compactly supported steady or traveling states. As we work in $ 1$ dimension, ODE methods apply; however, the models considered have formally conserved Hamiltonian, Mass, and Momentum functionals, which allow for variational analysis as well.


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Additional Information

Pierre Germain
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
Email: pgermain@cims.nyu.edu

Benjamin Harrop-Griffiths
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
Email: benjamin.harrop-griffiths@cims.nyu.edu

Jeremy L. Marzuola
Affiliation: Department of Mathematics, University of North Carolina, Phillips Hall, Chapel Hill, North Carolina 27599
Email: marzuola@math.unc.edu

DOI: https://doi.org/10.1090/qam/1538
Received by editor(s): August 16, 2018
Received by editor(s) in revised form: January 29, 2019
Published electronically: April 17, 2019
Additional Notes: The first author was supported by the NSF grant DMS-15010.
The second author was supported by a Junior Fellow award from the Simons Foundation.
The third author was supported in part by U.S. NSF Grants DMS–1312874 and DMS-1352353.
Article copyright: © Copyright 2019 Brown University