A comment on the series solution of the solitarywave problem
Abstract
The formal series solution of the solitary wave problem is an expansion of the surface elevation in powers of the function sech(∊ x), where ∊ is a parameter and x the coordinate in the horizontal direction. Previous numerical calculations have indicated that this series is not convergent. In this note it is shown, purely analytically, that it must necessarily be so. Furthermore, a simple method to calculate the coefficients in this series is given; the coefficients are easily obtained by means of the computer algebra system Maple. Also, it is shown that z = x + i y as a function of the complex potential w has a pole where β w = i π when β tends to zero. This is a proof of a previous conjecture which was put forward as a result of extensive numerical calculations.
 Publication:

Zeitschrift Angewandte Mathematik und Physik
 Pub Date:
 2003
 DOI:
 10.1007/PL00012631
 Bibcode:
 2003ZaMP...54..179E
 Keywords:

 Surface waves;
 shallowwater waves;
 solitary waves.