Steady solution of the velocity field of steep solitary waves

•The wave profile, wave speed and velocity field of steep solitary waves are studied.•The IGN-5 equations could be used to study the solitary waves of H/h ≤ 0.79.•Higher IGN equations could calculate highest solitary wave. The steady solutions of solitary waves are studied through the use of the Irrotational Green–Naghdi (IGN) equations for an incompressible and inviscid fluid. The steady solutions are obtained by Newton–Raphson method. We consider the solitary gravity waves of H/h = 0.30, 0.50, 0.70, 0.79, 0.8296, 0.833199 (where H is the wave amplitude and h is the water depth). Some experiments are conducted to test the results of the IGN equations. In particular, we focus on the wave profile and velocity field. We find that for the cases of the solitary waves of H/h ≤ 0.79, the IGN-5 results agree very well with the Euler solutions. For the solitary waves that close to the maximum amplitude, the converged IGN results are presented. It is also shown that high level IGN results on wave speed agree well with the results obtained by others.