Exact Two-Dimensional Multi-Hump Waves on Water of Finite Depth with Small Surface Tension

  • Shengfu DengEmail author
  • Shu-Ming Sun


This paper concerns the existence of multi-hump traveling waves propagating on the free surface of a two-dimensional water channel under the influence of gravity and small surface tension force. The fluid of constant density is assumed to be inviscid and incompressible and the flow is irrotational. It was known that the exact governing equations, called Euler equations, possess a generalized solitary-wave solution of elevation that consists of a single crest (or hump) at the center and a much smaller oscillation at infinity. This paper provides the first proof of the existence of multi-hump waves using the Euler equations. It is shown that when the wave speed is near its critical value and the surface tension is small, the Euler equations have a two-hump solution which consists of two crests, that are spaced far apart, and a smaller oscillation at infinity. Moreover, the ideas and methods may be used to study \(2^m\)-hump solutions.



The authors would like to thank the referees for their valuable comments and suggestions. Deng is partially supported by the National Natural Science Foundation of China (Nos. 11371314, 11771197), the Natural Science Foundation of Fujian Province of China (No. 2019J01064), the Guangdong Natural Science Foundation of China (No. 2017A030313030), and the Scientific Research Funds of Huaqiao University. Sun is partially supported by U.S. National Science Foundation (No. DMS-1210979).

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Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaqiao UniversityQuanzhouChina
  2. 2.Department of MathematicsLingnan Normal UniversityZhanjiangChina
  3. 3.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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