On the steady solitary-wave solution of the Green–Naghdi equations of different levels

• Solution of the high-level GN equations are obtained for steady solitary waves.
• The classical nonlinear Green–Naghdi theories are solved numerically.
• The solutions at each level are studied for convergence and then compared with existing experimental data and other predictions.
• The agreement between the converged GN solution (trajectories, velocities and surface elevation) and experiments and Euler’s equations is very good.

The steady-state solitary wave solution of high-level Green–Naghdi (GN) equations is obtained by use of the Newton–Raphson method. Four aspects of solitary waves are studied: the wave speed, wave profile, velocity field and particle trajectory. A convergence study is performed for each individual case. Results of the converged model are compared with the existing laboratory experiments and other theoretical solutions for an inviscid and incompressible fluid, including the solutions of the Euler equations. Particle trajectories, predicted by the GN model, show close agreement with the laboratory measurements and provide a new approach to understanding the movement of the particles under a solitary wave. It is further shown that high-level GN equations can predict the solitary wave of the highest height.