An integral description of the propagation of steady-state perturbations of the heavy liquid surface

The system of equations of motion describing the gravity wave propagation in a perfect heavy liquid layer is transformed into a new integral equation for the free surface elevations. In the limit cases, this integral equation describes the linear and nonlinear periodic waves as well as the known types of solitary waves. In this case a dispersion equation arises because perturbations of the second and higher orders of smallness are neglected. The integral equation allows for the propagation of invariable surface perturbations of arbitrary forms if their spatial spectrum is concentrated near small wave numbers (compared to the inverse wave amplitude). Several examples of solutions are presented.