Details
| Original language | English |
|---|---|
| Pages (from-to) | 1247-1261 |
| Number of pages | 15 |
| Journal | Proceedings of the American Mathematical Society |
| Volume | 151 |
| Issue number | 3 |
| Early online date | 15 Dec 2022 |
| Publication status | Published - Mar 2023 |
Abstract
Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed–wave height space, and reaches a singular highest wave at ϕ(0) = μ2 . The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).
ASJC Scopus subject areas
- Mathematics(all)
- Mathematics(all)
- Applied Mathematics
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Proceedings of the American Mathematical Society, Vol. 151, No. 3, 03.2023, p. 1247-1261.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A Direct Construction of a Full Family of Whitham Solitary Waves
AU - Ehrnström, Mats
AU - Nik, Katerina
AU - Walker, Christoph
N1 - Funding Information: The first author was supported by grant no. 250070 from the Research Council of Norway. The second author was partially supported by the Austrian Science Fund (FWF) project F 65. The authors would like to thank E. Wahlén for drawing our attention to the investigation . Funding Information: Received by the editors April 12, 2022, and, in revised form, June 14, 2022. 2020 Mathematics Subject Classification. Primary 76B15, 76B25, 35Q35. The first author was supported by grant no. 250070 from the Research Council of Norway. The second author was partially supported by the Austrian Science Fund (FWF) project F65.
PY - 2023/3
Y1 - 2023/3
N2 - Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed–wave height space, and reaches a singular highest wave at ϕ(0) = μ2 . The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).
AB - Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed–wave height space, and reaches a singular highest wave at ϕ(0) = μ2 . The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).
UR - http://www.scopus.com/inward/record.url?scp=85146460552&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2204.03274
DO - 10.48550/arXiv.2204.03274
M3 - Article
AN - SCOPUS:85146460552
VL - 151
SP - 1247
EP - 1261
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
SN - 0002-9939
IS - 3
ER -