TY - JOUR

T1 - On weakly singular and fully nonlinear travelling shallow capillary–gravity waves in the critical regime

AU - Mitsotakis, Dimitrios

AU - Dutykh, Denys

AU - Assylbekuly, Aydar

AU - Zhakebayev, Dauren

N1 - Funding Information:
The work of D. MITSOTAKIS was supported by the Marsden Fund administrated by the Royal Society of New Zealand. A. ASSYLBEKULY and D. ZHAKEBAYEV acknowledge the support from Science Committee of MES of the Republic of Kazakhstan under the project No. 2018/GF4.
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2017/5/25

Y1 - 2017/5/25

N2 - In this Letter we consider long capillary–gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude relation, which coincides with Scott–Russel's empirical formula for solitary waves, while their decay rate is the same regardless their amplitude. Moreover, they can be of depression or elevation type independent of their speed. The dynamics of these solutions are studied as well.

AB - In this Letter we consider long capillary–gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude relation, which coincides with Scott–Russel's empirical formula for solitary waves, while their decay rate is the same regardless their amplitude. Moreover, they can be of depression or elevation type independent of their speed. The dynamics of these solutions are studied as well.

KW - Capillary–gravity waves

KW - Nonlinear dispersive waves

KW - Peakons

UR - http://www.scopus.com/inward/record.url?scp=85016610935&partnerID=8YFLogxK

U2 - 10.1016/j.physleta.2017.03.041

DO - 10.1016/j.physleta.2017.03.041

M3 - Article

AN - SCOPUS:85016610935

SN - 0375-9601

VL - 381

SP - 1719

EP - 1726

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

IS - 20

ER -