Two improved displacement shallow water equations and their solitary wave solutions

  • Feng WuEmail author
  • Zheng Yao
  • Wanxie Zhong
Original Article


Two improved displacement shallow water equation (DSWE) are constructed by applying the Hamilton variational principle under the shallow water approximation. The first one is the extended DSWE (EDSWE), which contains a higher order nonlinear term omitted in the original DSWE; and the second one is the fully nonlinear DSWE (FNDSWE) which contains all the nonlinear terms. By using the exp-function method, the EDSWE is analyzed and different types of waves, including the regular solitary wave, the loop solitary wave, the solitary wave with a sharp peak, and the periodic wave, are obtained. The exact solitary wave solution of the FNDSWE is also obtained. It is proved that under the shallow water approximation, the particle trajectory of the solitary wave is a parabolic curve.


Solitary wave solution Displacement shallow water equation Hamilton variational principle Particle trajectory Exp-function method 



The authors are grateful for the support of the Natural Science Foundation of China (Nos. 51609034, 11472067) and Fundamental Research Funds for the Central Universities (No. DUT17RC(3)069).


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis of Industrial Equipment, Faculty of Vehicle Engineering and MechanicsDalian University of TechnologyDalianPeople’s Republic of China
  2. 2.Transportation Equipments and Ocean Engineering CollegeDalian Maritime UniversityDalianPeople’s Republic of China

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