Journal of Nonlinear Mathematical Physics

Volume 26, Issue 1, December 2018, Pages 107 - 132

Rational solutions to Q3δ in the Adler-Bobenko-Suris list and degenerations

Authors
Song-lin Zhao*
Department of Applied Mathematics, Zhejiang University of Technology Hangzhou, 310023, Zhejiang, P.R. China, songlinzhao@zjut.edu.cn
Da-jun Zhang*
Department of Mathematics, Shanghai University Shanghai, 200444, P.R. China, djzhang@staff.shu.edu.cn
*Corresponding author
Corresponding Author
Song-lin Zhao
Received 15 October 2017, Accepted 25 August 2018, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2019.1544793How to use a DOI?
Keywords
NQC equation, ABS list, Casoratian, rational solutions
Abstract

We derive rational solutions in Casoratian form for the Nijhoff-Quispel-Capel (NQC) equation by using the lattice potential Korteweg-de Vries (lpKdV) equation and two Miura transformations between the lpKdV and the lattice potential modified KdV (lpmKdV) and the NQC equation. This allows us to present rational solutions for the whole Adler-Bobenko-Suris (ABS) list except Q4. The known Miura transformation for soliton solutions between the NQC equation and Q3δ and the known degenerations for solitons from Q3δ to Q2, Q1δ, H3δ, H2 and H1 in the ABS list are used. We show that the Miura transformation and degenerations are valid as well for rational solutions which are usually considered as “long-wave-limit” of solitons. All the rational solutions can be expressed in terms of {zj} which are linear functions of (n, m).

Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 1
Pages
107 - 132
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2019.1544793How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Song-lin Zhao
AU  - Da-jun Zhang
PY  - 2021
DA  - 2021/01
TI  - Rational solutions to Q3δ in the Adler-Bobenko-Suris list and degenerations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 107
EP  - 132
VL  - 26
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1544793
DO  - https://doi.org/10.1080/14029251.2019.1544793
ID  - Zhao2021
ER  -