This paper considers a progressive solitary wave of permanent form in an ideal fluid of constant depth and explores Davies' approximation [Proc. R. Soc. Lond. A, 208 (1951), pp. 475--486] with high-order corrections to Levi-Civita's surface condition for the logarithmic hodograph variable. Using a complex plane that was originally introduced by Packham [Proc. R. Soc. Lond. A, 213 (1952), pp. 234--249], it is shown that a singularity at infinity can be regularized. Therefore, the solutions in Packham's complex plane under high-order Davies' approximation maintain two critical properties of a solitary wave, the correct exponential decay in the outskirt of wave and the harmonic property of a solution, that are often violated in classical long wave approximations. After introducing an accurate numerical method to compute solitary wave solutions in Packham's complex plane, we compare high-order Davies' approximate solutions with fully nonlinear solutions as well as long wave approximate solutions. The results demonstrate that high-order Davies' approximation produces rapidly converging series solutions even for relatively large amplitude waves and that Davies' approximate solutions compare much better with the fully nonlinear solutions than the long wave approximate solutions.


  1. solitary waves
  2. gravity water waves
  3. approximation in the complex domain

MSC codes

  1. 30E10
  2. 76B15
  3. 76B25

Get full access to this article

View all available purchase options and get full access to this article.


J. G. B. Byatt-Smith, An exact integral equation for steady surface waves, Proc. R. Soc. Lond. A, 315 (1970), pp. 405--418.
D. Clamond and D. Dutykh, Fast accurate computation of the fully nonlinear solitary surface gravity waves, Comput. & Fluids, 84 (2013), pp. 35--38.
T. V. Davies, The theory of symmetrical gravity waves of finite amplitude. I, Proc. R. Soc. Lond. A, 208 (1951), pp. 475--486.
T. V. Davies, Gravity waves of finite amplitude. III. Steady, symmetrical, periodic waves in a channel of finite depth, Quart. Appl. Math., 10 (1952), pp. 57--67.
D. Dutykh and D. Clamond, Efficient computation of steady solitary gravity waves, Wave Motion, 51 (2014), pp. 86--99.
W. A. B. Evans and M. J. Ford, An exact integral equation for solitary waves (with new numerical results for some “internal” properties), Proc. R. Soc. Lond. A, 452 (1996), pp. 373--390.
J. Fenton, A ninth-order solution for the solitary wave, J. Fluid Mech., 53 (1972), pp. 257--271.
K. O. Friedrichs and D. H. Hyers, The existence of solitary waves, Comm. Pure Appl. Math., 7 (1954), pp. 517--550.
P. Henrici, Applied Computational Complex Analysis, Vol. 3, John Wiley & Sons, New York, 1986.
J. K. Hunter and J.-M. Vanden-Broeck, Accurate computations for steep solitary waves, J. Fluid Mech., 136 (1983), pp. 63--71.
D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag., 39 (1895), pp. 422--443.
H. Lamb, Hydrodynamics, 6th ed., Dover, Mineola, NY, 1945.
C. W. Lenau, The solitary wave of maximum amplitude, J. Fluid Mech., 26 (1966), pp. 309--320.
R. R. Long, Solitary waves in the one- and two-fluid systems, Tellus, 8 (1956), pp. 460--471.
M. S. Longuet-Higgins and J. D. Fenton, On the mass, momentum, energy and circulation of a solitary wave. II, Proc. R. Soc. Lond. A., 340 (1974), pp. 471--493.
L. M. Milne-Thomson, Theoretical Hydrodynamics, 5th ed., Dover, Mineola, NY, 1968.
S. Murashige, Numerical use of exterior singularities for computation of gravity waves in shallow water, J. Engrg. Math., 77 (2012), pp. 1--18.
S. Murashige, Davies' surface condition and singularities of deep water waves, J. Engrg. Math., 85 (2014), pp. 19--34.
B. A. Packham, The theory of symmetrical gravity waves of finite amplitude. I. The solitary wave, Proc. R. Soc. Lond. A, 213 (1952), pp. 234--249.
S. A. Pennell and C. H. Su, A seventeenth-order series expansion for the solitary wave, J. Fluid Mech., 149 (1984), pp. 431--443.
S. A. Pennell, On a series expansion for the solitary wave, J. Fluid Mech., 179 (1987), pp. 557--561.
G. G. Stokes, The outskirts of the solitary wave, Math. and Phys. Papers, 5 (1905), p. 163.
J, Strutt (Lord Rayleigh), On waves, Phil. Mag. 5th Ser., 1 (1876), pp. 257--279.
M. Tanaka, The stability of solitary waves, Phys. Fluids, 29 (1986), pp. 650--655.
J.-M. Vanden-Broeck, Gravity-Capillary Free-Surface Flows, Cambridge University Press, Cambridge, UK, 2010.
T. Y. Wu, J. Kao, and J. E. Zhang, A unified intrinsic functional expansion theory for solitary waves, Acta Mech Sin., 21 (2005), pp. 1--15.
T. Y. Wu, X. Wang, and W. Qu, On solitary waves. Part $2$: A unified perturbation theory for higher order waves, Acta Mech Sin., 21 (2005), pp. 515--530.
T. Y. Wu and S. Murashige, On tsunami and the regularized solitary wave theory, J. Engrg. Math., 70 (2011), pp. 137--146.
H. Yamada, On the highest solitary wave, Rep. Res. Inst. Appl. Mech. Kyushu Univ., 5 (1957), pp. 53--67.

Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 189 - 208
ISSN (online): 1095-712X


Submitted: 4 August 2014
Accepted: 16 December 2014
Published online: 12 February 2015


  1. solitary waves
  2. gravity water waves
  3. approximation in the complex domain

MSC codes

  1. 30E10
  2. 76B15
  3. 76B25



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.