Two improved displacement shallow water equations and their solitary wave solutions


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2


  1. 1.

    Verschaeve JCG, Pedersen GK, Tropea C (2018) Non-modal stability analysis of the boundary layer under solitary waves. J Fluid Mech 836:740–772

    Article  Google Scholar 

  2. 2.

    Benjamin TB (1992) A new kind of solitary wave. J. Fluid Mech 245(−1):401

    Article  Google Scholar 

  3. 3.

    Jamshidzadeh S, Abazari R (2017) Solitary wave solutions of three special types of Boussinesq equations. Nonlinear Dyn 88(4):2797–2805

    Article  Google Scholar 

  4. 4.

    Zou L, Zong Z, Wang Z et al (2010) Differential transform method for solving solitary wave with discontinuity. Phys Lett A 374(34):3451–3454

    Article  Google Scholar 

  5. 5.

    Berloff NG (2005) Solitary wave complexes in two-component condensates. Phys Rev Lett 94(12):120401

    Article  Google Scholar 

  6. 6.

    Schwämmle V, Herrmann HJ (2003) Geomorphology: solitary wave behaviour of sand dunes. Nature 426(6967):619–620

    Article  Google Scholar 

  7. 7.

    Goodman RH (2008) Chaotic scattering in solitary wave interactions: a singular iterated-map description. Chaos 18(2):270–281

    Article  Google Scholar 

  8. 8.

    Sokolow A, Bittle EG, Sen S (2007) Solitary wave train formation in Hertzian chains. Europhys Lett 77(77):24002

    Article  Google Scholar 

  9. 9.

    Lu DQ, Dai SQ, Zhang BS (1999) Hamiltonian formulation of nonlinear water waves in a two-fluid system. Appl Math Mech Engl 20(4):343–349

    Article  Google Scholar 

  10. 10.

    Chun C (2008) Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method. Phys Lett A 372(16):2760–2766

    Article  Google Scholar 

  11. 11.

    Ertekin RC, Hayatdavoodi M, Kim JW (2014) On some solitary and cnoidal wave diffraction solutions of the Green-Naghdi equations. Appl Ocean Res 47:125–137

    Article  Google Scholar 

  12. 12.

    Zhao BB, Ertekin RC, Duan WY et al (2014) On the steady solitary-wave solution of the Green-Naghdi equations of different levels. Wave Motion 51(8):1382–1395

    Article  Google Scholar 

  13. 13.

    Madsen PA, Schaffer HA (1998) Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Philos Trans R Soc A 356(1749):3123–3184

    Article  Google Scholar 

  14. 14.

    Constantin A, Ehrnström M, Villari G (2008) Particle trajectories in linear deep-water waves. Nonlinear Anal Real World Appl 9(4):1336–1344

    Article  Google Scholar 

  15. 15.

    Constantin A, Villari G (2008) Particle trajectories in linear water waves. J Math Fluid Mech 10(1):1–18

    Article  Google Scholar 

  16. 16.

    Constantin A (2006) The trajectories of particles in Stokes waves. Invent Math 166(3):523–535

    Article  Google Scholar 

  17. 17.

    Henry D (2006) The trajectories of particles in deep-water Stokes waves. Int Math Res Not.

    Article  Google Scholar 

  18. 18.

    Chen Y, Hsu H, Chen G (2010) Lagrangian experiment and solution for irrotational finite-amplitude progressive gravity waves at uniform depth. Fluid Dyn Res 42(4):045511

    Article  Google Scholar 

  19. 19.

    Clamond D (2007) On the Lagrangian description of steady surface gravity waves. J Fluid Mech 589:433–454

    Article  Google Scholar 

  20. 20.

    Lagrange JL (2013) Analytical mechanics. Springer, Berlin

    Google Scholar 

  21. 21.

    Zhong WX, Yao Z (2006) Shallow water solitary waves based on displacement method. J Dalian Univ Technol 46(1):151–156

    Google Scholar 

  22. 22.

    Liu P, Lou SY (2008) A (2 + 1)-dimensional displacement shallow water wave system. Chin Phys Lett 25(9):3311–3314

    Article  Google Scholar 

  23. 23.

    Wu F, Yao Z, Zhong W (2017) Fully nonlinear (2 + 1)-dimensional displacement shallow water wave equation. Chin Phys B 26(0545015):253–258

    Google Scholar 

  24. 24.

    Morrison PJ, Lebovitz NR, Biello JA (2009) The Hamiltonian description of incompressible fluid ellipsoids. Ann Phys N Y 324(8):1747–1762

    Article  Google Scholar 

  25. 25.

    Wu F, Zhong W (2017) On displacement shallow water wave equation and symplectic solution. Comput Method Appl Mech 318:431–455

    Article  Google Scholar 

  26. 26.

    Wu F, Zhong W (2017) A shallow water equation based on displacement and pressure and its numerical solution. Environ Fluid Mech 17(6):1099–1126

    Article  Google Scholar 

  27. 27.

    Wu F, Zhong WX (2016) Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm. Appl Math Mech Engl 37(1):1–14

    Article  Google Scholar 

  28. 28.

    Suzuki Y, Koshizuka S, Oka Y (2007) Hamiltonian moving-particle semi-implicit (HMPS) method for incompressible fluid flows. Comput Methods Appl Mech 196(29–30):2876–2894

    Article  Google Scholar 

  29. 29.

    Alemi Ardakani H (2016) A symplectic integrator for dynamic coupling between nonlinear vessel motion with variable cross-section and bottom topography and interior shallow-water sloshing. J Fluid Struct 65:30–43

    Article  Google Scholar 

  30. 30.

    Liu P, Li Z, Luo R (2012) Modified (2 + 1)-dimensional displacement shallow water wave system: symmetries and exact solutions. Appl Math Comput 219(4):2149–2157

    Google Scholar 

  31. 31.

    Liu P, Fu PK (2011) Modified (2 + 1)-dimensional displacement shallow water wave system and its approximate similarity solutions. Chin Phys B 20(0902039):90203

    Article  Google Scholar 

  32. 32.

    Marsden JE, Pekarsky S, Shkoller S et al (2001) Variational methods, multisymplectic geometry and continuum mechanics. J Geom Phys 38(3–4):253–284

    Article  Google Scholar 

  33. 33.

    Kinnmark I (1986) The shallow water wave equations: formulation, analysis and application. Springer, Berlin

    Google Scholar 

  34. 34.

    He JH, Wu XH (2006) Exp-function method for nonlinear equation. Chaos Solitons Fract 30:700–708

    Article  Google Scholar 

  35. 35.

    Constantin A, Escher J (2007) Particle trajectories in solitary water waves. Bull Am Math Soc 44(3):423–431

    Article  Google Scholar 

Download references


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).

Author information



Corresponding author

Correspondence to Feng Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wu, F., Yao, Z. & Zhong, W. Two improved displacement shallow water equations and their solitary wave solutions. Environ Fluid Mech 20, 5–18 (2020).

Download citation


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